Math 1A: Calculus Worksheets, Exercises of Calculus

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Math 1A: Calculus Worksheets

7 th^ Edition

Department of Mathematics, University of California at Berkeley

i Math 1A Worksheets, 7 th^ Edition

Preface

This booklet contains the worksheets for Math 1A, U.C. Berkeley’s calculus course. Christine Heitsch, David Kohel, and Julie Mitchell wrote worksheets used for Math 1AM and 1AW during the Fall 1996 semester. David Jones revised the material for the Fall 1997 semesters of Math 1AM and 1AW. The material was further updated by Zeph Grunschlag and Tom Insel, with help from the comments and corrections provided by David Lippel, Max Oks, and Sarah Reznikoff. Tom Insel coordinated the 1998 edition with much assistance and new material from Cathy Kessel and in consultation with William Stein. Cathy Kessel and Michael Wu have further revised the 1999 and 2000 edition respectively. Michael Hutchings made tiny changes in 2012. In 1997, the engineering applications were written by Reese Jones, Bob Pratt, and Pro- fessors George Johnson and Alan Weinstein, with input from Tom Insel and Dave Jones. In 1998, applications authors were Michael Au, Aaron Hershman, Tom Insel, George Johnson, Cathy Kessel, Jason Lee, William Stein, and Alan Weinstein.

About the worksheets

This booklet contains the worksheets that you will be using in the discussion section of your course. Each worksheet contains Questions, and most also have Problems and Ad- ditional Problems. The Questions emphasize qualitative issues and answers for them may vary. The Problems tend to be computationally intensive. The Additional Problems are sometimes more challenging and concern technical details or topics related to the Questions and Problems. Some worksheets contain more problems than can be done during one discussion section. Do not despair! You are not intended to do every problem of every worksheet.

Why worksheets?

There are several reasons to use worksheets:

• Communicating to learn. You learn from the explanations and questions of the students in your class as well as from lectures. Explaining to others enhances your understanding and allows you to correct misunderstandings.
• Learning to communicate. Research in fields such as engineering and experimental science is often done in groups. Research results are often described in talks and lectures. Being able to communicate about science is an important skill in many careers.
• Learning to work in groups. Industry wants graduates who can communicate and work with others.

• 1. Graphing a Journey.. Contents
• 2. Graphical Problems
• 3. Tangent Lines and ε–δ Preliminaries
• 4. Calculating Limits of Functions
• 5. The Precise Definition of a Limit
• 6. Continuity
• 7. Limits at Infinity and Horizontal Asymptotes
• 8. Derivatives.
• 9. Differentiation
• 10. The Chain Rule
• 11. Implicit Differentiation and Higher Derivatives
• 12. Using Differentiation to do Approximations
• 13. Exponential Functions
• 14. Inverse Functions
• 15. Logarithmic Functions and their Derivatives.
• 16. Inverse Trigonometric Functions.
• 17. Hyperbolic Functions
• 18. Indeterminate Forms and l’Hospital’s Rule
• 19. Falling Objects and Limits Involving Logarithms and Exponentials
• 20. Maximum and Minimum Values.
• 21. The Mean Value Theorem.
• 22. Monotonicity and Concavity
• 23. Applied Optimization
• 24. Antiderivatives
• 25. Sigma Notation and Mathematical Induction
• 26. Area iii Math 1A Worksheets, 7 th Edition
• 27. The Definite Integral
• 28. The Fundamental Theorem of Calculus.
• 29. The Substitution Rule
• 30. The Logarithm Defined as an Integral
• 31. Areas Between Curves
• 32. Volume
• 33. Volumes by Cylindrical Shells
• 34. Integration and Optimization

1. Graphing a Journey

Questions

1. Before you came to UC Berkeley you probably lived somewhere else (another country, state, part of California, or part of Berkeley). Sketch a graph that shows the speed of your journey to UC Berkeley as a function of time. (For example, if you came by car this graph would show speedometer reading as a function of time.) Label the axes to show speed.

Ask someone outside of your group to read your graph. See if that person can tell from your graph what form (or forms) of transportation you used.

6 v

t

2. Using the same labeling on the x-axis, sketch the graph of the distance you traveled on your trip to Berkeley as a function of time. (For example, if you traveled by car, this would be the odometer reading as a function of time—if you’d set the odometer to zero at the beginning of your trip.)

Ask someone else outside of your group to read your graph. See if that person can tell from your graph what form (or forms) of transportation you used.

6 x

t

2. Graphical Problems

Questions

1. Is there a function all of whose values are equal to each other? If so, graph your answer. If not, explain why.

Problems

1. (a) Find all x such that f (x) ≤ 2 where

f (x) = −x^2 + 1 f (x) = (x − 1)^2 f (x) = x^3

Write your answers in interval notation and draw them on the graphs of the functions. (b) Using the functions in part a, find all x such that |f (x)| ≤ 2. Write your answers in interval notation and draw them on the graphs of the functions. (c) Can you find upper bounds for the functions in part a? That is, for each function f is there a number M such that for all x, f (x) ≤ M? (d) What about lower bounds for the functions in part a? That is, for each f can you find a number m such that for all x, f (x) ≥ m? (e) What about finding upper and lower bounds for these functions restricted to the interval [− 1 , 1]? That is, for each f can you find numbers M and m such that for all x in [− 1 , 1], m ≤ f (x) ≤ M? (f) True or false? If M is an upper bound for the function f and M ′^ is an upper bound for the function g, then for all x which are in the domains of both f and g, |f (x) + g(x)| ≤ M + M ′.

2. (a) Graph the functions below. Find their maximum and minimum values, if they exist. You don’t need calculus to do this! y = −x^2 + 1 y = x^2 − 1 y = (x − 1)^2 y = sin x − 1 y = sin(x − 1) (b) Suppose f (x) = x^2 and g(x) = sin x. i. Write the functions in part a in terms of f and g. (For example, if h(x) = 2x^2 you can write h in terms of f as h(x) = 2f (x).) If you find more than one way of writing these functions in terms of f and g, show that they are equivalent. ii. How can you change the graph of f to obtain the graphs of the first three functions? Use your work from part a to help you.

iii. How can you change the graph of g to obtain the graphs of the last two functions?

3. This problem is like problem 2 except that we are asking that x not be equal to a.

Suppose a is a real number and δ is a positive real number. How do you describe all real numbers x that are within δ of a, but not equal to a:

(a) Graphically. (b) Using inequalities. (c) Using absolute value notation. (d) Using interval notation.

4. Let f (x) be x^2.

(a) Find all the positive numbers x such that f (x) is within 1 of 9. (“Within” means the same thing it did in Problems 1 and 2, but here it refers to numbers on the y-axis.) Give your answer: i. On a graph of y = x^2. ii. Using inequalities. iii. Using interval notation. (b) It’s difficult to express all numbers x such that f (x) is within 1 of 9 using absolute value notation. (Why?) Instead: i. Find a real number δ such that whenever x is within δ of 3, f (x) is within 1 of 9. Write this number using the min notation (“min” is for “minimum”). If a and b are two numbers, then min{a, b} is the smaller of a and b. For example, min{ 5 , 4 } = 4. If a and b are equal, then min{a, b} is just a (or b). For example, min{

ii. Using absolute value notation and the value of δ that you have found, write an expression for x such that x is within δ of 3. (c) i. Find a real number δ such that whenever x is within δ of 3, f (x) is within 1 /2 of 9. Write this number using the min notation. ii. Using absolute value notation and the value of δ that you have found, write an expression for x such that x is within δ of 3. (d) Is it true that for any positive number ε, there is a positive δ so that f (x) is within ε of 9 whenever x is within δ of 3? (ε is the lowercase Greek letter epsilon and stands for “error.”)

If your answer is yes, show how you can write an expression in terms of ε for a δ that works. Explain why your δ works.

If your answer is no, show that there is a positive number ε for which the statement above is not true.

Additional Problems

1. Graph y = 2x − x^2. For which points a is the tangent line to the curve at the point (a, 2 a − a^2 ) a horizontal line? How is the value of 2x − x^2 at these points related to the values of 2x − x^2 at other points?
2. Look back at Problem 1 above. The point (2, 0) is on the curve y = 2x − x^2.

(a) Draw the tangent line to the curve at the point (2, 0). (b) Try to determine the equation of this tangent line. What information do you need in order to determine the equation of a line? What information do you know in this situation? What information do you need?

Additional Problems

1. Below are listed three limits and three methods for evaluating limits. Match the best method to each limit and evaluate each limit. (Find the best method if you did not have a graphing calculator or computer.) - lim x→ 0

x^1000 − 703 x^506 + πx^17 + 480 − 372 x^75 + 39x^14 +

7 x^5 − 24

• graphing
• lim x→ 4

x^2 + 2x − 24 x − 4

• algebra
• lim x→ 0 x^2 sin

x

• direct substitution

Explain why the method you chose is the best of the three possibilities.

2. Using a calculator, try to guess limn→∞ n sin

(π n

. (Remember that n is in radians, not degrees!)

3. Draw a circle of radius 1. Inscribe a regular n-gon inside it. Draw line segments from the vertices to the center.

(a) What is the measure of the angles formed around the center of the circle? (b) What is the area of one triangle? What is the area of the polygon? (c) Calculate limn→∞ (Area of n-gon). Does this answer make sense intuitively?

5. The Precise Definition of a Limit

Questions

1. An important aspect of mathematical statements is the order of the words in the statement.

(a) “If 4 divides a number, then that number is even.” (“a divides b” means that b is divisible by a.) (b) “If a number is even, then 4 divides that number.”

Determine if each statement is true or false. If true, explain why. If false, give a counterexample. Why is the order of the two parts in each statement important?

2. Another important concept in mathematical statements is a quantifier : a phrase such as “for every” or “there exists.” (These are sometimes written ∀ and ∃.)

(a) “Every number is even.” (b) “There exists a number which is even.” Determine whether statements a and b are true or false. If true, explain why. If false, give a counterexample. (c) Now look back at the definition of limx→a f (x) = L. What are the quantifiers?

3. Suppose a, L, ε, and δ are real numbers, and that ε and δ are positive.

(a) What interval is determined by |x − a| < δ? What is the left endpoint? The right endpoint? The midpoint? Sketch the interval on the x-axis. (b) Sketch the set given by 0 < |x − a| < δ. How does this set of real numbers differ from the interval given in part a?

(c) On the y-axis, sketch the interval for |y − L| < ε. What are the endpoints and the midpoint? (d) Now sketch the region in the xy-plane determined by |x − a| < δ and |y − L| < ε. (e) Graph a function f so that |f (x) − L| < ε whenever 0 < |x − a| < δ.

(f) On the same axes and using the same L and a, graph a function g which does not satisfy the statement |g(x) − L| < ε whenever 0 < |x − a| < δ.

(a) Graph the function f (x) =

x^2 − 1 x + 1

x 6 = − 1 2 x = − 1

(b) Now, suppose that a = −1 and that L is chosen to be f (−1) = L = 2. If ε = 12 , show that there is no possible δ so that if |x − (−1)| < δ, then |f (x) − 2 | < 12. (Hint: Given any δ > 0, find a point x such that |x − 2 | < δ but |f (x) − 2 | 6 < 12 ). (c) From the graph, make a better guess for what L = limx→− 1 f (x) should be. With this new L and ε = 12 , find a value of δ so that if |x−(−1)| < δ, then |f (x)−L| < 12.

(d) In general, for an arbitrary value of ε, what δ would you choose so that if |x − (−1)| < δ, then |f (x) − L| < ε? (Hint: express δ as a function of ε.)

6. Continuity

Questions

1. There are at least three different types of discontinuities. Give a graphical example of each discontinuity: removable, jump, and infinite.
2. What three properties does a function f (x) need if it’s going to be continuous at a point a?
3. What does it mean for a function to be continuous on an interval?
4. (a) Continuous functions are quite common. What are two basic types of continuous functions? (b) What operations allow you to build more continuous functions from already- known continuous functions? List as many of the operations as you can.
5. When can you “pull the limit inside a function”? That is, what do you need to know about f and g, so that

lim x→a f (g(x)) = f (b) = f (lim x→a g(x))?

Problems

1. Let f (x) =

1 − x^2 if 0 ≤ x ≤ 1 1 + x 2 if 1 < x ≤ 2

(a) Show that f is not continuous on [0, 2]. (b) Show that f does not take on all values between f (0) and f (2), in other words that there’s a number between f (0) and f (2) that is not a value of f on the interval [0, 2].

2. (a) Show that if f is a continuous function on an interval, then so is |f |. (b) If |f | is continuous, must f be continuous? If so, prove it. If not, find a coun- terexample.
3. Assume that f (x) and g(x) are continuous at a number a and that c is a constant.

(a) Prove that cf (x) is also continuous at a. (b) Prove that f (x) · g(x) is also continuous at a.

7. Limits at Infinity and Asymptotes

Questions

1. (a) Draw the graphs of three functions with vertical asymptotes at x = a which are as different as possible. Describe these differences using limits, e.g., limx→a+^ f (x) = ∞, but limx→a+ g(x) = −∞. How many different vertical asymptotes can the graph of a function have? (b) Draw three different graphs of functions with horizontal asymptotes that are as different as possible. How many different horizontal asymptotes can the graph of a function have?
2. Let f , g, h, j, and k be functions. Assume that

i. limx→∞ f (x) = ∞, ii. limx→∞ g(x) = −∞, iii. limx→∞ h(x) = c > 0 (where c is a constant), iv. limx→∞ j(x) = 0, v. limx→∞ k(x) = 0+.

Have each person in the group explain to the group how to do two of the following problems. Simplify all expressions that you can. Indicate which limits you can’t evaluate. Explain your reasoning and explain when you use (i) through (v) above.

(a) limx→∞[f (x) + j(x)] = (b) limx→∞[g(x) + h(x)] = (c) limx→∞[f (x) + g(x)] = (d) limx→∞[f (x) − g(x)] = (e) limx→∞[h(x)j(x)] = (f) limx→∞[h(x)g(x)] = (g) limx→∞[k(x)g(x)] = (h) limx→∞[f (x)g(x)] = (i) limx→∞[j(x)/f (x)] = (j) limx→∞[f (x)/j(x)] = (k) limx→∞[f (x)/g(x)] = (l) limx→∞[k(x)/j(x)] =

Problems

1. Make a rough sketch of the curve y = xn, where n is an integer, for the following five cases: (i) n = 0; (ii) n > 0, n odd; (iii) n > 0, n even; (iv) n < 0, n even; and (v) n < 0, n odd. In each case, find the following limits.

(a) limx→ 0 +^ xn (b) limx→ 0 − xn (c) limx→∞ xn (d) limx→−∞ xn

2. Let P and Q be polynomials with leading coefficients a and b respectively. Find limx→∞ P Q^ ((xx)) if the degree of P is

(a) less than the degree of Q. (b) equal to the degree of Q. (c) greater than the degree of Q.