AP CalculusUnit 4

Applications of Differentiation

Section 1Extrema on an Interval

Definition of ExtremaLet f be defined on an interval I containing c.

1. f(c) is the minimum of f on I when _____________ for all x in I

2. f(c) is the maximum of f on I when_____________ for all x in I

The minimum and maximum of a function on an interval are the

extreme values, or extrema, of the function on the interval. The

minimum and maximum of a function on an interval are also called

the absolute minimum and absolute maximum, or the global

minimum and global maximum, on the interval. Extrema can

occur at _______________________ or ___________________ of

an interval. Extrema that occur at the endpoints are called

endpoint extrema.

The Extreme Value TheoremIf f is continuous on a closed interval [a,b], then f has both a

______________________________________________________

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ExplorationUse a graphing calculator to find the extrema of the function.a. f(x)=x2-4x+5 on the interval [-1,3]

b. f(x)=x3-2x2-3x-2 on the interval [-1,3]

Definition of Relative Extrema1. If there is an open interval containing c on which f(c) is a

_____________________, then f(c) is called a relative maximum

of f, or you can say that f has a relative maximum at (c,f(c)).

2. If there is an open interval containing c on which f(c) is a

_____________________, then f(c) is called a relative minimum

of f, or you can say that f has a relative minimum at (c,f(c)).

Relative maximum and relative minimum are sometimes called

local maximum and local minimum.

Example 1.1Find the derivative at each relative extremum shown below:

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Critical NumberLet f be defined at c. If _____________ or if f ________________

at c, then c is a critical number of f.

Theorem: Relative Extrema Occur Only at Critical NumbersIf f has a relative minimum or relative maximum at x=c, then c is a critical number of f.

Example 1.2Find the extrema of f(x)=3x4-4x3 on the interval [-1,2].

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Example 1.3Find the extrema of f(x)=2x-3x2/3 on the interval [-1,3].

Example 1.4Find the extrema of f(x)=2sin(x)-cos(2x) on the interval [0,2π].

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Section 2Rolle’s Theorem and the Mean Value Theorem

Rolle’s TheoremLet f be _________________ on the closed interval

[a,b] and ____________________ on the open

interval (a,b). If f(a)=f(b), then there is at least one

number c in (a,b) such that ________________.

Example 2.1Find the two x-intercepts of f(x)=x2-3x+2 and show that f'(x)=0 at some point between the two x-intercepts.

Example 2.2Let f(x)=x4-2x2. Find all values of c in the interval (-2,2) such that f'(c)=0.

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The Mean Value TheoremIf f is continuous on the ______________________________ and

differentiable on the________________________________, then

there exists a number c in (a,b) such that

f'(c)=

Example 2.3For f(x)=5-(4/x), find all values c in the open interval (1,4) such that

f ' (c )=f ( 4 )−f (1)

4−1

Example 2.4Two stationary patrol cars equipped with radar are 5 miles apart on a highway, as shown. As a truck passes the first patrol car, its speed is clocked at 55 miles per hour. Four minutes later, when the truck passes the second patrol car, its speed is clocked at 50 miles per hour. Prove that the truck must have exceeded the speed limit (of 55 miles per hour) at some time during the 4 minutes.

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Section 3Increasing and Decreasing Functions and the First Derivative

Test

Increasing and Decreasing Functions

A function f is increasing on an interval when, for any two

numbers x1 and x2 in the interval, x1<x2 implies that ___________

A function f is decreasing on an interval when, for any two

numbers x1 and x2 in the interval, x1<x2 implies that ___________

Theorem: Test for Increasing and Decreasing FunctionsLet f be a function that is continuous on the closed interval [a,b] and differentiable on the open interval (a,b)1. If ____________ for all x in (a,b), then f is __________________________ on [a,b]2. If ____________ for all x in (a,b), then f is __________________ on [a,b]3. If ____________ for all x in (a,b), then f is __________ on [a,b]

Example 3.1Find the open intervals on which f(x)=x3-3/2x2 is increasing or decreasing.

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Strictly Monotonic: a function that is either _________________ on the entire interval or _________________ on the entire interval

The First Derivative TestLet c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows.1. If f'(x) changes from _______________________________ at c, then f has a relative minimum at (c,f(c)).2. If f'(x) changes from _______________________________ at c, then f has a relative maximum at (c,f(c)).3. If f'(x) is ________________________ of c or ______________ ____________________ of c, then f(c) is neither a relative maximum nor a relative minimum.

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Example 3.2Find the relative extrema of f(x)=1/2x-sin(x) in the interval (0,2π).

Example 3.3Find the relative extrema of f(x)=(x2-4)2/3

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Example 3.4Find the relative extrema of

f ( x )= x4+1x2

Example 3.5Neglecting air resistance, the path of a projectile that is propelled at an angle 𝛉 is

y= gsec2θ

2v02 x2+tanθx+h

for 0≤𝛉≤π/2 where y is height, x is the horizontal distance, g is the acceleration due to gravity, v0 is the initial velocity, and h is the initial height. Let g=-32 feet/second, v0=24 feet/second, and h=9 feet. What value of 𝛉 will produce a maximum horizontal distance?

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Section 4Concavity and the Second Derivative Test

ConcavityLet f be differentiable on an open interval I. The graph of f is _________________________ on I if f' is increasing on the interval and _______________________ on I when f' is decreasing on the interval.

Theorem: Test for ConcavityLet f be a function whose second derivative exists on an open

interval I.

1. If ______________ for all x in I, then the graph of f is

________________________ on I.

2. If ______________ for all x in I, then the graph of f is

____________________________________ on I.

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Example 4.1Determine the open intervals on which the graph of

f ( x )= 6x2+3 is concave upward or downward.

Example 4.2Determine the open intervals on which the graph of

f ( x )= x2+1x2−4 is concave upward or concave downward.

Points of InflectionLet f be a function that is continuous on an open interval, and let c

be a point in the interval. If the graph of f has a tangent line at the

point (c,f(c)), then this point is a point of inflection of the graph of

f when the _____________________________________________

_______________________ (or downward to upward) at the point.

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Theorem: Points of InflectionIf (c,f(c)) is a point of inflection of the graph of f, then either

____________ or __________________ at x=c.

Example 4.3Determine the points of inflection and discuss the concavity of the graph of f(x)=x4-4x3

The Second Derivative TestLet f be a function such that f'(c)=0 and the second derivative of f

exists on an open interval containing c.

1. If _______________, then f has a relative minimum at (c,f(c)).

2. If _______________, then f has a relative maximum at (c,f(c)).

If _______________, then the test fails. That is, f may have a

relative maximum, a relative minimum, or neither.

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Example 4.4Find the relative extrema of f(x)-3x5+5x3

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Section 5A Summary of Curve Sketching

Example 5.1 f (x)=2(x2−9)

x2−4First Derivative:

Second Derivative:

x-intercepts:

y-intercept:

Vertical Asymptotes:

Horizontal Asymptotes:

End Behavior:

Critical Numbers:

Possible Points of Inflection:

Domain:

Test Intervals:

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Example 5.2 f ( x )= x

2−2 x+4x−2

Example 5.3 f ( x )= x

√ x2+2

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Example 5.4f(x)=2x5/3-5x4/3

Example 5.5f(x)=x4-12x3+48x2-64x

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Example 5.6 f ( x )= cos (x )

1+sin (x)

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Section 6Optimization Problems

Example 6.1A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume?

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Example 6.2Which points on the graph of y=4-x2 are closest to the point (0,2)?

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Example 6.3A rectangular page is to contain 24 square inches of print. The margins at the top and bottom of the page are to be 1.5 inches, and the margins on the left and right are to be 1 inch. What should the dimensions of the page be so that the least amount of paper is used?

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Example 6.4Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed by two wires, attached to a single stake, running from ground level to the top of each post. Where should the stake be placed to use the least amount of wire?

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Example 6.5Four feet of wire is to be used to form a square and a circle. How much of the wire should be used for the square and how much should be used for the circle to enclose the maximum total area?

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Section 7Newton’s Method

Newton’s Method for Approximating the Zeros of a FunctionLet f(c)=0, where f is differentiable on an open interval containing c. Then, to approximate c, use these steps:1. Make an initial estimate x1 that is close to c.2. Determine a new approximation

xn+1=xn−f (xn)f ' (xn)

3. When |xn-xn+1| is within the desired accuracy, let xn+1 serve as the final approximation. Otherwise, return to step 2 and calculate a new approximation.

Iteration: each successive application of the procedure

Example 7.1Calculate three iterations of Newton's Method to approximate a zero of f(x)=x2-2. Use x1=1 as the initial guess.

Example 7.2Use Newton's Method to approximate the zeros of f(x)=2x3+x2-x+1. Continue the iterations until two successive approximations differ by less than 0.0001.

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Converge: when successive numbers in a sequence approach a limit

This is the idea behind Newton's Method, but what happens if the approximations do not converge?

Example 7.3The function f(x)=x1/3 is not differentiable at x=0. Show that Newton's Method fails to converge using x1=0.1

*If this is not satisfied, Newton's Method is not helpful!

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Section 8Differentials

Tangent Line Approximation of f at c*Also called Linear Approximation*as x→c, y→f(c) so,

Example 8.1Find the tangent line approximation of f(x)=1+sin(x) at the point (0,1). Then use a table to compare the y-values of the linear function with those of f(x) on an open interval containing x=0.

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DifferentialsLet y=f(x) represent a function that is differentiable on an open interval containing x. The differential of x (denoted by dx) is any ___________________________. The differential of y (denoted by dy) is ______________________

Example 8.2Let y=x2. Find dy when x=1 and dx=0.01. Compare this value with ∆y for x=1 and ∆x=0.01.

Propagated ErrorIf the measured value x is used to compute another value f(x), then the difference between f(x+∆x) and f(x)

Example 8.3The measured radius of a ball bearing is 0.7 inch. The measurement is correct to within 0.01 inch. Estimate the propagated error in the volume V of the ball bearing.

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Calculating Differentials

Differential Form: let u and v be differential function of x, so

___________________ and ____________________

Example 8.4Find the differential of each equation:a. y=x2

b. y=√x

c. y=2sin(x)

d. y=xcos(x)

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e. y=1/xLeibniz Notation: notation for derivative and differentials using dy, dx, etc.

Example 8.5y=f(x)=sin(3x)

Example 8.6y=f(x)=(x2+1)1/2

To approximate function values:

Example 8.7Use differentials to approximate √16.5

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