CALCULUS: Graphical,Numerical,Algebraic by Finney,Demana ... · 1 2 ( ) 4 Find the Horizontal...
Transcript of CALCULUS: Graphical,Numerical,Algebraic by Finney,Demana ... · 1 2 ( ) 4 Find the Horizontal...
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CALCULUS: Graphical,Numerical,Algebraic by Finney,Demana,Watts and Kennedy Chapter 3: Derivatives 3.3: Derivative of a function pg. 116-126
A) Using a definition of the derivative find the derivative of y =x2 at x = a
B) Using a definition of the derivative find the derivative of y =x3 at x = a
C) Using a definition of the derivative find the derivative of y =x2 + 4 at x = a
What you'll Learn About How to find the derivative of: Functions with positive and negative integer powers Functions with products and quotients
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Find the derivative using the power rule
D) f(x) = 3 + x2 - x3 + x5 E) 74
35
xx
y
F) 3 xy G)
x
xxy
1
43
35
H) xx
xxf21
4)(
Find the Horizontal Tangents of each curve
I) x3 + 2x2 = 0 J) 032
5
3
2 23 xxx
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Using Algebra and the Product Rule to take a derivative
J) y = (x2 + 3)(x3 - x) J) y = (x2 + 3)(x3 - x)
Using Algebra and the Quotient Rule to take a derivative
K) x
xxf
9)(
3 K)
x
xxf
9)(
3
Take the Derivative of the function
L) y = (x2 + x + 2)(x5 + x3 + 5x)
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Take the Derivative of the function
M) 2
4
2)(
x
xxf
N) 12 35)(
xxxf
O) 31
43)(
xx
xxxf P)
1
1)(
3
3
x
xxf
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Find the equation for the tangent line at the given point
Q) 2
5 2
x
xxy
at x = 1 R) 35 2 xy at x = 3
S) Find an equation of the line perpendicular to the tangent to the curve
y =4x3 - 6x + 2 at the point (2, 22).
T) Find the points on the curve y = x3 - 3x2 - 9 where the tangent is
parallel to the x-axis
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U) Suppose u and v are differentiable functions at x = 2 and
u(2) = 3, ,3)2( u v(2)=1, 2)2( v
i) Find uvdx
d
ii) Find
v
u
dx
d
iii) Find uvvudx
d223
V) Find the derivative of y = x with respect to x
W) Find the derivative of y = x with respect to t
X) Find the derivative of y = x with respect to P
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CALCULUS: Graphical,Numerical,Algebraic by Finney,Demana,Watts and Kennedy Chapter 3: Derivatives 3.5: Derivatives of Trig Functions pg. 141-147
A) y = 5 + x2 - tanx B) y = xsinx
C) cot
4y C)
cot
4y
D)
cscsec
cossin
y
What you'll Learn About How to find the derivative of a trig function
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Find equations for the lines that are tangent and normal to the
graph of y = 2cosx at 4
x
Find the points on the curve y = cot x, 2
0
x , where the
tangent line is parallel to the line y = -2x.
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CALCULUS: Graphical,Numerical,Algebraic by Finney,Demana,Watts and Kennedy Chapter 3: Derivatives 3.6: Chain Rule pg.148-156
xsyD
xosyC
yBxyA
cotcxc )
3c )
4xsin ) sin )
2
2
2
What you'll Learn About How to find the derivative of a composite function
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cossin
1 )
cossin )
)2cos()2sin(5 )
433
2
xxyF
xxyE
xxyE
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5 )
1
)
1
)
4-3
3
2
3
2
xxyH
x
xyG
x
xyG
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21
1 )
6-3x )
3
54
xyJ
xyI
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csc3 )
csc3 )
xxyM
xxyL
3
cot9 if y Find N)
xy
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Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3.
Evaluate the derivatives with respect to x
2 at x g(x)
f(x) ) 3 at x f(x)g(x) )
3 at x g(x)f(x) ) 2 at x 2f(x) )
DC
BA
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2 at x (x)g(x)f )
3 at x g
1 )
2 at x f(x) ) 2 at x g(x)f )
22
2
F
xG
FE
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CALCULUS: Graphical,Numerical,Algebraic by Finney,Demana,Watts and Kennedy Chapter 3: Derivatives 3.8: Derivatives of Inverse Trig Functions pg.165-171
Find the derivative of the inverse sine function using implicit differentiation
What you'll Learn About How to find the derivative of inverse functions
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sin5xrccscf(x) ) xarctanx-1x )
sinxarccosx )
x
1arccos ) xarcsin )
332
2
2
xaEyD
yC
yByA
Find the equation of the tangent line
2at x x csc ) -1 yF
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G) Find the derivative of f(x) = sinx at 6
x
H) Find the derivative of f(x) = arcsinx at 2
1x
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-144. f (2) -3, f (2) , and g(x) f ( ),
3
what is the equation of the tangent line to g(x)
at x -3?
-3 -3A) -2 x 3 B) 2 x 3
4 4
3 3) -2 x 3 D) 3 x 2
4 4
E) -
I f x
y y
C y y
y
4
2 x 33
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-1-45. f (2) -3, f (2) , and g(x) f ( ),
3
what is the equation of the tangent line to g(x)
at x -3?
-3 -3A) -2 x 3 B) 2 x 3
4 4
3 4) -2 x 3 D) 2 x 3
4 3
E)
I f x
y y
C y y
y
4
-2 x 33
-1-36. f (2) -3, f (2) , and g(x) f ( ),
4
what is the equation of the tangent line to g(x)
at x -3?
-3 -4A) -2 x 3 B) 3 x 2
4 3
3 4) -2 x 3 D) 2 x 3
4 3
E)
I f x
y y
C y y
y
-4
-2 x 33
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CALCULUS: Graphical,Numerical,Algebraic by Finney,Demana,Watts and Kennedy Chapter 3: Derivatives 3.9: Derivatives of Exponential and Logarithmic Functions
7 ) ex- )
) 5 )
5 ) )
6 ) 5 )
7 ) 5 )
22x443
4
35
5
3arctansin
2
xx
xx
xx
xx
xx
yBexyI
eyHeyG
eyFeyE
yDyC
yByA
What you'll Learn About How to take the derivative of exponential and logarithmic functions
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arcsinxlnln-xn )
5ln ) ln )
ln ) ln )
xlog
5 )
x
4log )
xlog ) xlog )
23
4
4
2
7
5
36
3
5
lxyI
xyHxyG
xyFxyE
yDyC
yByA
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CALCULUS: Graphical,Numerical,Algebraic by Finney,Demana,Watts and Kennedy Chapter 3: Derivatives 3.6/10.1: Derivatives of Parametric Equations
Graph the parametric function given
0 t ty 3-t ) 2 xA
B) Find the derivative of the function at t=5
C) Find the equation of the tangent line at t=1
9t y 3 2 tx
D) Find the equation of the tangent line at 4
sin y cos x
E) Find the equation of the tangent line at t
2tan y 1 - 2sec2 ttx
What you'll Learn About How to take the derivative of functions in Parametric Form
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A curve C is defined by the parametric equations
x = t2 – 4t +1 and y = t3. Determine the equation of the
line tangent to the graph of C at the point (1, 64)?
Determine the horizontal and vertical tangents for the
parametric curve
2sin2 cos x)
4t -tyt -1 ) 2
yB
xA
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The derivative of ex is: (Itself)(Derivative of the power)
The derivative of ax is:
(Itself)(ln of the base)(Derivative of the power)
The derivative of ax is: (Itself)(ln of the base)(Derivative of the power)
The derivative of ln u is:
(one over what you are taking the ln of) times now you should be in the
numerator (Derivative of what you are taking the ln of)
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When you do the power rule the base does not change
only the power
- Once you have done the power rule, you are done
with the powers
When you do the derivative of a trig function the angle
does not change
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