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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 ...
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7 th^ Edition
Department of Mathematics, University of California at Berkeley
i Math 1A Worksheets, 7 th^ Edition
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:
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
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
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 ′.
iii. How can you change the graph of g to obtain the graphs of the last two functions?
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.
(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.
(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?
x^1000 − 703 x^506 + πx^17 + 480 − 372 x^75 + 39x^14 +
7 x^5 − 24
x^2 + 2x − 24 x − 4
x
Explain why the method you chose is the best of the three possibilities.
(π n
. (Remember that n is in radians, not degrees!)
(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?
(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?
(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?
(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.)
lim x→a f (g(x)) = f (b) = f (lim x→a g(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].
(a) Prove that cf (x) is also continuous at a. (b) Prove that f (x) · g(x) is also continuous at a.
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)] =
(a) limx→ 0 +^ xn (b) limx→ 0 − xn (c) limx→∞ xn (d) limx→−∞ xn
(a) less than the degree of Q. (b) equal to the degree of Q. (c) greater than the degree of Q.