Chapter 4 Applications of Derivatives

MATH 160 Study Guide for Chapter 4 Applications of Derivatives Fall, 2008

This Study Guide describes everything you are expected to know, understand, and be able to do from Chapter 4. Exam questions on this material will ask you to do one or more of the tasks described in this Study Guide. When you have mastered an item on the Study Guide, check it off. When every item is checked, you are ready for any question from Chapter 4 that could appear on a midterm exam or the final exam. (Items from Secs 4.1 and 4.3 are repeated from a Study Guide handed out previously.)

Sec. 4.1 Extreme Values of Functions

• Write complete definitions of the following words and phrases:

a) absolute maximum (aka global maximum) of a function (pg 244)
b) absolute minimum (aka global minimum) of a function (pg 244)
c) local maximum (aka relative maximum) (pg 247)

e) local minimum (aka relative minimum) (pg 247)
f) critical point (of a function) (pg 248).

Illustrate each of the above by sketching a graph.

Given a sketch of the graph of a function, identify and label all of the above.
(Notice that whether a function has absolute and local extrema depends on the domain of the function
as well as the equation/formula for the function.)

Study Suggestion: Study Sec. 4.1.
Representative homework problems; Page 252; #1 – 10

• Write a complete statement of the Extreme Value Theorem (pg 246).
Illustrate the Extreme Value Theorem graphically.
Give graphical examples to show that when any of the hypotheses of the Extreme Value Theorem are not satisfied, the function may or may not have an absolute maximum or minimum.

Study Suggestion: Study Theorem 1 in Sec. 4.1. Find examples among exercises 1 – 14, pg 252.

• Write a complete statement of the First Derivative Theorem for Local Extreme Values (Theorem 2, pg 247).
Use a graph to explain in terms of secant and tangent lines how to see that this theorem is plausible
(short of a formal proof).
Give examples to show that a function might or might not to have a local extremum at a point where its derivative is 0.

Study Suggestion: Study Theorem 2 and the discussion of Finding Extrema in Sec. 4.1.

• Explain the connection between critical points and local extrema of a function.

Representative homework problems: Page 252; #11 – 14.

• Describe a step-by-step procedure for finding the absolute extrema of a continuous function on a closed interval of finite length (summarized in the box top of page 249).
Use this procedure to find the absolute maximum and minimum values of a given function and the points where these extreme values are attained.

Representative homework problems: Page 253; #15 – 34 and #35 – 44.

• Give examples to show that a function can fail to have a local extremum at a critical point and at an end point of its domain.

Representative homework problems: Pages 253 – 254; #53, 65, 68, 69, 70.

Sec. 4.2 The Mean Value Theorem

• (a) Write a complete statement of the Mean Value Theorem (pg 257).

(b) Explain what the Mean Value Theorem says graphically by sketching the graph of a function and the relevant secant and tangent lines, and then explaining how these lines are related, and

(b) Explain what the Mean Value Theorem says physically by interpreting the hypotheses and the conclusion of the theorem in terms of an object moving along a straight line path.

Study Suggestion: Graphical interpretation is discussed on page 257 – 258. Physical interpretation is discussed on pages 258.
Representative homework problems: Page 260; #1 – 4 (think graphically).
Page 261; #45 – 49 (think physically).
Page 262; #56 (think about h(x) = f(x) – g(x)) and #59.

• Explain what the connection between Rolle’s Theorem and the Mean Value Theorem is.

• Given the graph of a function over a closed interval, determine whether the Mean Value Theorem is applicable or not. In both cases, EITHER locate the point or points that satisfy the conclusion of the Mean Value Theorem (and explain how to see that the conclusion is satisfied at these points) OR explain how to see that there are no such points.

Study Suggestion: This is just the graphical interpretation of the Mean Value Theorem in the other direction. Read Sec. 4.2.

• Given a function defined on a given closed interval [a, b] by an expression y = f(x), determine whether the Mean Value Theorem is applicable or not. In both cases, use calculus (and algebra) to EITHER locate the point or points that satisfy the conclusion of the Mean Value Theorem OR to explain how to see that there are no such points.

Representative homework problems: Page 260; #5 – 9 (In these exercises, if there is a point c that satisfies the conclusion of the Mean Value Theorem, find it). Mathophiles, try Page 262, #51, 52, & 58.

• Apply the fact that two functions that have the same derivative must differ by a constant, your knowledge of the basic differentiation formulas, and your skill with algebra and trigonometry to

(a) find all functions that have a given polynomial, power function, or simple trigonometric function as first derivative,

(b) show that two seemingly different functions differ only by a constant.

Representative homework problems: Page 261; #27 – 40. Page 262, #53. Page 319; #17, 18.

Sec. 4.3 Monotonic Functions and the First Derivative Test

• Write a complete definition of what it means for a function to be increasing and for a function to be decreasing on an interval (pg 263).
Explain how to use the derivative to determine the intervals on which a function is increasing and on which it is decreasing. Explain how to use the first derivative to determine whether a function has a local maximum, a local minimum, or neither at its critical points and the end points of its domain. Illustrate your explanations with graphs (pg 263).

Study Suggestion: Study Sec. 4.3.

Related homework problems: Page 267; #43 – 46.

• Given a function defined by an expression y = f(x), calculate and analyze its first derivatives (using algebra, not a calculator) to find the critical points of the function.
Knowing the critical points of a function, use the First Derivative Test for Monotonic Functions (page 263) to determine the intervals where the function is increasing and the intervals where the function is decreasing.
Use the First Derivative Test for Local Extrema (page 265) to determine whether the function has a local maximum, a local minimum, or neither at each critical point and, if applicable, at the end point of its domain.

Representative homework problems: Page 266; #1 –36; Page 267; #37, 38, 41, 42, 47, 48.

Sec. 4.4 Concavity and Curve Sketching

• Given the graph of a function, identify and indicate

(a) the intervals where the function is increasing,
(b) the intervals where the function is decreasing,
(c) the critical points,
(d) the intervals where the function is concave up,
(e) the intervals where the function is concave down, and
(f) the inflection points.

Study Suggestion: Read the examples Section 4.4.
Representative homework problems: Page 274 – 275; #1 – 8, and page 276; #71, 72, 74.

• Given a function defined by an expression y = f(x) (which may involve constants designated by letters), analyze its first and second derivatives to determine

Representative homework problems: Page 275; #9 – 40. Page 277; 79, 80, 81, 83, 84.

• State the Second Derivative Test for Local Extrema.
Explain how to see that the sign of the second derivative reliably indicates whether a function has a local maximum or local minimum at a point where the first derivative is 0.
Give examples showing that if the first and second derivative are both zero at a point x = c, then the function may or may not have a local extremum at the point x = c.

Representative homework problems: Page 277; #82.

• Sketch the graph of a function whose first and/or second derivatives have certain properties.
Properties may be given symbolically, algebraically, or graphically or you may have to figure them out from an expression y = f(x) for the function or for its derivative.

Representative homework problems: Pages 275 – 276; #63 – 70, 75, 76, 77, 78.

Sec. 4.5 Applied Optimization Problems

• Solve optimization problems similar to the exercises in this section of the textbook.

Representative homework problems: Page 253; #55 – 64. Page 285; #1– 37. Also look at Page 320, #63 – 71.

NOTE: You are required to know the Pythagorean Theorem; properties of similar triangles; area formulas for simple plane figures; volume formulas for spheres, cylinders, cones, and rectangular parallelepipeds (boxes); and the definitions of the trigonometric functions.

Sec. 4.6 Indeterminate Forms and L’Hospital’s Rule

State and prove Theorem 6 (L’Hospital’s Rule, first form). What is the difference between Theorem 6 (L’Hospital’s Rule, first form) and Theorem 7 (L’Hospital’s Rule, stronger form)?

Study Suggestion: Study pages 292 – 293 of Sec 4.6.

Recognize situations where limits can be evaluated by using L’Hospital’s Rule. Correctly apply L’Hospital’s Rule to evaluate these limits.

Study Suggestion: Study Sec 4.6 from the middle of page 295 through page 297 (omit Theorem 8 and the proof of Theorem 7)
Representative Homework Problems: Page 298; #1 – 31, 33, 35.

Sec. 4.7 Newton’s Method (This topic is explored in Calculator Lab 4. It may or may not be discussed in class.)

Given the graph of a function y = f(x) and an initial approximation xo for a zero of the function, find the next three Newton approximates graphically and an illustrate the process of finding the these approximates on the graph.

Study Suggestion: Study sec 4.7 and the Calculator Lab on Solving Equations by Newton’s Method.

Given a specific function y = f(x) implement Newton’s method on a calculator to calculate an estimated value for a specified zero or zeros of the function to a specified number of decimal places (initial estimate xo may or may not be given). Sketch a graph of the function and illustrate/explain the calculations graphically.

Study Suggestion: Page 305; #1 – 10, 12.

Sec. 4.8 Antiderivatives

1. Explain what is meant by an antiderivative and by the indefinite integral for a function.
What is the difference, if any, between an antiderivative and the indefinite integral of a function?
Explain why an arbitrary constant is added in the indefinite integral.

2. Determine whether a given function y = g(x) or family of functions y = g(x) + C is an antiderivative or the indefinite integral for a given function y = f(x).

Representative Homework Problems: Page 315; #55 – 60 and 61 – 64

3. Find the indefinite integral of functions that can be written as sums/differences of constant multiples of functions whose antiderivatives are known (i.e. functions listed in Table 4.2, pg 308) by algebraic manipulations, perhaps using trigonometric identities.
(You are expected to know the basic trigonometric identities, how to represent all six trig functions in terms of the sine and cosine, the Pythagorean identity, and the double angle formula for the sine.)

Representative Homework Problems: Page 314 - 315; #1 – 16 and 17 – 54.

4. Interpret the general solution to a first order initial value problem = g(x) graphically.
Explain graphically how an initial value selects one solution from the infinite number of possibilities.

Representative Homework Problems: Page 315 – 316 ; #65 & 66, #89 – 92.

5. Solve initial value problems = g(x), 1 n 3 where g(x) can easily be antidifferentiated.

Representative Homework Problems: Page 316; #67 – 86, 93 – 97, & 99

Page 2 March 26, 2008