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PreCalculus - Ahlborn Def. P - 3
Chapters P - 3
DEFINITIONS, THEOREMS, AND FORMULAS
Review Geometry Formulas
Note: Definitions and theorems marked with an * should be carefully memorized. You may be
asked to state the definition or theorem on a test or quiz.
1. Thm. Circumference
Circle: rC 2=
2. Thm. Area of a Plane Figure
a. Rectangle: lwA = b. Parallelogram: bhA =
c. Triangle: bhA21= d. Trapezoid: )(
21 bBhA +=
e. Circle:2
rA =
3. Thm. Surface Area of a Solid Figure
a. Sphere: 24 rSA = b. Cylinder: 222 rrhSA +=
c. Rectangular solid: whlhlwSA 222 ++=
4. Thm. Volume of a Solid Figure
a. Rectangular solid: lwhV= b. Cylinder: hrV2
=
c. Cone: hrV 2
31 = d. Sphere: 3
34 rV =
Review Measurement Equivalencies
5. Linear Measure:
12 inches = 1 foot
3 feet = 1 yard
5280 feet = 1 mile
100 meters = 1 kilometer
100 centimeters = 1 meter
1000 millimeters = 1 meter
6. Degree measure:
360 = 1 revolution 2 radians = 1 revolution
7. Weight:
16 ounces = 1 pound 2000 pounds = 1 ton
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Chapter P Fundamental Concepts of Algebra
8. Def. Rational Exponents: ( )mnnm
aa = , providedn
mis a rational number, 1>n and n a is a
real number.
9. Prop. Radicals of a Power:
=oddisif
evenisif
na
naa
n n .
10. Prop. Powers of a Radical: ( ) aa nn = .
11. Prop. Product Property of Radicals: nnn baab = , provided 0>a and 0>b whenever n
is even.
12. Prop. Quotient Property of Radicals:n
n
n
b
a
b
a= , provided 0>a and 0>b whenever n is
even.
13. Prop. Reducing an Index: n mmn aa = , provided 0>a whenever mn is even.
14. Thm. Quadratic Formula: If 02
=++ cbxax , thena
acbbx
2
42
= , provided 0a .
15. Thm. Square Root Property: If dx =2 , then dx = , provided 0>d .
16. Def. Root of an Equation: A root of an equation is another name for a solution to the equation.
17. Def. And/Or: The word and means an intersection of two sets and may be written as ; the
word or means the union of two sets and may be written as .
18. Def. Imaginary Unit: The imaginary unit i is 1 .
19. Def. Principal Square Root of a Negative Real Number: The principal square root of b ,
where b is any positive real number, is defined to be bib = .
20. Def. *Complex Numbers: The set of Complex Numbers is the set of numbers bia + where a
and b are real numbers.
21. Def. *Discriminant: The expression acb 42 is called the discriminant of a quadratic equation.
22. Thm. Nature of the Roots of a Quadratic Equation: The quadratic equation 02 =++ cbxax
where a, b, and c are any real numbers has
a. two real roots if the discriminant is positive.
b. one real root if the discriminant is zero.c. two complex roots if the discriminant is negative.
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23. Def. Interval Notation: Let a and b be any two numbers on the number line where ba < .
Intervals on a number line can be expressed as follows:
a. bxa is written [ ]ba, and is called a closed interval.
b. bxa
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36. Def. Line Symmetry: A graph has line symmetry if there is a line for which the points on oneside of the line are a mirror refection of the points on the other side of the line.
37. Thm. Line Symmetry: A curve is symmetric about the:
a. x-axis if for every point ),( yx on the curve, there is also a point ( )yx , on the curve.
b. y-axis if for every point ),( yx on the curve, there is also a point ( )yx, on the curve.
c. diagonal line xy = if for every point ),( yx on the curve, there is also a point ( )xy,on the curve.
38. Def. Point Symmetry: A graph has point symmetry if it is possible to pair the points on the graph
so that one single point is the midpoint of the segment joining every such pair of points.
39. Thm. Point Symmetry About the Origin: A curve is symmetric about the origin if for every point
),( yx on the curve, there is also a point ),( yx on the curve.
40. Skill. Toolbox functions: You should be able to name these functions and sketch each graph
immediately on sight:
a. Constant Function:cy = .
b. Identity Function:xy = .
c. Absolute ValueFunction: xy = .
d. Square Function:2xy = .
e. Square Root Function:
xy = .
f.
Cube Function: 3xy = .
g. Cube Root Function:3 xy = .
h. Reciprocal Function:
xy
1= .
i.
Greatest Integer
Function: [ ]xy = .
41. Def. Greatest Integer Function: [ ]xy = , where y is the greatest integer less than or equal to x.
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42. Def. *Increasing Function: A function is increasing on an open interval if for any pair of values
1x and 2x where 21 xx < , then )()( 21 xfxf < .
43. Def. *Decreasing Function: A function is decreasing on an open interval if for any pair of values
1x and 2x where 21 xx < , then )()( 21 xfxf > .
44. Def. Constant Function: A function is constant on an open interval if for any pair of values 1x and 2x , )()( 21 xfxf = .
45. Def. Relative Maximum (or Local Maximum): )(cf is a relative maximum off if there exists
an open interval containing c for which )()( xfcf for all x in the interval.
46. Def. Relative Minimum (or Local Minimum): )(cf is a relative minimum off if there exists
an open interval containing c for which )()( xfcf for all x in the interval.
47. Def. *Even Function: A function f is an even function if )()( xfxf = for all x in the
domain off.
48. Def. *Odd Function: A function f is an odd function if )()( xfxf = for all x in the
domain off.
49. Thm. Even andOdd Functions: A function is even if and only if its graph is symmetric about the
y-axis. A function is odd if and only if its graph is symmetric about the origin.
50. Thm. *Graphing Transformations:
a. The graph of )()( hxfxg = results from translating (or sliding) the graph of )(xf h
units horizontally.
b. The graph of kxfxg += )()( results from translating (or sliding) the graph of )(xf k
units vertically.
c. The graph of )()( cxfxg = results from taking the graph of )(xf and compressing or
stretching it horizontally by a factor ofc
1.
d. The graph of )()( xcfxg = results from taking the graph of )(xf and compressing or
stretching it vertically by a factor of c.
e. The graph of )()( xfxg = results from reflecting the graph of )(xf about they-
axis.
f. The graph of )()( xfxg = results from reflecting the graph of )(xf about thex-axis.
g. The graph of )()( xfxg = results from the graph of )(xf as follows: points in
quadrants I and IV are identical to the original. Points in quadrants II and III are a mirror
reflection of those in quadrants I and IV, respectively.
h. The graph of )()( xfxg = results from the graph of )(xf as follows: points above
thex-axis remain in place and points below thex-axis are reflected about thex-axis.
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Chapter 1B Function Operations and Coordinate Geometry
51. Def. Operations on Functions:
a. )()())(( xgxfxgf +=+ b. )()())(( xgxfxgf =
c. )()())(( xgxfxgf = d.
)(
)()(
xg
xfx
g
f=
provided 0)( xg
52. Def. Composite Function: Given two functions f and g, the composite function is written
( ) )(xgf and is found by ))(())(( xgfxgf = . Its domain is the set of numbers in thedomain of g for which )(xg is in the domain off.
53. Skill. Finding the Domain of a Composite Function:
a. First find the domain of the inner function. I will refer to this as the preliminary domain
because the domain of the composite function will either be this set or a subset of it.
b. Next form the composite function. Simplify and examine this composite function to seeif there are any additional values that need to be excluded from the domain.
c. Exclude any additional values found in part b. from the preliminary domain. The
remaining set is then the domain of the composite function.
54. Def. Identity Function: )(xI represents the function xy = .
55. Def. *Inverse of a Function: The inverse of a function )(xfy = is the function )(1
xf for
which xxIxffxff === )())(())(( 11 .
56. Def. *One-To-One Function: A function is 1:1 if no two different ordered pairs have the same
second element.
57. Thm. Horizontal Line Test: A function graphed in thexy-plane is 1:1 if and only if a horizontalline intersects the graph in at most one point.
58. Thm. Existence of an Inverse Function: A function f has an inverse that is also afunction if and
only iff is one to one (or its graph passes a horizontal line test).
59. Thm. Properties of Inverse Functions:
a. If ( )ba, is on the graph of a function )(xfy = , then ( )ab, is on the graph of
)(1 xfy = .
b. The domain of )(xfy = is the range of )(1 xfy = . The range of )(xfy = is the
domain of )(1 xfy = .
c. The graph of )(1
xfy
= is the mirror image of )(xfy = about the line xy = .
60. Skill Finding the Equation of an Inverse Function: To find the inverse of a function, reverse the
x and y variables and solve for y.
61. Skill Finding the Range of a Function: To find the range of )(xf , find )(xf 1 and
determine its domain. The domain of )(xf 1 is the range of )(xf .
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62. Def. Slope-Intercept Form of a Line: The slope intercept form of a non-vertical line with slope
m andy-intercept b is bmxy += .
63. Def. *Point-Slope Form of a Line: The point-slope form of a non-vertical line with slope m
that passes through point ( )11,yx is ( )11 xxmyy = .
64. Def. Two Intercept Form of a Line: The two-intercept form of a non-vertical line with
intercepts ( )0,a and ( )b,0 is 1=+b
y
a
x.
65. Def. Step Function: A step function is a function whose graph consists of discontinuous
horizontal line segments whose endpoints are increasing or decreasing, forming the visual
pattern of steps. The two most common step functions are the floor and ceiling functions.
a. Floor Function (or Greatest Integer Function): xy = or [ ]xy = or )int(xy = ,where y is the greatest integer less than or equal to x.
b. Ceiling Function: xy = , where y is the smallest integer greater than or equal to x.
66. Skill Writing a Step Function:
a. Determine if fractional parts of the input values need to be rounded up or down. If up,
choose a ceiling function; if down, choose a floor function.
b. Find one pair of starting values to use as an ordered pair.
c. Since slope is change in y over change in x, determine the value of each
that corresponds to the steps of the function. Do notreduce this slope.
d. Write a linear equation using the Point-Slope Form of a Line.
e. Pull the denominator of the slope under the x factor and put that fraction in
the appropriate ceiling or floor bracket. Keep the numerator of the slope
outside the bracket.
f. Solve for y to obtain the equation in function form.
67. Thm. Distance Formula: The distance between points ),( 11 yx and ),( 22 yx is
( ) ( )212
2
12yyxxd += .
68. Thm. Midpoint Formula: The midpoint between points ),(11
yx and ),(22
yx is the point
++=
2,
2
2121yyxx
M .
69. Def. Locus: A locus is the set of all points satisfying a given condition.
70. Geom Def Circle: A circle is the set of all points in a plane that are equidistant from a fixed point.
71. Alg Def. Circle: The set of points ),( yx such that222 )()( kyhxr += form a circle
whose center is at ),( kh and whose radius has value r.
72. Thm Distance From Point to Line: The distance from a point ( )00 , yx to the line
0=++ cbyax is22
00
ba
cbyaxd
+
++= .
1
11
11
11
(
),(
ybxxay
b
xxayy
xxb
ayy
b
am
yx
+
=
=
=
=
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Chapter 2A Modeling and Polynomial Functions
1. Def. Standard Form for a Quadratic Function: khxay +=2)( .
2.
Def. General Form for a Quadratic Function: cbxaxy++=
2
, where 0
a .
3. Thm. General Form for a Quadratic Function: If cbxaxy ++=2 , then:
a. The absolute value of a causes a vertical compression or stretch.
b. The sign of a determines whether or not the graph is reflected about thex-axis.
c. Thex-coordinate of the vertex can be found bya
bx
2
= . The value corresponds to the
average of thex-intercepts (when they exist).
4. Def. *Power Function: A function of the formn
axy = , where n is a constant natural number,
is called a power function.
5. Thm. Graphing a Power Function: Given nxy = .
a. If n is even, the graph decreases in the second quadrant and increases in the first
quadrant.
b. If n is odd, the graph increases over all real numbers.
c. The graph always passes through the origin and the point )1,1( .
6. Def. *Polynomial Function: A polynomial function has the form
011
1 ... axaxaxayn
nn
n ++++=
, where n is a nonnegative integer and 0na . (The
degree of the polynomial is the degree of its highest power term. The coefficient na is
referred to as the leading coefficient.)
7. Def. Cubic Polynomial Function: 012
23
3 axaxaxay +++= , where 0na .
8. Def. Turning Points: A turning point of a polynomial function is a point at which the graphchanges from increasing to decreasing, or vice versa. If the function changes from increasing
to decreasing, the turning point is called a relative (or local) maximum. If the function
changes from decreasing to increasing, the turning point is called a relative (or local)
minimum.
9. Def. Multiplicity of Roots (or Zeros): The multiplicity of a root r for a polynomial function
( )xf is the maximum number of times the factor rx occurs in ( )xf .
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10. Thm. Graphing a Cubic Function: Given a cubic function 012
23
3 axaxaxay +++= :
a. Its graph is a smooth continuous curve with either 2 turning points or no turning points.
b. Its domain is all real numbers; range is all real umbers.
c. For large absolute values of the variable x, the highest power term becomes dominant,
causing the graph of the cubic polynomial function to resemble the graph of the cubic
powerfunction3
3xay = .
d. If the cubic polynomial function has a root r of multiplicity two (or a double root) then
the shape of the graph near rx = will resemble the vertex of a quadratic function
(meaning the graph will be tangent to thex-axis at this point).
11. Def. Quartic Polynomial Function: 012
23
34
4 axaxaxaxay ++++= , where 04 a .
12. Thm. Graphing a Polynomial Function: Given a polynomial function
011
1 ... axaxaxayn
nn
n ++++=
, where 0na :
a. Its graph is a smooth continuous curve with a maximum of 1n turning points.
b. Its domain is all real numbers. If n is odd, its range is also all real numbers.
c. For large absolute values of the variable x, the highest power term becomes dominant,
causing the graph of the polynomial to resemble the graph of the power functionn
nxay = .
d. If the polynomial function has a root r of multiplicity m, then the shape of the graph
near rx = will resemble the graph of the function mn bxay )( = .
Chapter 2B Rational Root Theorem and Rational Functions
13. Thm. Remainder Theorem: If a polynomial )(xf is divided by cx , then the remainder is
)(cf .
14. Thm. Factor Theorem: cx is a factor of a polynomial )(xf if and only if 0)( =cf .
15. Thm. Rational Root (or Zero) Theorem: Let )(xf be a simplified polynomial with integral
coefficients. If the equation 0)( =xf has a rational rootq
pthat is in lowest terms, then p
must be an integral factor of the constant term, and q must be an integral factor of the
leading coefficient.
16. Thm. Real Roots (or Zeros) of a Polynomial Equation:
a. If a polynomial has degree n, then it has at most n real roots.
b. If a polynomial has degree n where n is odd, then it has at least one real root.
17. Def. Variation in Sign: If the terms of a polynomial )(xf are written in decreasing order
according to the powers ofx (ignoring missing terms), each pair of successive coefficients
with opposite signs is called a variation of sign.
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18. Thm. Descartes Rule of Signs: If )(xf is a polynomial with real coefficients, then:
a. the number of positive real roots of 0)( =xf is either equal to the number of variations
of sign of )(xf or is less than this number by a positive even integer.
b. the number of negative real roots of 0)( =xf is either equal to the number of variations
of sign of )( xf or is less than this number by a positive even integer.
19. Thm. Fundamental Theorem of Algebra: If )(xf is a polynomial of degree n, where 1n ,
then the equation 0)( =xf has at least one root within the set of complex numbers.
(Remember that real numbers are part of the set of complex numbers.)
20. Thm. *Properties of Polynomial Equations:
a. If )(xf is a polynomial of degree n, then (counting multiple roots separately) the
equation 0)( =xf has n roots within the set of complex numbers.
b. If )(xf is a polynomial with real coefficients and bia + is root of the equation
0)( =xf , then bia is also a root.
c. If )(xf is a polynomial with rational coefficients and ba + is root of the equation
0)( =xf , then ba is also a root.
21. Thm. Sum and Product of the Roots of a Quadratic Function: If 1r and 2r are the roots of a
quadratic equation, then the equation can be written as 0)( 21212
=++ rrxrrx .
22. Thm. Sum and Product of the Roots of a Cubic Equation: If 1r , 2r , and 3r are the roots of a
cubic equation, then the equation can be written as
0)()( 3211332212
3213
=+++++ rrrxrrrrrrxrrrx .
23. Def. *Rational Function: A rational function is a function that can be written as the quotient of
two polynomials (the denominator not zero).
24. Thm. Graphing the Reciprocal of Power Function: Given a functionnx
y1
= :
a. Its graph is a smooth curve with two separate parts.
b. If n is even the graph lies in quadrants I and II. If n is odd the graph lies in quadrants I
and III.
c. Its domain is all real numbers except zero. It always passes through the point )1,1( .
d. Its graph has a horizontal asymptote along thex-axis and a vertical asymptote along the
y-axis.
25. Def. *Vertical Asymptote: The line ax = is a vertical asymptote of a graph if the y values
approach infinity (positive or negative) as the x values approach a.
26. Def. *Horizontal Asymptote: The line Ly = is a horizontal asymptote of a graph if the y
values approach L as the x values approach infinity (positive or negative).
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Chapter 3 Exponential and Logarithmic Functions
27. Def. *Exponential Function: xby = for 0>b , and 1b is called the exponential function.
(Note: More general functions of the form xaby = are also referred to as exponential.)
28. Thm. Properties of the Exponential Function: Given xby = .
a. The domain is all real numbers; the range is allpositive real numbers.
b. The graph is a smooth curve that always goes through the point )1,0( .
c. It is a one-to-one function that increases when 1>b and decreases when 10 b , 1b and 0>x .
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36. Thm. Properties of the Logarithm Function: Given xy blog= .
a. The domain is allpositive real numbers; the range is all real numbers.
b. The graph is a smooth curve that always goes through the point )0,1( .
c. It is a one-to-one function that increases when 1>b and decreases when 10
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PreCalculus - Ahlborn Def. 4 10P
Chapters 4 10P
DEFINITIONS, FORMULAS, AND PROPERTIES
Chapter 4 Trigonometric Functions
Note: Definitions and theorems marked with an * should be carefully memorized. You may be
asked to state the definition or theorem on a test or quiz.
1. Def. Degree: A degree is a unit of measure which divides a full circular rotation into 360 parts.
2. Def. Radian: A radian is a unit of measure which divides a full circular rotation into 2 parts.
3. Thm. Arc length: Let s be the length of the arc and C the circumference of the circle.
a. In degrees: Cs =360
. b. In radians: C
s =
2.
4. Def. Apparent Size: When an object is being observed, the angle which the object makes with
the eye is called the apparent size of the object.
5. Def. Linear Speed (Velocity): The linear speed of an object traveling around a circle is thequotient of the arc length traveled and the time.
6. Def. Angular Speed (Velocity): The angular speed of an object traveling around a circle is thequotient of the arc measure (or central angle measure) traveled and the time.
7. Def. Sine and Cosine Functions: Given a circle of radius r and a point ),( yx on the circle. If
is the angle formed by the positivex-axis and a ray with endpoint at the center of the circle
and passing through ),( yx , thenr
y =sin and
r
x=cos .
8. Def. Sine and Cosine Functions: On a unit circle, y=sin and x =cos .
9. Def. Sine and Cosine Functions: On a right triangle,hypotenuse
oppositesin = and
hypotenuse
adjacentcos = .
10. Def. Secant and Cosecant Functions: On a circle,
cos
1sec = (where 0cos ) and
sin
1csc = (where 0sin ).
11. Def. Secant and Cosecant Functions: On a right triangle,adjacent
hypotenusesec = and
opposite
hypotenusecsc = .
12. Def. Periodic Function: A function f is periodic if there exists a positive number p such that
( ) )(xfpxf =+ for all x in the domain of the function. The smallest positive number pfor which f is periodic is called theperiodoff.
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13. Def. Tangent and Cotangent Functions: On a circle,
cos
sintan = (where 0cos ) and
sin
coscot = (where 0sin ).
14. Def. Tangent and Cotangent Functions: On a right triangle, adjacent
opposite
tan = and
opposite
adjacentcot = .
15. Thm. Reciprocal Relationships:
a.
csc
1sin = b.
sec
1cos = c.
cot
1tan =
d.
sin
1csc = e.
cos
1sec = f.
tan
1cot =
16. Thm. Even/Odd Relationships:a. sin)sin( = b. cos)cos( = c. tan)tan( =
d. csc)csc( = e. sec)sec( = f. cot)cot( =
17. Thm. Periodic Relationships: (Note that k is an integer and is in radians.)
a. k sin)2(sin =+ b. cos)2(cos =+ k
c. sec)2(sec =+ k d. k csc)2csc( =+
e. tan)(tan =+ k f. cot)(cot =+ k
18. Def. Reference Angle: The reference angle is the number of degrees (always positive) from the
terminal ray to thex-axis.
19. Def. Amplitude: The amplitude of a function is half the difference between the maximum andminimum values of the function (provided it has a maximum and minimum value.)
20. Thm. Amplitude and Period: If )(xf has period p and amplitude A, then )(cxfa has
periodc
pand amplitude Aa .
21. Def. Phase Shift: If )(xfy = is a periodic function whose graph is transformed to
( ))( hxcfy = , then the value h is called the phase shift.
Chapters 5A Trigonometric Graphs and Sum/Difference Formulas
22. Def. Inverse Sine Function: xy 1sin= (also written xy arcsin= ) means yx sin= , where
22
y
.
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23. Def. Inverse Cosine Function: xy 1cos= (also written xy arccos= ) means yx cos= ,
where y 0 .
24. Def. Inverse Tangent Function: xy 1tan= (also written xy arctan= ) means yx tan= ,
where
22
y
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Chapter 5B Trigonometric Equations and Applications
32. Thm: Double Angle Formulas:
a. cossin22sin = b.
c.
2tan1
tan22tan
=
33. Thm. Half Angle Formulas: (Do NOT Memorize.)
a.2
cos1
2sin
= b.
2
cos1
2cos
+=
c.
sin
cos1
cos1
sin
2
tan
=
+
=
34. Thm. Law of Sines: IfA, B, and C are measures of the angles of a triangle, and a, b, and c
are the lengths of the sides opposite these angles, respectively, thenc
C
b
B
a
A sinsinsin== .
35. Thm. Area of a Triangle: IfA, B, and C are measures of the angles of a triangle, and a, b,
and c are the lengths of the sides opposite these angles, then the area of the triangle can be
found by Cabarea sin2
1= .
36. Thm. Herons Formula: If a, b, and c are the lengths of the sides of a triangle and s is one-half of its perimeter, then the area of the triangle can be found by
))()(( csbsassarea = .
37. Thm. Law of Cosines: IfA, B, and C are measures of the angles of a triangle, and a, b, and c
are the lengths of the sides opposite these angles, then Cabbac cos2222 += .
38. Def. Surveying Compass Readings: In surveying, a compass reading is given as an acute anglefrom the north-south line toward the east or west.
39. Def. Navigational Compass Readings: The course of a ship or plane is described as an angle,
measured clockwise, from the north direction to the direction of the ship or plane. This angleis referred to as the bearing of the ship or plane and is written with three digits to the left of
the decimal point.
Chapter 6A Polar Equations and Complex Numbers
40. Def. Polar Coordinate System: A polar coordinate system is defined by a fixed point (called the
pole) and a ray (called thepolar axis) with its vertex at the pole and extending horizontally to
the right. Each point is designated by an ordered pair ),( r where r is the distance of the
point from the pole and is the angle formed by the polar axis and a ray from the polethrough the point.
1cos2
sin21
sincos2cos
2
2
22
=
=
=
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41. Thm. Polar Circles:
a. ar sin= defines a circle passing through the pole with a vertical diameter of length a.
b. ar cos= defines a circle passing through the pole with a horizontal diameter of length
a.
c. ar=
defines a circle centered at the origin with radius a.42. Thm. Polar Lines:
a. a= defines a line through the pole that forms an angle of measure a with the polar
axis.
b. ar sec= or ra cos= defines a vertical line a units from the pole.
c. ar csc= or ra sin= defines a horizontal line a units from the pole.
43. Thm. Polar Spirals:
a. ar= , called the Archimedes Spiral, defines a spiral about the pole for which thedistance between the windings is always equal.
b. baer= , called the Logarithmic Spiral, defines a spiral about the pole for which theangle between the tangent and the radius is constant.
44. Thm. Polar Roses:
a. bar sin= defines a polar rose centered at the origin with petals of length a. Thepetals are evenly spaced around a circle with the tip of the first petal lying at an angle of
b
90. If b is even there are b2 petals. If b is odd there are b petals.
b. bar cos= defines a polar rose centered at the origin with petals of length a. The
petals are evenly spaced around a circle with the tip of the first petal lying at an angle of
0. If b is even there are b2 petals. If b is odd there are b petals.
45. Thm. Polar Cardioids and Limaons (do not memorize):
a. sinbar = , where 0>a and 0>b , defines a limaon symmetric about they-axis,with a maximum extension of ba + . If ba < the limaon has an inner loop. If ba = ,
the limaon is heart shaped and is called a cardioid. If ba > , the limaon is shaped like
a lima bean and does not reach the pole.
b. bar cos= where 0>a and 0>b , defines a limaon symmetric about thex-axis
with the same features as described above.
46. Thm. Converting Between Polar and Rectangular Coordinates:
a.
r
x=cos
b.r
y =sin
c.x
y =tan
d. 22 yxr += x
r y
( )yx, ( )r,
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47. Def. Rectangular Form for a Complex Number: biaz += , where a and b are real numbers
and i is the imaginary unit, is called the rectangular form for a complex number.
48. Def. Complex Number Plane: A coordinate system where the horizontal axis is the real axis andthe vertical axis is the imaginary axis is called a complex number plane. A complex number
bia + is graphed as a point with coordinates ),( ba .
49. Def. Polar Form for a Complex Number: ( ) sincos irz += , where r is a radius length and
is a polar angle, is called the polar form for a complex number. ( ) sincos ir + ,sometimes abbreviated as rcis .
50. Def. Conjugate of a Complex Number: The conjugate of a complex number bia + is the
number bia . Notation: z represents the conjugate ofz.
51. Thm. Conjugate of a Complex Number in Polar Form: If ( ) sincos irz += , then
( ))sin()cos( += irz .
52. Def. Magnitude (or Absolute Value) of a Complex Number: The magnitude of a complex
number is its distance from the origin. The distance is designated as z .
53. Thm. Magnitude of a Complex Number: rbaz =+= 22 .
54. Thm. Product of Complex Numbers in Polar Form: If ( )1111 sincos irz += and
( )2222 sincos irz += , then ( ) ( )( )21212121 sincos +++= irrzz .
55. Thm. Quotient of Complex Numbers in Polar Form: If ( )1111 sincos irz += and
( )2222 sincos irz += , then ( ) ( )( )21212
1
2
1 sincos += ir
r
z
z.
56. Thm. De Moivre's Theorem: If ( ) sincos irz += , then ( ) ( )( ) ninrz nn sincos += .
Chapter 6B Vectors and Conic Sections
57. Def. Vector: A vector is a quantity which is described by a direction and a magnitude. It is
represented by a ray
AB , where A is called the initial point(or tail of the vector) and B is
called the terminal point (or head of the vector).
58. Def. Equal Vectors: Two vectors are equal if they have the same magnitude and the same
direction.
59. Def. Zero Vector: The zero vector is a vector of magnitude zero, i.e. a point. (It is assigned nodirection.)
60. Def. Sum of Two Vectors: If
AB and
BC represent two vectors, then
=+ ACBCAB .
Vector
AC is called the resultantvector for the two given vectors.
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61. Def. Magnitude (or Absolute Value) of a Vector:
v is called the magnitude or absolute value
of the vector. Its value corresponds to the length of the ray, with no regard for direction.
62. Def. Scalar: A scalar is any real number which is used in a system of vectors.
63. Def. Scalar Multiple of a Vector:
vk (where 0>k ) represents a vector with the same
direction as
v , but whose magnitude is k times as large.
64. Def. Opposite of a Vector:
AB means
BA .
65. Def. Difference of Two Vectors:
wv means
+
wv .
66. Def. Component Form of a Vector: Let the tail of a vector be at ( )11, yx and its head be at
( )22 , yx . Then the vector
v can be expressed as bav ,=
where 12 xxa = and
12 yyb = .
67. Def. Position Vector: Any vector whose initial point is at the origin is called a position vector.
68. Def. Unit Vector: Any vector with a magnitude of one is called a unit vector.
69. Def. Unit Vectors
i and
j : 0,1=
i and 1,0=
j .
70. Thm. Converting Component Form to Unit Vector Form: Any vector bav ,=
can be
expressed as
+= jbiav .
71. Thm. Vector Operations in Components and Unit Vectors: For bav ,=
, dcw ,=
and
scalar k:
a. 22 bav +=
b.
+== jkbikakbkavk ,
c.
= vkvk d.
+++=++=+ jdbicadbcawv )()(,
e.
+== jdbicadbcawv )()(,
72. Thm. Properties of Vector Addition and Scalar Multiplication:
a.
+=+ vwwv b.
++=+
+
uwvuwv
c. ( )
=
vkjvjk d.
+=
+ wkvkwvk
e.
+=+ vjvkvjk )(
73. Def. Parallel Vectors: Two vectors are parallel if they have the same or opposite direction.
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74. Thm. Parallel Vectors:
v and
w are parallel if and only if
= vkw .
75. Def. Orthogonal Vectors: Two vectors are orthogonal (perpendicular) if the angle between them
is 90.
76.
Def. Dot Product: The dot product of two vectors bav ,=
and dcw ,=
is
bdacwv +=
.
77. Thm. Orthogonal Vectors:
v and
w are orthogonal if and only if 0=
wv .
78. Thm. Properties of the Dot Product:
a.
vu is a
scalar.
b.
= uvvu c.
2
= uuu
79. Thm. Angle Between Two Vectors: If
u and
v are any two non-zero vectors, then the angle
( 0 ) between
u and
v is determined by the formula
=
vu
vu cos .
80. Thm. Area of a Triangle: If 111 , bav =
and 222 , bav =
are two vectors which form the
sides of a triangle in thexy-plane, then the area of the triangle is:
=
22
11
2
1
ba
baabsA .
81. Thm. Displacement Formula: Displacement equals velocity times time: tvd
= .
82. Def. *Ellipse: An ellipse is the set of all points in a plane the sum of whose distances from twofixed points (called thefoci) is constant. (The midpoint of the segment connecting the foci is
called the centerof the ellipse. The longer axis of the ellipse is called the major axis, the
shorter is the minor axis. The endpoints of the major axis are the vertices; the endpoints of
the minor axis are the covertices.)
83. Thm. Standard Form of an Ellipse: 12
2
2
2
=+b
y
a
xis the standard form for an ellipse with center
at the origin. The major axis is vertical if ba < , and horizontal if ba > . Intercepts are
)0,(a , )0,( a , ),0( b , and ),0( b .
84. Def. *Eccentricity: The eccentricity of an ellipse or hyperbola is a numerical value that
corresponds to the shape of the curve. Its value e is found by
vertexcenter tofromdistance
focuscenter tofromdistance=e .
85. Def. *Hyperbola: A hyperbola is the set of all points in a plane the difference of whose distancesfrom two fixed points (called thefoci) is constant. (A line through the foci intersects the
hyperbola in two points called the vertices. The line segment connecting the vertices is called
the transverse axis. The midpoint of the transverse axis is the centerof the hyperbola.)
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86. Thm. Standard Form of a Hyperbola: 12
2
2
2
=b
y
a
xis the standard form for a hyperbola with
center at the origin that opens horizontally. It has vertices )0,(a and )0,( a .
12
2
2
2
=a
x
b
yis the standard form for a hyperbola with center at the origin that opens
vertically. It has vertices ),0( b and ),0( b . Both types of hyperbolas have asymptotes
xa
by = .
87. Def. *Parabola: A parabola is the set of all points in a plane that are equidistant from a fixedpoint (called thefocus) and a fixed line (called the directrix).
88. Thm. Focus of a Parabola: Given ( ) khxay += 2 . Ifp is the directed distance from the
vertex to the focus, thena
p4
1= .
89. Def. Latus Rectum: The latus rectum of a parabola is a chord that passes through the focus and is
parallel to the directrix.
90. Thm. Latus Rectum Theorem: The length of the latus rectum of a parabola is p4 , where p is
the directed distance from the vertex to the focus.
91. Def. Conic Section: A conic section is a curve generated by the intersection of a circular conical
surface with a plane. It can be expressed in the general form:
022 =+++++ FEyDxCyBxyAx . (Note: We have only considered conic sections where
0=B .)
92. Thm. Identifying a Conic Section: For a quadratic equation of the form
022 =++++ FEyDxCyAx , the graph is a(n):
a. circle if CA = .
b. ellipse if CA and both have the same sign.
c. parabola if 0=A or 0=C , but notboth.
d. hyperbola ifA and C have opposite signs.
93. Def. Eccentricity of a Conic: Given a fixed point F not on a fixed line D. The set of all points
P in a plane such thatPD
PFe = is:
a. an ellipse if 10 e .
94. Thm. Linear Position Formula: ( ) ( ) tvyxyx
+= 00 ,, is a vector equation that describes the
position of an object ( )yx, moving with constant velocity
v from a given starting point
( )00 , yx .
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95. Def. Parametric Equations: Parametric equations express the x and y components of a vectorequation in separate equations, using a third variable called the parameter.
Chapter 10 Sequences and Series
96. Def. Sequence: A sequence is a function whose domain is a set of consecutive (generally
positive) integers. (The terms of a sequence can be represented by: ,...,...,,, 4321 naaaaa
Note that 1a is often written as a.)
97. Def. Explicitly Defined Sequence: An explicitly defined sequence is one in which each term isdefined in terms of its number n.
98. Def. Recursively Defined Sequence: A recursively (or implicitly) defined sequence is one inwhich each new term is defined in terms of the preceding term.
99. Def. Factorial Notation: n! is defined to have the value: 123)...3)(2)(1(! = nnnnn .
100. Def. Zero Factorial: 1!0 = .101. Def. Series: A series is the indicated sum of the terms of a sequence.
102. Def. Summation (or Sigma) Notation: The symbol =
n
mi
ia represents the sum of the series
nmmm aaaa ...21 +++ ++ , where m and n are integers and nm . (The is the
capitalized form of the Greek letter sigma and stands for a sum in mathematics.)
103. Thm. Properties of Summations:
a. cnc
n
i
=
=1
b. ==
=
n
i
i
n
i
i acac
11
c. [ ] = = =
=
n
i
n
i
n
i
iiii baba
1 1 1
104. Def. Arithmetic Sequence: An arithmetic sequence is a sequence in which each term after thefirst is obtained by adding a fixed number, called the common difference, to the preceding
term.
105. Thm. Nth Term of an Arithmetic Sequence: For any arithmetic sequence whose first term is a
and whose common difference is d, the nth
term can be found by dnaan )1( += .
106. Thm. Arithmetic Sequence Interpreted as a Function: An arithmetic sequence is a linearfunction whose domain is the set of natural numbers and whose slope is the common
difference of the terms.
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107. Def. Arithmetic Means: The terms between two given terms of an arithmetic sequence are calledarithmetic means. (A single arithmetic mean of two given numbers is called the arithmetic
mean of the given numbers.
108. Thm. The Arithmetic Mean: The arithmetic mean between two numbers is the average of the
numbers.
109. Def. Arithmetic Series: An arithmetic series is the sum of an arithmetic sequence.
110. Thm. Sum of an Arithmetic Series: The sum of n terms of an arithmetic series is:
2
)( nn
aanS
+= .
111. Def. Geometric Sequence: A geometric sequence is a sequence in which each term after the firstis obtained by multiplying the preceding term by a fixed nonzero constant, called the common
ratio.
112. Thm. Nth Term of a Geometric Sequence: For any geometric sequence whose first term is a and
whose common ratio is r, the nth
term can be found by 1= nn ara .
113. Def. Geometric Means: The terms between two given terms of a geometric sequence are calledgeometric means. (A single geometric mean of the same sign as the two given numbers iscalled the geometric mean or the mean proportional of the given numbers.)
114. Thm. Geometric Sequence Interpreted as a Function: A geometric sequence with a positive
common ratio (other than 1) is an exponential function whose domain is the set of natural
numbers. The base of the exponent is the common ratio of the terms.
115. Thm. The Geometric Mean: The geometric mean between a and b is ab .
116. Def. Geometric Series: A geometric series is the indicated sum of a geometric sequence.
117. Thm. Sum of a Geometric Series: The sum of the first n terms of a geometric series is found by:
r
ra
S
n
n
=
1
)1(
where 1
r .
118. Def. *Infinite Sequence: An infinite sequence is a sequence whose number of terms is infinite.
119. Def. *Limit of an Infinite Sequence: The limit of an infinite sequence is a single real number towhich the terms of the sequence get progressively closer.
120. Def. Limit Notation: The symbol Lann
=
lim means that L is the limit (as n gets infinitely
large) of the sequence whose terms are defined by na .
121. Def. *Converge: An infinite sequence is said to converge if it has a limit.
122. Def. *Diverge: An infinite sequence is said to diverge if it does not have a limit.
123. Thm. Convergent Sequences: An arithmetic sequence is never convergent (unless the commondifference is zero). A geometric sequence converges if the common ratio has an absolute
value less than or equal to 1.
124. Def. *Sum of an Infinite Series: The sum S of an infinite series is the limit of the sequence of
partial sums nS , if it exists. Notation: nn
SS
= lim .
125. Def. *Convergent Series: A series is said to converge if its sequence of partial sums has a limit.
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126. Thm. Sum of an Infinite Geometric Series: If na is a geometric sequence for which 1
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Chapters P - 4
DEFINITIONS, THEOREMS, AND FORMULAS
Chapter P&1 Preview of Calculus and Limits
Indicates that the item should be memorized in exact detail. You may be asked to quote it on a quiz
or test.
1. Def. Even Function: )(xf is even if )()( xfxf = .
2. Def. Odd Function: )(xf is odd if )()( xfxf = .
3. Def. Limit: Lxfcx
=
)(lim means that )(xf becomes increasingly close to L as x gets closer
to c. (Note that L must be a single real number.)
4. Def. Right Hand Limit: Lxfcx
=+
)(lim means that )(xf becomes increasingly close to L as
x gets closer to c from the right.
5. Def. Left Hand Limit: Lxfcx
=
)(lim means that )(xf becomes increasingly close to L as x
gets closer to c from the left.
6. Thm. Limit Existence: If )(xf is a function and c and L are real numbers, then
Lxfcx
=
)(lim if and only if Lxfcx
=
)(lim and Lxfcx
=+
)(lim .
7. Thm. Properties of Limits: Let b and c be real numbers; let n be a positive integer; and let f
and g be functions whose limit at c exists. Then
a. Constant Function: bbcx=
lim .
b. Scalar Multiple: [ ] )(lim)(lim xfbxfbcxcx
= .
c. Sum or Difference: [ ] )(lim)(lim)()(lim xgxfxgxfcxcxcx
= .
d. Product: [ ] )(lim)(lim)()(lim xgxfxgxfcxcxcx
= .
e. Quotient: 0)(limprovided)(lim
)(lim
)(
)(lim =
xgxg
xf
xg
xf
cxcx
cx
cx.
f. Power: [ ]n
cx
n
cxxfxf
=
)(lim)(lim .
8. Thm. Squeeze Law: If )()()( xgxfxh for all x in an open interval containing c, except
possibly at c itself, and if )(lim)(lim xgLxhcxcx
== , then Lxfcx
=
)(lim .
9. Thm. Trig Limit for the Sine: 1sin
lim0
= x
x
xfor x measured in radians.
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10. Thm. Trig Limit for the Cosine: 0cos1
lim0
=
x
x
x.
11. Def. Continuity at a Point: A function ( )xf is continuous at c if )()(lim cfxfcx
=
. (Note
that this statement requires that both ( )cf and )(lim xfcx
exist.)
12. Def. Continuity on an Open Interval: A function is continuous on an open interval ( )ba, if itis continuous at each point in the interval.
13. Def. Continuity on a Closed Interval: A function is continuous on a closed interval [ ]ba, if it
is continuous on ( )ba, , )()(lim afxfax
=+
, and )()(lim bfxfbx
=
.
14. Def. Discontinuity: A function f has a discontinuity at c iff is defined on an open interval
containing c (except possibly at c) and f is not continuous at c.
15. Def. Removable Discontinuity: A discontinuity at cx = is called removable iff can be made
continuous by appropriately defining (or redefining) only ( )cf .
----------------------------------------------------End for Chapter 1 Quiz-----------------------------------------------
16. Thm. Properties of Continuity: If b is a real number and ( )xf and ( )xg are continuous atcx = , then each of the following functions are also continuous at c:
a. Scalar Multiple: )(xfb
b. Sum or Difference: )()( xgxf
c. Product: )()( xgxf
d. Quotient: 0)(provided)(
)(cg
xg
xf
17. Thm. Continuity for Polynomial and Rational Functions: Polynomial functions are everywhere
continuous. Rational functions are continuous on their domain.
18. Thm. Continuity for Composite Functions: If ( )xg is continuous at c and ( )xf is continuous
at ( )cg , then the composite function given by ( )( )xgf is continuous at c= .
19. Thm. Limit of a Composite Function: If ( ) Lxgcx
=
lim and ( )xf is continuous at L, then
( )( ) ( ) ( )Lfxgfxgfcxcx
=
=
limlim .
20. Thm. Intermediate Value Theorem: If ( )xf is continuous on [ ]ba, and k is any number
between ( )af and ( )bf , then there is at least one number c between a and b such thatkcf =)( .
Geometric Application: Under the given conditions, if ( )af and ( )bf have oppositesigns, thenthere is a point in the open interval where the graph crosses thex-axis.
21. Def. Infinite Limit: =
)(lim xfcx
(or ) means that )(xf increases (or decreases) without
bound as x approaches c.
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22. Def. Vertical Asymptote: A vertical asymptote is a line cx = such that =
)(lim xfcx
.
23. Def. Limit at Infinity: Lxfx
=
)(lim means that f x( ) approaches L as x increases without
bound.
24. Def. Horizontal Asymptote: A horizontal asymptote is a line Ly=
such that Lxfx=
)(lim or
Lxfx
=
)(lim .
----------------------------------------------------End for Chapter 1 Test-----------------------------------------------
Chapter 2 Differentiation
25. Def. Average Velocity: The average velocity of an object over an interval of time is the net
change in position during the interval divided by the change in time. For a function ( )ts ,
that is12
12
tt
tstsv
=
)()(.
26. Def. Instantaneous Velocity: The instantaneous velocity of an object at time 1t is given by the
limit of the average velocity as 2t approaches 1t . For the function ( )ts , that is
12
121
12 tt
tststv
tt
=
)()(lim)( (provided the limit exists).
27. Def. Difference Quotient: The expression12
12
xx
xfxf
)()(is called a difference quotient and
represents the average rate of change of ( )xf over the interval [ ]21 xx , .
28. Def. Difference Quotient, Alternate Forms:h
xfhxf )()( +or
x
xfxxf
)()( +
29. Def. Derivative:12
121
)()(lim)('
12 xx
xfxfxf
xx
=
(provided the limit exists) is called the
derivative of ( )xf at x1
and represents the instantaneous rate of change of ( )xf at the
point x1.
30. Def. Derivative, Alternate Forms:h
xfhxfxf
h
)()(lim)('
0
+=
(provided the limit exists)
orx
xfxxfxfx
)()(lim)('
0
+=
(provided the limit exists)
31. Def. Tangent Line: If )(xf is defined on an open interval containing c, and if the derivative
)(' cf exists, then the line passing through ( ))(, cfc with slope )(' cf is the tangent line
to the graph of )(xf at the point ( ))(, cfc .
32. Def. Normal Line: A normal line to a curve at a point is a line perpendicular to the tangent line at
the point.
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33. Def. Local Linearity: A curve is called locally linear over an interval when zooming in on thecurve causes it to look like a straight line.
34. Def. Slope of a Curve: The slope of a curve at a point is the slope of the tangent line at the point.
35. Def. Differentiable: A function is said to be differentiable at a point if it has a derivative at the
point.
36. Def. Vertical Tangent Line: If )(xf is continuous at cx = and +=+
h
xfhxf
h
)()(lim
0
(or ), then the line cx = is called the vertical tangent to the curve at ( ))(, cfc .
37. Thm. Local Linearity and Differentiability: If a curve is locally linear at a point cx = and the
tangent line is not vertical there, then the function is differentiable at cx = .
38. Thm. Continuity and Differentiability: If a curve is differentiable at a point cx = , then it is
continuous at cx = .
----------------------------------------------------------End for Chapter 2 Quiz------------------------------------------------------
39. Thm. Properties of Derivatives:
a. If cy = , then 0='y .
b. If )(xfcy = , then )('' xfcy = .
c. If )()( xgxfy = , then )(')('' xgxfy = .
d. If nxy = , then 1= nnxy' .
40. Thm. Derivative of the Sine: If xy sin= , then xy cos' = .
41. Thm. Derivative of the Cosine: If xy cos= , then xy sin' = .
42. Def. Right-Hand Derivative: The right-hand derivative of )(xf ish
xfhxf
h
)()(lim
+
+ 0
,
provided the limit exists.
43. Def. Left-Hand Derivative: The left-hand derivative of )(xf ish
xfhxf
h
)()(lim
+
0
,
provided the limit exists.
44. Def. Speed: Speed is the absolute value of the velocity of a moving object.
45. Thm. Changing Speed: If the velocity and acceleration of a moving object have the same sign,
then the speed of the object is increasing. If the velocity and acceleration have oppositesigns, then the speed is decreasing.
46. Thm. Product Rule: If ( )xf and ( )xg are differentiable functions at x, then
[ ] )(')()()(')()( xgxfxgxfxgxfdx
d+= .
47. Thm. Extended Product Rule: ''')'( fghhfgghffgh ++= .
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48. Thm. Quotient Rule: If ( )xf and ( )xg are differentiable functions at x, and 0)( xg , then
[ ]2)(
)(')()()('
)(
)(
xg
xgxfxgxf
xg
xf
dx
d =
.
49. Thm. Chain Rule: If ( ))(xgfy = is a differentiable function of )(xg , and )(xg is a
differentiable function ofx, then ( )[ ] ( ) )(')(')( xgxgfxgfdxd = .
50. Thm. Alternate Form of Chain Rule: If )(ufy = is a differentiable function of u, and
)(xgu = is a differentiable function ofx, thendx
du
du
dy
dx
dy= .
51. Thm. Derivative of the Tangent: If xy tan= , then xy2
sec' = .
52. Thm. Derivative of the Cotangent: If xy cot= , then xy2csc' = .
53. Thm. Derivative of the Secant: If xy sec= , then xxy tansec' = .
54. Thm. Derivative of the Cosecant: If xy csc= , then xxy cotcsc' = .
55. Thm. Absolute Value Rule: If xy = , thenx
xy =' .
----------------------------------------------------------End for Chapter 2 Test------------------------------------------------------
Chapter 3A Applications of Differentiation
56. Skill: Procedure for Solving Related Rate Problems:
a. Draw and label an appropriate figure.b. Write each rate (that is given or asked for) as a derivative.
c. Write an equation that relates all of the variables involved in the above derivatives.
d. Differentiate the equation with respect to time.e. Substitute each given rate and given quantity into the equation.
f. Solve for the remaining rate.
57. Def. Maximum (or Absolute Maximum or Global Maximum): )(cf is a maximum off if
)()( xfcf for every x in the domain off.
58. Def. Minimum (or Absolute Minimum or Global Minimum): )(cf is a minimum off if
)()( xfcf for every x in the domain off.
59. Def. Relative Maximum (or Local Maximum): )(cf is a relative maximum off if there exists
an open interval containing c for which )()( xfcf for all x in the interval.
60. Def. Relative Minimum (or Local Minimum): )(cf is a relative minimum off if there exists
an open interval containing c for which )()( xfcf for all x in the interval.
61. Thm. Extreme Value Theorem: If )(xf is continuous on a closed interval, then f has both a
maximum and minimum value on the interval.
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62. Def. Critical Point: A point in the domain of a function f at which 0' =f or 'f does not
exist is called a critical point off.
63. Thm. Finding Relative Extrema: Iff has a relative maximum or minimum at cx = , then c isa critical point off.
64. Thm. Finding Absolute Extrema: Iff is a continuousfunction on a closed interval and ( )cf isan absolute maximum or minimum on that interval, then c is either a critical point or one of
the endpoints.
65. Skill: Procedure for Finding Absolute Extrema:
a. Identify any points of discontinuity. Divide the domain of the function up into intervals
on which the function is continuous. Consider each interval separately.
b. Find all critical points and calculate the functional value at each.
c. If there are endpoints on the domain interval(s), calculate the functional value at each
endpoint. If there are no endpoints, find the limit of the y values as x approaches the
left or right end of the domain interval(s).
d. Select the largesty-coordinate as the maximum and the smallesty-coordinate as the
minimum. (Remember that a limit value does NOT represent a point and can therefore
never represent a maximum or minimum value. However, it can rule out the presence ofan absolute maximum or absolute minimum.)
66. Thm. Rolle's Theorem: Iff is continuous on [ ]ba, , differentiable on ( )ba, , and
)()( bfaf = , then there is at least one number c in ( )ba, such that 0)(' =cf .
67. Thm. Mean Value Theorem: Iff is continuous on [ ]ba, and differentiable on ( )ba, , then
there exists a number c in ( )ba, such thatab
afbfcf
=
)()()(' .
Geometric Interpretation: Under the given conditions, there is a point in the open interval
where the tangent to the curve is the same as the slope of the line joining the endpoints.
Application: Under the given conditions, there is a point in the open interval where theinstantaneous rate of change is the same as the average rate of change on the interval. If the
function is a position function, then there is a point in the open interval where the
instantaneous velocity is the same as the average velocity on the interval.
----------------------------------------------------------End for Chapter 3A Quiz----------------------------------------------------
68. Def. Increasing Function: A function f x( ) is increasing on an interval if for any two numbers
x1
and x2
in the interval, 21 xx < implies )()( 21 xfxf < .
69. Def. Decreasing Function: A function f x( ) is decreasing on an interval if for any two numbers
x1
and x2
in the interval, 21 xx < implies )()( 21 xfxf > .
70. Def. Strictly Monotonic: A function is called strictly monotonic on an interval if it is eitherincreasing on the entire interval or decreasing on the entire interval.
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71. Thm. Interpreting the Derivative: Let f x( ) be a function that is differentiable on the open
interval ),( ba . Then:
a. If 0)(' >xf for all x in ),( ba , then )(xf is increasing on ),( ba .
b. If 0)(' xf for all x in ),( ba , then the graph of )(xf is concave upward.
b. If 0)(''
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Chapter 3B More Applications of Differentiation
79. Def. Differential ofy: Let )(xfy = represent a differentiable function and let dx be any
nonzero change in x. Then the differential ofy, written dy is given by dxxfdy )('= .
80. Def. Linear Approximation of y : dy is called the linear approximation of the actual
increment, y .
81. Def. Linear Approximation off(x): The expression )())((')( afaxafxf + is called the
linear approximation to ( )xf near ax = .
----------------------------------------------------------End for Chapter 3B Quiz----------------------------------------------------
82. Def. Cost Function: The cost function ( )xC represents the total cost of producing x number ofunits of some item.
83. Def. Revenue Function: The revenue function ( )xR represents the total money taken in when
selling x number of units of some item.
84. Def. Profit Function: The profit function ( )xP is the difference between the revenue functionand the cost function.
85. Def. Marginal Cost: The derivative of the cost function is called the marginal cost and representsthe additional cost of producing 1 more item.
86. Def. Marginal Revenue: The derivative of the revenue function is called the marginal revenue
and represents the additional money taken in upon selling 1 more item.
Chapter 4 Integration
87. Thm. Area Existence: Iff is a continuous function on [ ]ba, , then the limits as n of boththe lower and upper sums exist and are equal to each other. That is
( ) ( )
=
=
=
xMfxmf
n
i
in
n
i
in
11
limlim wheren
abx
= and ( )imf and ( )iMf
represent the minimum and maximum values, respectively, off on the interval.
88. Def. Area Under a Curve: Let f be a continuous function on the interval [ ]ba, . The area of theregion bounded by the graph off, thex-axis, and the vertical lines x a= and x b= is:
=
=n
i
in
xcfarea1
)(lim where n
abx
= and iii xcx 1 .
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89. Thm. Properties of Summations:
a. cnc
n
i
==1
b.
===
n
i
n
i
ifcifc11
)()(
c. [ ] = = =
=
n
i
n
i
n
i
igifigif
1 1 1
)()()()(
90. Def. Riemann Sum: Let f be defined on the closed interval [ ]ba, which is partitioned by the
set },,,,{ 210 bxxxxa n == . If ],[ 1 iii xxc and x x xi i i= 1, then the sum
=
n
i
ii xcf
1
)( is called a Riemann sum of f for the given partition.
91. Def. Norm of the Partition: The length of the largest subinterval of a partition is called the norm
of the partition and is denoted by .
92. Def. Regular Partition: If every subinterval in a partition is of equal length, the partition is called
regular.
93. Def. Definite Integral: Iff is defined on the interval [ ]ba, and the limit of the Riemann sum
=
n
i
ii xcf
10
)(lim exists, then this limit is called the definite integral of f on [ ]ba, and
is denoted by ( )
=
=
n
i
ii
b
a
xcfdxxf
10
)(lim . (The values a and b are called the lower
and upper limits of the integral, respectively.)
94. Def. Integrable Function: A function is said to be integrable on an interval if it has a definite
integral on the interval.
95. Thm. Continuity and Integrability: If a function f is continuous on the closed interval [ ]ba, ,
then f is integrable on [ ]ba, .
96. Thm. Area Under a Curve: Let f be a continuous, function on the interval [ ]ba, . The area ofthe region bounded by the graph off, thex-axis, and the vertical lines ax = and bx = is
found by:
( )=b
a
dxxfarea .
97. Def. Definite Integral at a Point: ( ) 0=a
a
dxxf .
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98. Def. Definite Integral over a Reverse Interval: Iff is an integrable function on [ ]ba, , then
( ) ( ) =b
a
a
b
dxxfdxxf .
99. Thm. Definite Integral over Two Adjacent Intervals: Iff is an integrable function on the three
intervals indicated, then ( ) ( ) ( ) =+b
a
b
c
c
a
dxxfdxxfdxxf
100. Thm. Definite Integral of a Constant Times a Function: Iff is an integrable function on [ ]ba,
and k is a constant, then ( ) ( ) =b
a
b
a
dxxfkdxxfk .
101. Thm. Definite Integral of a Sum or Difference of Functions: Iff and g are both integrable
functions on [ ]ba, , then ( ) ( )[ ] ( ) ( )=
b
a
b
a
b
adxxgdxxfdxxgxf .
102. Thm. Comparing Definite Integrals: Iff and g are both integrable functions on [ ]ba, and
)()( xgxf for bxa , then ( ) ( ) b
a
b
a
dxxgdxxf .
103. Thm. Integration of an Odd Function: Iff is an odd function which is integrable on [ ]aa, ,
then ( ) 0=
a
a
dxxf .
104. Thm. Integration of an Even Function: Iff is an even function which is integrable on [ ]aa, ,
then ( ) ( ) =
aa
a
dxxfdxxf
0
2 .
105. Thm. Distance versus Displacement: If the continuous function v(t) represents the velocity of a
function over an interval of time [ ]ba, , then:
Displacement (or change in position) = ( )b
a
dttv .
Total Distance Traveled = ( )b
a
dttv .
106. Thm. Fundamental Theorem of Calculus: If a function f has a continuous derivative on the
interval [ ]ba, , then ( )=b
a
dxxfafbf ')()( .
Restatement: The total change in a function is the definite integral of its rate of change.
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107.Def. Average Value of a Function: Iff is a continuous function on [ ]ba, , then the average
value off on [ ]ba, is found by ( )=b
a
dxxfab
y1
.
108. Thm. Mean Value Theorem for Integrals: If )(xf is continuous on [ ]ba, , then there exists
a number c in ( )ba, (a, b) such that ( )=b
a
dxxfab
cf1
)( .
109. Thm. Trapezoidal Rule: Iff is continuous on [a, b], then
( ) [ ])()(2)(2)(2)(2
1310 nn
b
a
xfxfxfxfxfn
abdxxf +++++
.
110. Thm. Simpson's Rule: Iff is continuous on [a, b] and T represents the approximation of
( )
b
adxxf using the Trapezoidal Rule and M represents the same integral approximated
with the same number of subdivisions using the Midpoint Rule, then the value3
2 TMS
+=
is the Simpson's Rule approximation for the integral.
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Chapters P - 3
DEFINITIONS, THEOREMS, AND FORMULAS - Review
Chapter P&1 Preview of Calculus and Limits
Indicates that the item should be memorized in exact detail. You may be asked to quote it on a quiz
or test.
1. Def. Even Function: )(xf is even if )()( xfxf = .
2. Def. Odd Function: )(xf is odd if )()( xfxf = .
3. Def. Limit: Lxfcx
=
)(lim means that )(xf becomes increasingly close to L as x gets closer
to c. (Note that L must be a single real number.)
4. Def. Right Hand Limit: Lxfcx
=+
)(lim means that )(xf becomes increasingly close to L as
x gets closer to c from the right.
5. Def. Left Hand Limit: Lxfcx
=
)(lim means that )(xf becomes increasingly close to L as x
gets closer to c from the left.
6. Thm. Limit Existence: If )(xf is a function and c and L are real numbers, then
Lxfcx
=
)(lim if and only if Lxfcx
=
)(lim and Lxfcx
=+
)(lim .
7. Thm. Properties of Limits: Let b and c be real numbers; let n be a positive integer; and let f
and g be functions whose limit at c exists. Then
a. Constant Function: bbcx=
lim .
b. Scalar Multiple: [ ] )(lim)(lim xfbxfbcxcx
= .
c. Sum or Difference: [ ] )(lim)(lim)()(lim xgxfxgxfcxcxcx
= .
d. Product: [ ] )(lim)(lim)()(lim xgxfxgxfcxcxcx
= .
e. Quotient: 0)(limprovided)(lim
)(lim
)(
)(lim =
xgxg
xf
xg
xf
cxcx
cx
cx.
f. Power: [ ]n
cx
n
cxxfxf
=
)(lim)(lim .
8. Thm. Squeeze Law: If )()()( xgxfxh for all x in an open interval containing c, except
possibly at c itself, and if )(lim)(lim xgLxhcxcx
== , then Lxfcx
=
)(lim .
9. Thm. Trig Limit for the Sine: 1sin
lim0
= x
x
xfor x measured in radians.
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10. Thm. Trig Limit for the Cosine: 0cos1
lim0
=
x
x
x.
11. Def. Continuity at a Point: A function ( )xf is continuous at c if )()(lim cfxfcx
=
. (Note
that this statement requires that both ( )cf and )(lim xfcx
exist.)
12. Def. Continuity on an Open Interval: A function is continuous on an open interval ( )ba, if itis continuous at each point in the interval.
13. Def. Continuity on a Closed Interval: A function is continuous on a closed interval [ ]ba, if it
is continuous on ( )ba, , )()(lim afxfax
=+
, and )()(lim bfxfbx
=
.
14. Def. Discontinuity: A function f has a discontinuity at c iff is defined on an open interval
containing c (except possibly at c) and f is not continuous at c.
15. Def. Removable Discontinuity: A discontinuity at cx = is called removable iff can be made
continuous by appropriately defining (or redefining) only ( )cf .
----------------------------------------------------End for Chapter 1 Quiz-----------------------------------------------
16. Thm. Properties of Continuity: If b is a real number and ( )xf and ( )xg are continuous atcx = , then each of the following functions are also continuous at c:
a. Scalar Multiple: )(xfb
b. Sum or Difference: )()( xgxf
c. Product: )()( xgxf
d. Quotient: 0)(provided)(
)(cg
xg
xf
17. Thm. Continuity for Polynomial and Rational Functions: Polynomial functions are everywhere
continuous. Rational functions are continuous on their domain.
18. Thm. Continuity for Composite Functions: If ( )xg is continuous at c and ( )xf is continuous
at ( )cg , then the composite function given by ( )( )xgf is continuous at c= .
19. Thm. Limit of a Composite Function: If ( ) Lxgcx
=
lim and ( )xf is continuous at L, then
( )( ) ( ) ( )Lfxgfxgfcxcx
=
=
limlim .
20. Thm. Intermediate Value Theorem: If ( )xf is continuous on [ ]ba, and k is any number
between ( )af and ( )bf , then there is at least one number c between a and b such thatkcf =)( .
Geometric Application: Under the given conditions, if ( )af and ( )bf have oppositesigns, thenthere is a point in the open interval where the graph crosses thex-axis.
21. Def. Infinite Limit: =
)(lim xfcx
(or ) means that )(xf increases (or decreases) without
bound as x approaches c.
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22. Def. Vertical Asymptote: A vertical asymptote is a line cx = such that =
)(lim xfcx
.
23. Def. Limit at Infinity: Lxfx
=
)(lim means that f x( ) approaches L as x increases without
bound.
24. Def. Horizontal Asymptote: A horizontal asymptote is a line Ly=
such that Lxfx=
)(lim or
Lxfx
=
)(lim .
----------------------------------------------------End for Chapter 1 Test-----------------------------------------------
Chapter 2 Differentiation
25. Def. Average Velocity: The average velocity of an object over an interval of time is the net
change in position during the interval divided by the change in time. For a function ( )ts ,
that is12
12
tt
tstsv
=
)()(.
26. Def. Instantaneous Velocity: The instantaneous velocity of an object at time 1t is given by the
limit of the average velocity as 2t approaches 1t . For the function ( )ts , that is
12
121
12 tt
tststv
tt
=
)()(lim)( (provided the limit exists).
27. Def. Difference Quotient: The expression12
12
xx
xfxf
)()(is called a difference quotient and
represents the average rate of change of ( )xf over the interval [ ]21 xx , .
28. Def. Difference Quotient, Alternate Forms:h
xfhxf )()( +or
x
xfxxf
)()( +
29. Def. Derivative:12
121
)()(lim)('
12 xx
xfxfxf
xx
=
(provided the limit exists) is called the
derivative of ( )xf at x1
and represents the instantaneous rate of change of ( )xf at the
point x1.
30. Def. Derivative, Alternate Forms:h
xfhxfxf
h
)()(lim)('
0
+=
(provided the limit exists)
orx
xfxxfxfx
)()(lim)('
0
+=
(provided the limit exists)
31. Def. Tangent Line: If )(xf is defined on an open interval containing c, and if the derivative
)(' cf exists, then the line passing through ( ))(, cfc with slope )(' cf is the tangent line
to the graph of )(xf at the point ( ))(, cfc .
32. Def. Normal Line: A normal line to a curve at a point is a line perpendicular to the tangent line at
the point.
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33. Def. Local Linearity: A curve is called locally linear over an interval when zooming in on thecurve causes it to look like a straight line.
34. Def. Slope of a Curve: The slope of a curve at a point is the slope of the tangent line at the point.
35. Def. Differentiable: A function is said to be differentiable at a point if it has a derivative at the
point.
36. Def. Vertical Tangent Line: If )(xf is continuous at cx = and +=+
h
xfhxf
h
)()(lim
0
(or ), then the line cx = is called the vertical tangent to the curve at ( ))(, cfc .
37. Thm. Local Linearity and Differentiability: If a curve is locally linear at a point cx = and the
tangent line is not vertical there, then the function is differentiable at cx = .
38. Thm. Continuity and Differentiability: If a curve is differentiable at a point cx = , then it is
continuous at cx = .
----------------------------------------------------------End for Chapter 2 Quiz------------------------------------------------------
39. Thm. Properties of Derivatives:
a. If cy = , then 0='y .
b. If )(xfcy = , then )('' xfcy = .
c. If )()( xgxfy = , then )(')('' xgxfy = .
d. If nxy = , then 1= nnxy' .
40. Thm. Derivative of the Sine: If xy sin= , then xy cos'= .
41. Thm. Derivative of the Cosine: If xy cos= , then xy sin' = .
42. Def. Right-Hand Derivative: The right-hand derivative of )(xf ish
xfhxf
h
)()(lim
+
+0
,
provided the limit exists.
43. Def. Left-Hand Derivative: The left-hand derivative of )(xf ish
xfhxf
h
)()(lim
+
0
,
provided the limit exists.
44. Def. Speed: Speed is the absolute value of the velocity of a moving object.
45. Thm. Changing Speed: If the velocity and acceleration of a moving object have the same sign,
then the speed of the object is increasing. If the velocity and acceleration have oppositesigns, then the speed is decreasing.
46. Thm. Product Rule: If ( )xf and ( )xg are differentiable functions at x, then
[ ] )(')()()(')()( xgxfxgxfxgxfdx
d+= .
47. Thm. Extended Product Rule: ''')'( fghhfgghffgh ++= .
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48. Thm. Quotient Rule: If ( )xf and ( )xg are differentiable functions at x, and 0)( xg , then
[ ]2)(
)(')()()('
)(
)(
xg
xgxfxgxf
xg
xf
dx
d =
.
49. Thm. Chain Rule: If ( ))(xgfy = is a differentiable function of )(xg , and )(xg is a
differentiable function ofx, then ( )[ ] ( ) )(')(')( xgxgfxgfdxd = .
50. Thm. Alternate Form of Chain Rule: If )(ufy = is a differentiable function of u, and
)(xgu = is a differentiable function ofx, thendx
du
du
dy
dx
dy= .
51. Thm. Derivative of the Tangent: If xy tan= , then xy2
sec'= .
52. Thm. Derivative of the Cotangent: If xy cot= , then xy2csc' = .
53. Thm. Derivative of the Secant: If xy sec= , then xxy tansec'= .
54. Thm. Derivative of the Cosecant: If xy csc= , then xxy cotcsc' = .
55. Thm. Absolute Value Rule: If xy = , thenx
xy =' .
----------------------------------------------------------End for Chapter 2 Test------------------------------------------------------
Chapter 3A Applications of Differentiation
56. Skill: Procedure for Solving Related Rate Problems:
a. Draw and label an appropriate figure.b. Write each rate (that is given or asked for) as a derivative.
c. Write an equation that relates all of the variables involved in the above derivatives.
d. Differentiate the equation with respect to time.e. Substitute each given rate and given quantity into the equation.
f. Solve for the remaining rate.
57. Def. Maximum (or Absolute Maximum or Global Maximum): )(cf is a maximum off if
)()( xfcf for every x in the domain off.
58. Def. Minimum (or Absolute Minimum or Global Minimum): )(cf is a minimum off if
)()( xfcf for every x in the domain off.
59. Def. Relative Maximum (or Local Maximum): )(cf is a relative maximum off if there exists
an open interval containing c for which )()( xfcf for all x in the interval.
60. Def. Relative Minimum (or Local Minimum): )(cf is a relative minimum off if there exists
an open interval containing c for which )()( xfcf for all x in the interval.
61. Thm. Extreme Value Theorem: If )(xf is continuous on a closed interval, then f has both a
maximum and minimum value on the interval.
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62. Def. Critical Point: A point in the domain of a function f at which 0'=f or 'f does not
exist is called a critical point off.
63. Thm. Finding Relative Extrema: Iff has a relative maximum or minimum at cx = , then c isa critical point off.
64. Thm. Finding Absolute Extrema: Iff is a continuousfunction on a closed interval and ( )cf isan absolute maximum or minimum on that interval, then c is either a critical point or one of
the endpoints.
65. Skill: Procedure for Finding Absolute Extrema:
a. Identify any points of discontinuity. Divide the domain of the function up into intervals
on which the function is continuous. Consider each interval separately.
b. Find all critical points and calculate the functional value at each.
c. If there are endpoints on the domain interval(s), calculate the functional value at each
endpoint. If there are no endpoints, find the limit of the y values as x approaches the
left or right end of the domain interval(s).
d. Select the largesty-coordinate as the maximum and the smallesty-coordinate as the
minimum. (Remember that a limit value does NOT represent a point and can therefore
never represent a maximum or minimum value. However, it can rule out the presence ofan absolute maximum or absolute minimum.)
66. Thm. Rolle's Theorem: Iff is continuous on [ ]ba, , differentiable on ( )ba, , and
)()( bfaf = , then there is at least one number c in ( )ba, such that 0)(' =cf .
67. Thm. Mean Value Theorem: Iff is continuous on [ ]ba, and differentiable on ( )ba, , then
there exists a number c in ( )ba, such thatab
afbfcf
=
)()()(' .
Geometric Interpretation: Under the given conditions, there is a point in the open interval
where the tangent to the curve is the same as the slope of the line joining the endpoints.
Application: Under the given conditions, there is a point in the open interval where theinstantaneous rate of change is the same as the average rate of change on the interval. If the
function is a position function, then there is a point in the open interval where the
instantaneous velocity is the same as the average velocity on the interval.
----------------------------------------------------------End for Chapter 3A Quiz----------------------------------------------------
68. Def. Increasing Function: A function f x( ) is increasing on an interval if for any two numbers
x1
and x2
in the interval, 21 xx < implies )()( 21 xfxf < .
69. Def. Decreasing Function: A function f x( ) is decreasing on an interval if for any two numbers
x1
and x2
in the interval, 21 xx < implies )()( 21 xfxf > .
70. Def. Strictly Monotonic: A function is called strictly monotonic on an interval if it is eitherincreasing on the entire interval or decreasing on the entire interval.
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71. Thm. Interpreting the Derivative: Let f x( ) be a function that is differentiable on the open
interval ),( ba . Then:
a. If 0)(' >xf for all x in ),( ba , then )(xf is increasing on ),( ba .
b. If 0)(' xf for all x in ),( ba , then the graph of )(xf is concave upward.
b. If 0)(''
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Chapter 3B More Applications of Differentiation
79. Def. Differential ofy: Let )(xfy = represent a differentiable function and let dx be any
nonzero change in x. Then the differential ofy, written dy is given by dxxfdy )('= .
80. Def. Linear Approximation of y : dy is called the linear approximation of the actual
increment, y .
81. Def. Linear Approximation off(x): The expression )())((')( afaxafxf + is called the
linear approximation to ( )xf near ax = .
----------------------------------------------------------End for Chapter 3B Quiz----------------------------------------------------
82. Def. Cost Function: The cost function ( )xC represents the total cost of producing x number ofunits of some item.
83. Def. Revenue Function: The revenue function ( )xR represents the total money taken in when
selling x number of units of some item.
84. Def. Profit Function: The profit function ( )xP is the difference between the revenue functionand the cost function.
85. Def. Marginal Cost: The derivative of the cost function is called the marginal cost and representsthe additional cost of producing 1 more item.
86. Def. Marginal Revenue: The derivative of the revenue function is called the marginal revenue
and represents the additional money taken in upon selling 1 more item.
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AP Calculus BC - Ahlborn Def. 4 -
Chapters 4 - 7
DEFINITIONS, THEOREMS, AND FORMULAS Semester 1
Chapter 4 Integration
Indicates that the item should be memorized in exact detail. You may be asked to quote it on a quiz
or test.
1. Thm. Area Existence: Iff is a continuous function on [ ]ba, , then the limits as n of boththe lower and upper sums exist and are equal to each other. That is
( ) ( )
=
=
=
xMfxmf
n
i
in
n
i
in
11
limlim wheren
abx
= and ( )imf and ( )iMf
represent the minimum and maximum values, respectively, off on the interval.
2. Def. Area Under a Curve: Let f be a continuous function on the interval [ ]ba, . The area of the
region bounded by the graph off, thex-axis, and the vertical lines x a= and x b= is:
=
=
n
i
in
xcfarea
1
)(lim wheren
abx
= and iii xcx 1 .
3. Thm. Properties of Summations:
a. cnc
n
i
==1
b. ==
=
n
i
n
i
ifcifc
11
)()(
c. [ ] = = =
=
n
i
n
i
n
i
igifigif
1 1 1
)()()()(
4. Def. Riemann Sum: Let f be defined on the closed interval [ ]ba, which is partitioned by the
set },,,,{ 210 bxxxxa n == . If c x xi i i [ , ]1 and x xi i i= 1, then the sum
=
n
i
ii xcf
1
)( is called a Riemann sum of f for the given partition.
5. Def. Norm of the Partition: The length of the largest subinterval of a partition is called the norm
of the partition and is denoted by .
6. Def. Regular Partition: If every subinterval in a partition is of equal length, the partition is called
regular.
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AP Calculus BC - Ahlborn Def. 4 -
2
7. Def. Definite Integral: Iff is defined on the interval [ ]ba, and the limit of the Riemann sum
=
n
i
ii xcf
10
)(lim exists, then this limit is called the definite integral of f on [ ]ba, and
is denoted by ( )
=
=n
i
ii
b
a
xcfdxxf1
0 )(lim . (The values a and b are called the lower
and upper limits of the integral, respectively.)
8. Def. Integrable Function: A function is said to be integrable on an interval if it has a definite
integral on the interval.
9. Thm. Continuity and Integrability: If a function f is continuous on the closed interval [ ]ba, ,
then f is integrable on [ ]ba, .
10. Thm. Area Under a Curve: Let f be a continuous, function on the interval [ ]ba, . The area ofthe region bounded by the graph off, thex-axis, and the vertical lines ax = and bx = is
found by:
( )=b
a
dxxfarea .
11. Def. Definite Integral at a Point: ( ) 0=a
a
dxxf .
12. Def. Definite Integral over a Reverse Interval: Iff is an integrable function on [ ]ba, , then
( ) ( ) =b
a
a
b
dxxfdxxf .
13. Thm. Definite Integral over Two Adjacent Intervals: Iff is an integrable function on the three
intervals indicated, then ( ) ( ) ( ) =+b
a
b
c
c
a
dxxfdxxfdxxf
14. Thm. Definite Integral of a Constant Times a Function: Iff is an integrable function on [ ]ba,
and k is a constant, then ( ) ( ) =b
a
b
a
dxxfkdxxfk .
15. Thm. Definite Integral of a Sum or Difference of Functions: Iff and g are both integrable
functions on [ ]ba, , then ( )