Unit 5 MC and FR Practice

Transcription

1 Name: Date:. If y = e, then y = ( )e ( )e ( )e. If y = e /, then y = e/ e / e / e /. If y = e cos, then dy d = e cos sin e cos sin e cos e cos sin. curve is defined by y = e sin. Find dy d. e sin cos sin cos sin e cos sin page

2 5. function f is defined by f () = e e. Find f (). e + e e e e + e e + e 6. Differentiate with respect to : y = e 7 (5/) dy d = e7 (5/) dy d = e7 (5/) dy d = 5 e7 (5/) dy d = e6 (5/) 7. Find y given y = e sin. cos esin (cos )e sin (sin )e sin (sin )e sin 8. d d eln 5 = 5 (ln 5) e (ln 5) 5 (ln 5 + e) 5 9. If y = e ln(sin ), then dy d = ln(cos ) cos cos ln sin ln page

3 0. Given f () = e ln( +), find f (). e e + e +. If y = ln( ), then dy d = ( ) ( ) ( ). If y = ln dy, then + d = ( + ) + ( + ) ( ) +. If y = ln(e 5), then dy d = e 5 5 e 5 e e 5 e 5 page

4 . Find dy d given y = ln(5 )6. (5 ) (5 ) Find the derivative of f () = ln( + ). 9( + ) ( + ) ( + ) + ( + ) ( + ) 9( + )( + ) 6. Find dy d for y = ln e e + 7. If f () = ln(sin( 8)), then f () = cos( 8) sin( 8) cos( 8) ln(cos( 8)) cot( 8) page

5 8. Find the derivative of f () = ln ( + ) ( 7) + + ( 7) ( 7) + + ( 7) 9. Let y =. Find dy d. [ + (ln )] [9 + ] [ ln ] 0. Find y given e y + = y. y e y e y y e y e y e y ln. Find y given e y = y. 0 y e y e y e y e y ln page 5

6 . Let ye = y. Find y. ye e y e + y e ( y) e y e y. Suppose f () = and let h() be the inverse of f. Find h ( 8). 6. Suppose f () = + and let h() be the inverse of f. Find h (). 5. tan d = ln cos + sec ln cos + ln cos + 6. sec d = ln sec + tan + sec + csc + ln tan + page 6

7 7. csc d = ln csc + cot + csc + csc + ln cot + 8. Which of the following is the indefinite integral for cos () sin() d? cos () + cos () ( cos () sin () ) + 0 cos5 () + 0 cos () + 9. Evaluate: sin(ln ) d sin(ln ) + tan(ln ) + cos(ln ) + sin(ln ) + 0. (6 5)e 5+ d = e + + e 5+ + e ( 5 + )e 5+ + page 7

8 . If dy d = , then y = ln ln arctan( + 5 ) + 6 ln( + 5 ) +. ln 7 d = (ln 7) + ln ln ln 7 + d. Find the indefinite integral for (ln()). ln ln() + ln ln() + ln + ln ln() +. If dy d = ln, then y = ln + ln + e ln + ln(ln ) + page 8

9 5. Which of the following is the definite integral for 0 d? 0 + ln ln 0 ln Evaluate: ( 5 ) d 5 ln ln ln 5 + e ln If dy d = e7, then y = 7e e 7 + 7e 7 + e Evaluate: d ln + 5 e + ln + ln + page 9

10 9. d = e ln e + ln e + e + ln + e + 0. Evaluate: 5e e + d 5e ln(+) + 5 ln(e + 5) + 5 ln(e + ) + ln(e + 5) +. Evaluate: + cos d sin ln cos + + tan + sec csc + sec tan +. Given a curve is defined by the equation f () = ( ln ). Find a point of inflection. (e, e) (, 0) (e, e) (e, ). Let f () = ln. Over what interval is the function increasing or decreasing? increasing 0 < < e ; decreasing > e increasing > e; decreasing < < e increasing > e ; decreasing 0 < < e increasing > e ; decreasing 0 < < e page 0

11 . Elvis drinks a 500 ml milkshake at Ye Olde Fountain Shoppe. Sadly, and unknown to Elvis, there is a crack in the glass that is getting wider with time, and the milkshake leaks out onto the counter at the rate r(t) = ln (t + ) ml/sec, for t seconds. Elvis drinks his milkshake at the constant rate of 0 ml/sec and begins drinking at the same time it starts leaking out (the instant he gets it), t = 0. a) How fast is milkshake leaving the glass at time t = 5 seconds? b) How much milkshake has left the mug at time t = 5 seconds? c) Write an epression for (t), the volume of milkshake left in the glass at time t. d) Elvis usually drinks a milkshake in 50 seconds. When will he actually finish it? 5. particle moves along the -ais with velocity given by v(t) = t t for t > 0. a) In which direction, left or right, is the particle moving at t = 0.5? Why? b) Find the acceleration of the particle at time t = 0.5. Is the velocity increasing at t = 0.5? Why or why not? c) Given that (t) is the position of the particle at time t and that () =, find (). d) Find the total distance traveled by the particle from t = to t =. 6. Suppose that the function f has a continuous second derivative for all, and that f (0) =, f (0) = 5, and f (0) = 0. Let g be a function whose derivative is given by g () = e (5f () + f ()). a) Write an equation of the line tangent to the graph of f when = 0. b) Is there sufficient information to determine whether or not the graph of f has a point of inflection when = 0?. Eplain your answer. c) Given that g(0) = 6, write an equation of the line tangent to the graph of g at the point where = 0. d) Show that g () = e (0f () + f () + f ()). Does g have a local maimum at = 0? Justify your answer. page

12 Problem-ttic format version.. c 0 05 Educide Software Licensed for use by Sarah Sammit Terms of Use at /9/ D D D

13 Teacher s Key Page.... D D.79, ml, 500 (t + ) ln (t + ) t, Left: v(0.5) < 0, a = yes, 9.8, y = 5 +, no, y = 6, yes