Math 1A: Calculus Worksheets

1 Math 1A: Calculus Worksheets7thEditionDepartment of Mathematics, University of California at BerkeleyiMath 1A Worksheets ,7thEditionPrefaceThis booklet contains the Worksheets for Math 1A, Berkeley s Calculus Heitsch, David Kohel, and Julie Mitchell wrote Worksheets used for Math 1 AMand 1AW during the Fall 1996 semester. David Jones revised the material for the Fall 1997semesters of Math 1AM and 1AW. The material was further updated by Zeph Grunschlagand Tom Insel, with help from the comments and corrections provided by David Lippel, MaxOks, and Sarah Reznikoff. Tom Insel coordinated the 1998 edition with much assistance andnew material from Cathy Kessel and in consultation with William Stein. Cathy Kessel andMichael Wu have further revised the 1999 and 2000 edition respectively. Michael Hutchingsmade tiny changes in 1997, the engineering applications were written by ReeseJones, Bob Pratt, and Pro-fessors George Johnson and Alan Weinstein, with input from Tom Insel and Dave Jones.

2 In1998, applications authors were Michael Au, Aaron Hershman, Tom Insel, George Johnson,Cathy Kessel, Jason Lee, William Stein, and Alan the worksheetsThis booklet contains the Worksheets that you will be using in the discussion sectionof your course. Each worksheet contains Questions, and mostalso have Problems and Ad-ditional Problems. The Questions emphasize qualitative issues and answers for them mayvary. The Problems tend to be computationally intensive. The Additional Problems aresometimes more challenging and concern technical details or topics related to the Questionsand Worksheets contain more problems than can be done during one discussion not despair! You are not intended to do every problem of every Worksheets ?There are several reasons to use Worksheets : Communicating to learn from the explanations and questions of the studentsin your class as well as from lectures. Explaining to others enhances your understandingand allows you to correct misunderstandings.

3 Learning to in fields such as engineering and experimentalscience is often done in groups. Research results are often described in talks andlectures. Being able to communicate about science is an important skill in manycareers. Learning to work in wants graduates who can communicateandworkwith 1A Worksheets ,7thEditioniiContents1. Graphing a Journey..12. Graphical Problems..33. Tangent Lines and Preliminaries .. 54. Calculating Limits of Functions .. 85. The Precise Definition of a Limit .. 106. Continuity .. 137. Limits at Infinity and Horizontal Asymptotes .. 158. Derivatives ..179. Differentiation .. 2010. The Chain Rule ..2211. Implicit Differentiation and Higher Derivatives .. 2412. Using Differentiation to do Approximations .. 2713. Exponential Functions .. 2914. Inverse Functions .. 3015. Logarithmic Functions and their Derivatives ..3216. Inverse Trigonometric Functions ..3417. Hyperbolic Functions.

4 3618. Indeterminate Forms and l Hospital s Rule .. 3819. Falling Objects and Limits Involving Logarithms and Exponentials .. 4020. Maximum and Minimum Values ..4321. The Mean Value Theorem ..4622. Monotonicity and Concavity .. 4823. Applied Optimization..5024. Antiderivatives ..5225. Sigma Notation and Mathematical Induction .. 54iiiMath 1A Worksheets ,7thEdition26. Area .. 5727. The Definite Integral ..6028. The Fundamental Theorem of Calculus ..6329. The Substitution Rule .. 6630. The Logarithm Defined as an Integral .. 6831. Areas Between Curves .. 6932. Volume .. 7233. Volumes by Cylindrical Shells .. 7534. Integration and Optimization .. 771 Math 1A Worksheets , a JourneyQuestions1. Before you came to UC Berkeley you probably lived somewhereelse (another country,state, part of California, or part of Berkeley). Sketch a graph that shows thespeedofyour journey to UC Berkeley as a function of time. (For example, if you came by carthis graph would show speedometer reading as a function of time.)

5 Label the axes toshow someone outside of your group to read your graph. See if that person can tell fromyour graph what form (or forms) of transportation you Using the same labeling on thex-axis, sketch the graph of thedistanceyou traveledon 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 odometerto zero at the beginning of your trip.)Ask someone else outside of your group to read your graph. See if that person can tellfrom your graph what form (or forms) of transportation you 1A Worksheets ,7thEdition2xypABCDL3. (a) In the graph above,Ahas coordinates (2,3) andBhas coordinates (4,8). Calcu-late the slope of the lineLthroughAandBand the value ofp.(b) The pointD(connected toB) moves towardC. What happens to the slope ofLand to the the value ofp?4. Graph(a)y=1x.(b)y= sinx. What are thex-intercepts of this graph?

6 ( , where does the graphcross thex-axis? A related question is: What are the zeros of the functiony= sinx?)(c)y= sin1x. What are thex-intercepts of this graph? What is the domain of sin1x?5. True or false? Between every two distinct rational numbers there is a rational your True or false? Between every two distinct rational numbers there is an irrationalnumber. Explain your True or false? For all real numbersaandb,|a+b| |a|+|b|.8. True or false? For all functionsfandg,|f(x) +g(x)| |f(x)|+|g(x)|for everyxinthe domains 1A Worksheets , ProblemsQuestions1. Is there a function all of whose values are equal to each other? If so, graph your not, explain (a) Find allxsuch thatf(x) 2 wheref(x) = x2+ 1f(x) = (x 1)2f(x) =x3 Write your answers in interval notation and draw them on the graphs of thefunctions.(b) Using the functions in part a, find allxsuch that|f(x)| 2. Write your answersin interval notation and draw them on the graphs of the functions.

7 (c) Can you findupper boundsfor the functions in part a? That is, for each functionfis there a numberMsuch that for allx,f(x) M?(d) What aboutlower boundsfor the functions in part a? That is, for eachfcan youfind a numbermsuch that for allx,f(x) m?(e) What about finding upper and lower bounds for these functions restricted to theinterval [ 1,1]? That is, for eachfcan you find numbersMandmsuch that forallxin [ 1,1],m f(x) M?(f) True or false? IfMis an upper bound for the functionfandM is an upperbound for the functiong, then for allxwhich are in the domains of bothfandg,|f(x) +g(x)| M+M .2. (a) Graph the functions below. Find their maximum and minimum values, if theyexist. You don t need Calculus to do this!y= x2+ 1y=x2 1y= (x 1)2y= sinx 1y= sin(x 1)(b) Supposef(x) =x2andg(x) = Write the functions in part a in terms offandg. (For example, ifh(x) = 2x2you can writehin terms offash(x) = 2f(x).) If you find more than one wayof writing these functions in terms offandg, show that they are How can you change the graph offto obtain the graphs of the first threefunctions?

8 Use your work from part a to help 1A Worksheets ,7thEdition4iii. How can you change the graph ofgto obtain the graphs of the last twofunctions?5 Math 1A Worksheets , Lines and PreliminariesQuestions1. Can the graph of a function have more than one tangent at a given point? If so, graphyour answer. If not, explain Is there a function whose graph doesn t have a tangent at some point? If so, graphyour answer. If not, explain Suppose is a positive real number ( is the lowercase Greek letter delta). How doyou describe all real numbersxthat are within of 0 as pictured on the line below? 0(a) Using inequalities (<, >).(b) Using absolute value notation.(c) Using interval This problem is like problem 1 except that we are usingainstead of a real number and is a positive real number. How do you describe allreal numbersxthat are within ofa?(a) Graphically, as on the line above.(b) Using are at least two ways to do this.

9 Find both.(c) Using absolute value notation.(d) Using interval 1A Worksheets ,7thEdition63. This problem is like problem 2 except that we are asking thatxnot be equal a real number and is a positive real number. How do you describe allreal numbersxthat are within ofa, but not equal toa:(a) Graphically.(b) Using inequalities.(c) Using absolute value notation.(d) Using interval Letf(x) bex2.(a) Find all the positive numbersxsuch thatf(x) is within 1 of 9. ( Within meansthe same thing it did in Problems 1 and 2, but here it refers to numbers on they-axis.) Give your answer:i. On a graph ofy= Using Using interval notation.(b) It s difficult to expressallnumbersxsuch thatf(x) is within 1 of 9 using absolutevalue notation. (Why?) Instead:i. Find a real number such that wheneverxis within of 3,f(x) is within 1of this number using the min notation ( min is for minimum ). Ifaandbare two numbers, then min{a, b}is the smaller ofaandb.

10 For example,min{5,4}= 4. Ifaandbare equal, then min{a, b}is justa(orb). Forexample, min{ 4,2}= Using absolute value notation and the value of that you have found, writean expression forxsuch thatxis within of 3.(c) i. Find a real number such that wheneverxis within of 3,f(x) is within1/2 of 9. Write this number using the min Using absolute value notation and the value of that you have found, writean expression forxsuch thatxis within of 3.(d) Is it true that for any positive number , there is a positive so thatf(x) iswithin of 9 wheneverxis within of 3? ( is the lowercase Greek letter epsilonand stands for error. )If your answer is yes, show how you can write an expression in terms of for a that works. Explain why your your answer is no, show that there is a positive number for which the statementabove is not 1A Worksheets ,7thEditionAdditional Problems1. Graphy= 2x x2. For which pointsais the tangent line to the curve at the point(a,2a a2) a horizontal line?