Calculus I Sample Exam #01

Transcription

1 Calculus I Sample Exam #01 1. Sketch the graph of the function and define the domain and range. 1 a) f( x) 3 b) g( x) x 1 x c) hx ( ) x x 1 5x6 d) jx ( ) x x x 3 6

2 . Evaluate the following. a) 5 sin 6 b) 7 tan 6 c) 4 sec 3 d) cos 1 3 e) sin 1 f) 4ln3 e 1 g) ln e eln e h) log Solve for x. a) x 1 ln 7 8 ln10x b) e x 3

3 4. Sketch the graph of csc and cot on the interval, separate set of axis. Clearly label the location of any vertical asymptotes.. Draw each graph on a 5. Express the function f x x 3 4 x 1 in piecewise form without the absolute values Define the Domain and Range of f x given f x x ln 3.

4 7. For the function f (x) graphed below, answer the following: a) lim f( x) b) x0 lim f( x) c) lim f( x) d) x0 x lim f( x) x e) f (0) f) ( 4) f g) lim f( x) x 4 h) lim f( x) x6 For parts i) through m) answer True or False. i) f (x) is continuous from the right at x 6. j) f (x) is not continuous at x 6. k) f (x) is continuous from the left at 0. l) f (x) is not continuous from the right at x.

5 8. Sketch a possible graph for a function f (x) with the specified properties: f 1 f 3 1; f x and f x lim lim 0 ; x3 x3 lim f x and lim f x ; lim f x and lim f x x1 x1 4 x x Find a value for the constant k that will make the function continuous at x for part a) and at x 3 for part b). x 3 x a) f( x) kx x b) x x 3 gx ( ) 4k x3

6 10. Find the following limits. Rationalize the denominator if necessary. a) lim x 4 3x x x 8 b) lim x 6 5x 3 x 3 x c) lim x x x 4 d) lim x 7x x 4x x

7 11. Find the following limits: x a) lim sin x 3 x b) lim t 0 1 t cos t c) lim 0 tan 7 d) tan 3x x lim x0 sin 6

8

9

10

11

12

13

14

15 Math3A Practice Exam Fall Sketch the graph of the function and define the domain and range. a) f ( x) 4 3x = ( x+ ) ( x ) b) h( x) = x ( x+ 3) ( x 6) 5 3

16 . Express the function f ( x) = 3 x+ + x 1 in piecewise form without the absolute values. 3. Evaluate the following. a) 5π sin 6 b) 1 3 cos

17 4. A robot moves in the positive direction along a straight line so after t minutes its distance s t is ( ) = t feet from the origin. Find the average velocity of the robot over the time interval [, 4 ]. Then find a formula for the instantaneous velocity at t o using the definition v inst ( + ) ( ) s t h s t = lim o o. h 0 h

18 5. For the function f (x) graphed below, answer the following: a) lim f( x) = b) x 0 lim f( x) = c) lim f( x) = d) x 0 x + lim f( x) = x 4 e) f ( 4) = f) f (0) = g) lim f( x) x = h) lim f( x) = x 6 For parts i) through l) answer True or False. i) f (x) is continuous from the left at x = 4. j) f (x) has a removable discontinuity at x = 6. k) f (x) is not continuous from the left at 0. l) f (x) is continuous from the right at x = 4.

19 6. Sketch a possible graph for a function f (x) with the specified properties: f ( 4) = f ( ) = 1; lim f ( x) 0 and lim f ( x) = = ; + x 3 x 3 lim f ( x) and lim f ( x) = + ; f ( x) and f ( x) = + x 1 x 1 lim =+ lim = x + x Find a value for the constant k that will make the functions continuous at x = 0. a) sin x x 0 f( x) = 3x + = b) 1 cos x + x 0 gx ( ) = 5x + = x k x k x x 0

20 8. Find the equation of the tangent line on f ( x) m tan = lim x xo ( ) ( ) f x f x x x o o to find the slope of the tangent line. = 1 at x o =. Use the definition x 9. The figures below show the position versus time curves of four different particles moving on a straight line. For each graph determine whether its instantaneous velocity is increasing or decreasing with time. s( t ) s( t ) s( t ) s( t ) t t t t

21 10. Find the following limits: a) lim x 5 x 5 x 5 b) 1 1 lim x+ h x h 0 h c) cot x 1 lim sin x cos x π x 4 d) ( ) + x h x lim h 0 h

22 11. Use the following definition m tan ( + ) ( ) tangent line on f ( x) = sin x at an arbitrary x value. f x h f x = lim to find the slope of a h 0 h Hint: Remember that lim sin h ( ) = 1 and 1 cos h ( ) h 0 h lim = 0. h 0 h

23

24

25

26

27

28

29

30

31

32 Math3A Exam #01 Spring 016 Solutions 1. Sketch the graph of the function and define the domain and range. Give the equations of any vertical or horizontal asymptotes if they exist. a) f x 5 6x 4 5 x x 3 h x x 4 b)

33 . Express the function f x x 4 3 x in piecewise form without the absolute values. 3. Given f x ax bx c answer the following questions. a) Given a 1, c 1, and 0 b, define the domain and range of the function. b) Given a 1, c 1, and 0 b, how many y-intercepts does the function have? c) Given a 1, c 1, and 0 b, how many x-intercepts does the function have?

34 4. A robot moves in the positive direction along a straight line so after t minutes its distance is 3t feet from the origin. Find the average velocity of the robot over the time interval s t Then find a formula for the instantaneous velocity at t o using the definition v inst s t h s t lim o o. h0 h 1,5. 5. Circle the limits below that represent the slope of the tangent line on the function g x at x b. g a g b lim a b a b g a g b lim a b b a g x g b lim x b x b g a g b lim b a a b g b g x lim b x b x

35 6. For the function f (x) graphed below, answer the following: a) lim f( x) b) x0 lim f( x) c) lim f( x) d) x0 x lim f( x) x3 e) f ( 3) f) f (0) g) lim f( x) x 4 h) lim f( x) x6 For parts i) through l) answer True or False. i) f (x) is continuous at x 1. j) f (x) has a removable discontinuity at x 6. k) f (x) is not continuous at 0. l) f (x) has a removable discontinuity at x 4.

36 7. Find a value for the constant k that will make the functions continuous at x 0. a) f( x) kx x sec x x b) sin x x 0 gx ( ) 3x 4k x 0 8. Answer True or False. a) If 1 f x f lim 3 x1 x 1 then 1 1 f h f lim 3. h0 h b) If f a f b lim ba ab then lim ba ab f a f b 1

37 9. Find the following limits: a) lim x 4 3x x x 8 b) lim x 3 x x x x 3x c) lim x x x 4 d) lim x cx x dx x

38 10. Use the following definition m line on f x tan x at an arbitrary x value. Hint: Note that tan h tan lim 1. h0 h f x h f x lim to find the slope of a tangent h0 h

39

40

41

42

43

44

45

46 Math3A Solution Exam #01 Fall Sketch the graph of the function and define the domain and range. a) f( x) x 4 x x b) hx ( ) x x6 3 x

47 . Given f x x 3, answer the following: x 1 a) lim f( x) x1 b) lim f( x) x c) What is the domain and range of f( x )? 3. Decide which of the polynomial functions in the list might have the given graph. More than one answer may be possible a) b) c) d) 4x y 4 5 y x x x x4 x5 4 3x y x x x y x x a) b) c) d) y y y y x x3 x6 x x3 x6 x x3 x6 x3 x6 5 3 x

48 4. Express the function f x x 3 x absolute values. 4 in piecewise form without the 5. Sketch a graph of the function. 1, x 0 x f x x x6, x 0 x 9

49 6. For the function f (x) graphed below, answer the following: a) lim f( x) b) x4 lim f( x) c) lim f( x) d) x4 x lim f( x) x3 e) f ( 6) f) f (0) g) lim f( x) x3 h) lim f( x) x6 For parts i) through m) answer True or False. i) The lim f( x) 0. x0 j) f (x) has a removable discontinuity at x. k) f (x) is not continuous at x. l) The lim f( x) 1. x

50 7. Sketch a possible graph for a function f (x) with the specified properties with f (x) having a domain of all real numbers. f 4 f 1; f x and f x lim lim 0 ; x3 x3 lim f x and lim f x ; f x and f x x 1 x1 4 lim lim x x Find a value for the constant k that will make the function continuous at x for part a) and at x 0 for part b). a) f( x) kx x sec x x b) sin 3x x 0 gx ( ) 4x 5k x 0

51 9. Answer the following questions. f ( x) f () a) Suppose the lim 4 x x must the ( ) ( ) h h 0 lim f h f. If f x is an even function, then what be equal to? b) Suppose f x is an odd function. If the slope of the tangent line on f x at x a is 4, then what must the f a h f a lim h 0 ( ) ( ) h be equal to? 10. Given a) t 0 t t 0 g t t 0 t, answer the following: t t lim gt ( ) b) lim gt ( ) c) t 1 lim gt ( ) t

52 11. Find the following limits. Rationalize the denominator if necessary. a) lim x x x x x x c) bx x ax x x lim b) 1 0 lim sin cos t t t

53

54

55

56

57

58

59