Unit 3Differentiation

Section 1The Derivative and The Tangent Line Problem

Secant Line: a line that crosses a curve at ___________________Call these points (c,f(c)) and (c+∆x,f(c+∆x)). The slope of the

secant line is:

Difference Quotient: the formula for finding the slope of a line that touches a curve

Change in x: Change in y:

Tangent Line with Slope m: If f is defined on an open interval containing c, and if the limit

exists, then the line passing through (c , f (c )) with slope m is the tangent line to the graph of f at the point (c , f (c )).

Example 1.1Find the slope of the graph of f(x)=2x-3 when c=2.

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Example 1.2Find the slope of f(x)=x2+1 at (0,1) and (-1,2).

Vertical Tangent LineIf f is continuous at c and

then the vertical line x=c passing through (c , f (c )) is the vertical tangent line to the graph of f.

The Derivative of a FunctionDifferentiation: one of two fundamental operations of calculus. It is the process of finding the derivative of a function.

Derivative of a Function:

Provided the limit exists. For all x for which this limit exists, f’ is a function of x.

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Instantaneous Rate of Change (Rate of Change): the rate of

change at a ______________________________ one variable with

________________________________

Differentiable (at x): a function when its derivative ___________

Differentiable on an Open Interval (a,b): a function when it is

differentiable __________________________________________

Notations of Derivatives:

Example 1.3Find the derivative of f(x)=x3+2x

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Example 1.4Find the slope of f ( x )=√x at (1,1) and (4,2) using the derivative of f(x).

Example 1.5Find the derivative of y=2

t

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Derivatives from the Left and from the Right: the one-sided limits

Differentiable on the Closed Interval [a,b]: a function f when it

is differentiable on __________ and when the derivative from the

________________ and the derivative from the

________________ both exist

**When a function is not continuous at x=c, it is also not differentiable at x=c.**

Differentiability implies continuity, but it is possible for a function to be continuous at x=c and not differentiable at x=c.

Example 1.6Find the derivative of f(x)=|x-2| (hint: use the derivatives from the left and right)

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Example 1.7Find the derivative of f(x)=x1/3 at x=0

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Section 2Basic Differentiation Rules and Rates of Change

Theorem 2.2: The Constant RuleThe derivative of a constant function is _____. That is, if c is a real number, then

Example 2.1Find the derivative of the following functions:y=7

f(x)=0

s(t)=-3

y=kπ2

ExplorationWithout any shortcuts, find the following derivatives and look for patterns.a. f(x)=x

b. f(x)=x2

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c. f(x)=x3

d. f(x)=x4

e. f(x)=x1/2

f. f(x)=x-1

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Theorem 2.3: The Power RuleIf n is a rational number, the then function f ( x )=xn is differentiable and

For f to be differentiable at x=0, n must be a number such that ________ is defined on an open interval containing 0.

Example 2.2Find the derivative using the power rule:a. f(x)=x3

b. g ( x )=3√ x

c. y=1x2

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Steps to Differentiation Problems

1. Given

2. Rewrite

3. Differentiate

4. Simplify

Example 2.3Find the slope of f(x)=x4 when (a) x=-1 and (b) x=0.

Example 2.4Find the equation of the tangent line to f(x)=x2 when x=-2

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Theorem 2.4: The Constant Multiple RuleIf f is a differentiable function and c is a real number, then cf is also differentiable and

Example 2.5Find the derivative:a. y=5x3

b. y=2x

c. f (t )=4 t2

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d. y=2√ x

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e. y=1

2 3√x2

f. y=−3 x2

Combination of Constant Multiple Rule and Power Rule

Example 2.6Find the derivative:

a. y= 52 x3

b. y=5

(2x )3

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c. y=7

3 x−2

d. y=7

(3x )−2

Theorem 2.5: The Sum and Difference RulesThe sum (or difference) of two differentiable functions f and g is also differentiable. Moreover, the derivative of f+g (or f-g) is the sum (or difference) of the derivatives of f and g.

Example 2.7Find the derivative:a. f(x)=x3-4x+5

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b. g ( x )=−x4

4+3 x3−2 x

c. y=3 x2−x+1x

Theorem 2.6: Derivatives of Sine and Cosine Functions

Example 2.8Find the derivative:a. y=2sin(x)

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b.y= sinx2

c. y=x+cos(x)

d. y=cosx−π3 sinx

Rates of ChangePosition Function: the function s that gives the position (relative to the origin) of an object as a function of time t

Average Velocity: change in distance divided by change in time:

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Example 2.9A billiard ball is dropped from a height of 100 feet. The ball's height s at time t is the position function s=-16t2+100 where s is measured in feet and t is measured in seconds. Find the average velocity over each time interval.a. [1,2]

b. [1,1.5]

c. [1,1.1]

Velocity: the derivative of the position function

Speed: the absolute value of the ___________________

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The Position Functions(t)=1/2gt2+v0t+s0

s0 is the

v0 is the

g is the

g= -32 ft/sec or g= -9.8 m/sec

Example 2.10At time t=0, a diver jumps frm a platform diving board that is 32 feet above the water. Because the initial velocity of the diver is 16 ft/sec, s(t)=-16t2+16t+32 where s is measured in feet and t is measured in seconds.a. When does the diver hit the water?

b. What is the diver's velocity at impact?

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Section 3Product and Quotient Rules and Higher Order Derivatives

Theorem 2.7: The Product RuleThe product of two differentiable functions f and g is itself differentiable. Moreover, the derivative of fg is the first function times the derivative of the second, plus the second function times the derivative of the first.

Extended version:

Find the derivative of f(x)=x2sin(x)cos(x)

Example 3.1Find the derivative: h(x)=(3x-2x2)(5+4x) (a) using the product rule and (b) by first multiplying.

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Example 3.2Find the derivative: y=3x2sin(x)

Example 3.3Find the derivative: y=2xcos(x)-2sin(x)

Theorem 2.8: The Quotient RuleThe quotient f/g of two differentiable functions f and g is itself differentiable at all values of x for which g(x)≠0. Moreover, the derivative of f/g is given by the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

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Example 3.4Find the derivative: y=

5 x−2x2+1

Example 3.5

Find the equation of a tangent line at (-1,1) of 3−1x

x+5

Example 3.6Find each derivative (hint: use the constant multiple rule).

a. y= x2+3x6

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b. y=5 x4

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c. y=−3(3x−2x2)7 x

d. y=9

5 x2

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Example 3.7Find the derivative:a. y=x-tan(x)

b. y=xsec(x)

Example 3.8Find the derivative of y=1−cos (x )

sin ( x) two different ways.

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Acceleration: derivative of the ____________________________

or second derivative of the ________________________________

Second Derivative: the derivative of the ____________________

_____________________, denoted by f"(x)

Higher-Order Derivative: a ______________________________

derivative of a function, must be a positive integer order

Third Derivative: the derivative of the _____________________,

denoted by f'''(x)

Example 3.9Because the moon has no atmosphere, a falling object on the moon encounters no air resistance. In 1971, astronaut David Scott demonstrated that a feather and a hammer fall at the same rate on the moon. The position function for each of these falling objects is s(t)=-0.81t2+2 where s(t) is the height in meters and t is the time in seconds. What is the ratio of the Earth's gravitational force to the moon's?

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Section 4The Chain Rule

Chain Rule: Differentiation rule that deals with composite

functions- if y changes ________ times as fast as u, and u changes

________ times as fast as x, then y changes ____________as fast

as x

Example 4.1Find dydu , dudx , and dydx and show that dydx=( dy

du)( dudx

)

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Theorem 2.10: The Chain RuleIf y=f (u) is a differentiable function of u and u=g (x) is a differentiable function of x, then y= f (g ( x )) is a differentiable function of x and

Example 4.2Decompose the following functions:

a. y= 1x+1

b. y=sin(2x)

c. y=√3 x2−x+1

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d. y=tan2(x)

Example 4.3Find the derivative:y=(x2+1)3

Theorem 2.11: The General Power RuleIf y=¿, where u is a differentiable function of x and n is a rational number, then

Example 4.4f(x)=(3x-2x2)3

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Example 4.5Find where f'(x)=0 and where f'(x) doesn't exist.

f ( x )=√(x2−1)2

Example 4.6g (t )= −7

(2 t−3)2

Example 4.7f ( x )=x2 √1−x2

Example 4.8f ( x )= x

3√ x2+4

Example 4.9

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y=( 3 x−1x2+3

)2

Trigonometric Functions and the Chain Rule

Example 4.10a. y=sin(2x)

b. y=cos(x-1)

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c. y=tan(3x)

Example 4.11a. y=cos(3x2)

b. y=cos(3)x2

c. y=cos(3x)2

d. y=cos2(x)

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e. y=√cos (x )

Example 4.12f(t)=sin3(4t)

Example 4.13Find the tangent line to f(x)=2sin(x)+cos(2x) at (π,1). Then find all points in (0,2π) with horizontal tangents.

Section 5

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Implicit Differentiation

Explicit Form: a function in which the dependent variable is

written in terms of the ___________________________________

Implicit Form: a function in which the dependent variable is not

_________________________________________ of the equation

Implicit Differentiation: process to find the derivative with

respect to to x of a function _______________________________

Implicit Differentiation

If a term involves only x's, _______________________________.

If a term involves y's, you must apply the ____________________

treating y as a differentiable function of x.

Example 5.1Differentiate with respect to x:

a. ddx [ x3 ]

b. ddx [ y3]

c. ddx [ x+3 y ]

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d. ddx [ x y2]

Guidelines for Implicit Differentiation

1. Differentiate both sides of the equation ___________________.

2. Collect all terms involving dydx on the _____________________

of the equation and move all other terms to the right side of the

equation.

3. Factor dydx out of the ______________________________________________.

4. Solve for ___________________.

Example 5.2Find dy/dx given that y3+y2-5y-x2=-4

Example 5.3

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Determine whether or not the function is differentiable. If it is differentiable, find the derivative.a. x2+y2=0

b. x2+y2=1

c. x+y2=1

Example 5.4

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Find the slope of the graph of x2+4y2=4 at the point (√2, -1/√2)

Example 5.5Find the slope of the graph of 3(x2+y2)2=100xy at the point (3,1).

Example 5.6Find dy/dx implicitly for sin(y)=x. Then find the largest interval for which y is a differentiable function of x.

Example 5.7

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Given x2+y2=25, find d2 yd x2

Example 5.8Find the tangent line to the graph of x2(x2+y2)=y2 at the point (√2/2, √2/2).

Section 6

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Related Rates

Related Rates

A use of the chain rule to find rates of change of two or more

related variables that are changing __________________________

For example, volume of a conical tank that is draining:

Exploration QuestionsIn a conical tank, height of the water is changing at -0.2 ft/min, and radius is changing at -0.1 ft/min. What is the rate of change of the volume when r=1 foot and h=2 feet?

Does dVdt depend on the values of r and h? Explain.

Example 6.1

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Given y=x2+3, find dydt when x=1, given that dxdt =2 when x=1.

Example 6.2A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area A of the disturbed water changing?

Guidelines for Solving Related-Rate Problems

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1. Identify all given quantities and quantities to be determined. Make a sketch and label all quantities.

2. Write an equation involving the variables whose rates of change are given or are to be determined

3. Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time.

4. After completing Step 3, substitute into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change.

Example 6.3Air is being pumped into a spherical balloon at a rate of 4.5 cubic feet per minute. Find the rate of change of the radius when the radius is 2 feet.

Example 6.4An airplane is flying on a flight path that will take it directly over a radar tracking station. The distance s is decreasing at a rate of 400 miles per hour when s=10 miles. What is the speed of the plane?

Example 6.5

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A television camera at ground level is filming the liftoff of a rocket that is rising vertically according to the position equation s=50t2, where s is measured in feet and t is measured in seconds. The camera is 2000 feet from the launch pad. Find the rate of change in the angle of elevation of the camera shown at 10 seconds after liftoff.

Example 6.6In the engine, a 7-inch connecting rod is fastened to a crank of radius 3 inches. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find the velocity of the piston when 𝛉=π/3. The velocity of a piston is related to the angle of the crankshaft.

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