AP Calculus.1 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The only way to guarantee the eistence of a it is to algebraically prove it. Describe the different ways you can investigate the eistence of a it.. Using words, eplain what is meant by the epression f q c q T.. How do you find the average speed of an object? 4. Suppose an object moves along the -ais with it s position function given by t 5t 7t, where t is measured in seconds. a) What is the average speed from t = to t = 4 seconds? b) How fast is the object moving at eactly t = 4 seconds? 5. An rover on another planet drops an object off a cliff. The object falls seconds after the object was dropped it lands 0 m below. y gt m in t sec, where g is a constant. Five a) Find the value of g. b) Find the average speed for the fall. c) With what speed did the rock hit the bottom? 6. Assume 7 and f b g. b a) f g b) f g b b c) 4g b d) f b g
7. When asked to evaluate the it of a function, what should be done first? 8. Evaluate the following its by using direct substitution. a) π sec 7 6 b) 4 4 c) 1 1 d) y y 5y6 y 6 f) e) 9. Eplain why you cannot use direct substitution to determine each of the following its. a) 1 b) 0 c) 4 16 0 10. If a it does not eist, there are possible reasons why. List all three possible reasons why a it may not eist. 11. Find each it, or eplain why the it does not eist. a) f ( ), if f ( ) = ln for 0 < ln for < 4 b) f, if f 1, 5, 1 c) 4 1 1 1 d) 4
1. Determine whether each statement about the graph below is True or False. a) f 1 b) f 1 does not eist c) f d) f 1 e) f 1 f) f 1 1 does not eist g) f f h) f 0 0 c eists at every c in ( 1, 1) i) f c eists at every c in (1, ) 1. Use the graph of f () to estimate the its and value of the function, or eplain why the it does not eist. a) f 1 e) f b) f 1 f) f c) f 1 g) f d) f (1) h) f () 14. For each of the following functions, (i) draw the graph, (ii) determine f what the value of f is or eplain why it doesn t eist. c c and c f, and (iii) eplain a) c =, 6, if f 4, if, if b) c = 1, f 1, if 1, if 1
15. Suppose 1, if 0 1 f, if 1. Draw a graph of f (), then answer the following questions. 1, if a) At what points c in the domain of f does f eist? c b) At what point(s) c does only the left-hand it eist? c) At what point(s) c does only the right-hand it eist? 16. A water balloon dropped from the roof of a small building falls y 4.9t m in t sec. Suppose you wanted to know the speed of the water balloon at eactly t = seconds. Originally, we used values of t really close to and found the average rate of change between them. Let s try something a little different a) Instead of using a numeric value close to, what would be the average speed of the balloon between t = and t = + h? (Simplify the epression as much as you can) b) To find the speed of the balloon at t =, it is tempting to simply plug in h = 0, however, this yields 0, which is 0 an indeterminate form. We CAN however, evaluate your simplified epression from part a using it as h 0. Evaluate this it. c) Now find and compare the speed of the balloon at t = like you did earlier in question #4.
AP Calculus.1 Worksheet Day All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. When evaluating its, what does it mean if direct substitution gives you 7 0?. When evaluating its, what does it mean if direct substitution gives you 0 0?. What are the methods (options) for dealing with the result 0 0? 4. Evaluate the following its algebraically. a) 6 b) 0 1 1 c) 0 11 d) 4 + 5 4 1 e) 1 1 f) 0 4 16 g) t t t t 4 h) 0 8
sin One of the its you should know is. This it ONLY works when the denominator matches the inside of the 0 sine function. If they do not match, you cannot change the inside of a sine function without a trig identity. Your goal will be to correctly show the algebra in order to use this it. 5. Evaluate each of the following its analytically. Be sure to show ALL steps in your evaluation. a) sin 0 5 b) sin 5 0 c) sin π 4 ( π 4) π 4 d) sin 4 0 sin 6. Evaluate each of the following by combining properties of its and your algebra skills. a) sin 0 b) tan 0 sin c) 0 d) sin 0 7. Consider 0 ( ) ( 0) f f = a) If you use direct substitution, what result do you get? b) Evaluate the it if 1 f. 8. If a 0, then a a a 4 4 =
9. Evaluate the following its analytically (all mied up): a) 0 4 4 b) 5 8 0 4 16 c) 1 4 d) 0 e) 1 f) sin 0 g) sin 7 0 h) 4 54 8 1. Evaluate h. h0 h : h is going to 0 not so treat this as if h is the variable your final answer will have a in it. 1. Suppose g, if 1. 1, if 1 a) g b) g c) g 1 1 1 d) g (1) =
AP Calculus. Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. Answer the following questions: a) How do you find horizontal asymptotes? b) Which of the parent functions have horizontal asymptotes? List the function(s) and asymptote(s) c) How do you find vertical asymptotes? d) Which of the parent functions have vertical asymptotes? List the function(s) and asymptote(s) e) When must you look for oblique (slanted) asymptotes? How do you find them?. For each of the following, find (i) f and (ii) f. Then (iii) identify all horizontal asymptotes, if any. a) f 5 b) f 4 1 1 c) f 5 4 e d) f e) f f) f sin. One of the functions in a c has a slanted (oblique) asymptote. Eplain why, and then find the asymptote.
4. For each of the following, (i) find the vertical asymptotes of the graph of f () and (ii) describe the behavior of the graph of f () to the left and right of each asymptote. 1 a) f b) f c) f 1 4 1 5 5. Find the it of g () as (i), (ii), (iii) 0, and (iv) 0 a) g 1 1 if 0 if 0 b) g 1 1 if 0 if 0 6. Sketch a function that satisfies the stated conditions. Include any asymptotes. f ( ) = f ( ) = 1 f + 5 5 ( ) = f ( ) = 1 ( ) = f ( ) f 0 + f ( ) = = 7. Sketch a function that satisfies the stated conditions. Include any asymptotes. f ( ) = 1 f ( ) = 4 f + 4 ( ) = f ( ) ( ) f = =
8. Eplain why there is no value L for which sin L. 9. Let f cos. a) Find the domain and range of f. b) Is f even, odd, or neither? Justify your response. c) Find f. Give a reason for your answer. k 10. If k is a positive integer, then =? Eplain your answer. e [Try letting k = what about k = 10? what about k = 1000? ] 11. Investigate using tables and graphs to determine the value of each it: + 1 and + 1 1. Evaluate each of the following its using all methods learned from this chapter. a) 5 1 1 + b) n 5n = n n n + 1 c) 5 = d) sec
e) e cos f) int 1 7 g) cos 1 1 1 h) 4n n + 10000 n = i) sin 0 4 j) 0 1 1 sin + sin k) l) 1 5
AP Calculus. Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. What is the definition of continuity (at a point)?. Sketch a possible graph for each function described. a) f (5) eists, but f does not eist. b) The f 5 5 eists, but f (5) does not eist.. Use the function g () defined and graphed below to answer the following questions. 1 if 1 1 if 10 g 1 if 0 1 if 1 if 1 a) Does g 1 eist? f) Is g continuous at = 1? b) Does g 1 eist? g) For what values of is g continuous? c) Does g g1 1? h) What value should be assigned to g ( 1) to make a new (etended) function continuous at = 1? d) Is g continuous at = 1? i) What value should be re-assigned to g (1) to make g continuous at = 1? e) Is g defined at = 1? j) Is it possible to etend g to be continuous at = 0? If so, what value should the etended function have there? If not, why not?
1 ; a ; Using the definition of continuity, justify your response. 4. Let f. Find a value of a so that the function f is continuous. ;. Find a value of k so that the function f is continuous. k 1 ; Using the definition of continuity, justify your response. 5. Let f a ;. Find all values of a that make f continuous at. 4 ; Using the definition of continuity, justify your response. 6. Let f + 5 + 7 7. If ( ) if f =, and if f is continuous at =, then k =? k+ if = Using the definition of continuity, justify your response. 8. If the function f is continuous for all real numbers and if f ( ) Use the definition of continuity to justify your response. 4 = +, when, then f ( ) =
9. Let f be the function defined by the following: sin, < 0, 0 < 1 f ( ) =, 1 <, For what values of is f NOT continuous? Use the definition of continuity to eplain why. 10. Write an etended function (see questions h and i) so that the given function is continuous at the indicated point. sin5 a) h at = 0 b) k 4 at = 4 11. Multiple Choice: Let f be the function given by f ( ) continuous for all real numbers? = ( 1)( 4) a. For what positive values of a is f A) None B) 1 only C) only D) 4 only E) 1 and 4
56 1. Let g. 7 10 a) Find the domain of g (). b) Find the g c for all values of c where g () is not defined. c) Find any horizontal asymptotes and justify your response. d) Find any vertical asymptotes and justify your response. e) Write an etension to the function so that g () is continuous at =. Use the definition of continuity to justify your response. 1. Without using a picture, give a written eplanation of why the function f 4 [, 4]. has a zero in the interval 14. Without using a picture, give a written eplanation of why the function f in the interval [0, ]. must equal at least once 4, if 15. Let h. 5 4, if a) What is h (0)? b) What is h (4)? c) On the interval [0, 4], there is no value of such that h () = 10 even though h (0) < 10 and h (4) > 10. Eplain why this result does not contradict the IVT.
AP Calculus.4 Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. What is a difference quotient?. How do you find the slope of a curve (aka slope of the tangent line to a curve) when = a?. What is a normal line? 4. What is the difference between the AVERAGE RATE OF CHANGE and INSTANTANEOUS RATE OF CHANGE? 5. Find the average rate of change of each function over the indicated interval. a) h( ) = e b) k( ) = + sin c) ( ) f = - on [, 0] on [ π/, π/] on [1, ] 6. Let f ( ) =. a) Write and simplify an epression for f (a + h). b) Find the slope of the curve at = a. c) When does the slope equal 1? d) Write the equation of the tangent line to the curve at = 4 e) Write the equation of the normal line to the curve at = 4
7. Let g( ) = a) Find the average rate of change from = 4 to = 9. b) Find the instantaneous rate of change at = 9. c) Write the equation of the tangent line when = 9 d) Write the equation of the normal line when = 9. 1 8. Let y =. Find the slope of the curve at =. Using this slope, write the equation of the tangent line and the - 1 equation of the normal line at that point. 9. Let y= --. Find the slope of the curve at = 0. Using this slope, write the equation of the tangent line and the equation of the normal line at that point.
10. Find an equation of the tangent line to the graph of f ( ) = at = 1. 11. An object is dropped from the top of a 150-m tower. It s height above the ground after t seconds is 150-4.9t m. How fast is the object falling seconds after it is dropped? 1. What is the rate of change of the area of a circle with respect to the radius when the radius is 4 in? 1. At what point is the tangent line to ( ) h = - 6 + 1 horizontal?