AP Practice: Derivatives For Part I, pick the best answer from the ...

6) If f x = x sin !! (x), then f′ x = a)!!!! ! b) sin !! (x) − x sin !! (x) !c) + !!! ! sin!! (x) !d) − !!! ! sin!! (x) !e) !!! ! __________________________________________________________________ 7) If y = x ! , then !"!" = a) x ∙ x !!! b) x ! ln x c) x ! 1 + ln x d) x ln(x) e) 1 + ln x __________________________________________________________________ 8) If y − x ! y ! = 6, then !"= !"!!!a)!!!!! ! !b) !!!!!!!! ! !!!! !c)d)!! ! !!!!!!"e) !!!!!!!!!! ! !__________________________________________________________________ 9) If x ! + y ! = 6, then !! !!! ! = a) − ! ! ! b) – !! !! !c)d)! !! ! ! ! ! !e) !!!! ! __________________________________________________________________ 10) lim !→!!" ! !!" !!!!= a) ! !!b)c)d) ! !!" !!!" !e) ! ! − ! !

11) If f x = e !! , then f !! ! x = !a)b)c)!!! ! !! ! !! !!d) − !! !! e) ! !For Part II, pick the best answer from the choices. You are permitted to use a calculator. Use the following table to answer items 1 through 6. 1) If H x = F x ! , then H ! 3 = x F(x) F′(x) F′′(x) G(x) G′(x) G′′(x) a) 0 3 5 4 −3 2 7 −2 b) 10 5 8 6 10 −6 −4 11 c) 25 d) 40 e) 100 __________________________________________________________________ 2) If H x = ! !, then ! ! H! 3 = a) − !"!b) − ! !c) 0 d) ! !e) !"!__________________________________________________________________ 3) If H x = F(x) ∙ G(x), then H !! 3 = a) −31 b) −16 c) 6 d) 40 e) 43 __________________________________________________________________ 4) If H x = G F x , then H ! 3 = a) −16 b) −6 c) −4 d) 28 e) 43 __________________________________________________________________ 5) If H x = G F x , then H !! 3 = a) −33 b) 0 c) 6 d) 56 e) 188

6) If H x = ln F x , then H ! 3 = a) 0.2 b) 0.25 c) 0.333 d) 0.621 e) 0.8 __________________________________________________________________ 7) If f and f !! are both differentiable for all x, with f 3 = 5 and f ! 3 = 7, then which of the following must be a line tangent to the graph of f !! ? a) y = 5 + 7(x − 3) b) y = ! + ! (x − 3) ! !c) y = 3 + 7(x − 5) d) y = ! + ! (x − 5) ! !e) y = 3 + ! !(x − 5) __________________________________________________________________ 8) At the moment that a rectangle is 8 feet long and 3 feet wide, its length is increasing at 0.5 feet/minute and its width is decreasing at 1.5 feet/minute. The area is a) decreasing at 10.5 square feet/minute. b) Increasing at 13.5 square feet/minute. c) Increasing at 8.5 square feet/minute. d) Decreasing at 0.5 square feet/minute. e) Decreasing at 0.75 square feet/minute. __________________________________________________________________ 9) A particle is traveling on the curve x ! − xy + y ! = 7. At the moment when the particle is at the point (2,3), its x-­‐coordinate is increasing at the rate of 5 units/minute. At this moment, the y-­‐coordinate of the particle is a) decreasing at 1.25 units/minute. b) decreasing at 0.625 units/minute. c) increasing at 0.5 units/minute. d) increasing at 20 units/minute. e) decreasing at 0.25 units/minute.

For Part III, show all of your work and clearly label any functions, graphs, tables, or other objects that you use. You are not permitted to use a calculator. 1) Consider the parabola y = x ! . a) Show that the line through the point (3, −7) with slope −2 is tangent to the parabola. b) Find another line through (3, −7) that is tangent to the parabola. c) Is there a third line through (3, −7) that is tangent to the parabola? Justify your answer. __________________________________________________________________ 2) Consider the curve xy ! − x ! y = 6. a) Find !"!" . b) Find all points on the curve where the tangent line is horizontal. Explain your reasoning. c) Find all points on the curve where the tangent line is vertical. Explain your reasoning. __________________________________________________________________ 3) Sand is falling from a rectangular box container whose base measures 40 inches by 20 inches at a constant rate of 300 cubic inches per minute. (Include units in all your answers.) a) How is the depth of the sand in the box changing? b) The sand is forming a conical pile (V = ! ! r! h). At a particular moment, the pile is 23 inches high and the diameter of the base is 16 inches. The diameter of the base at this moment is increasing at 1.5 inches per minute. At this moment, how fast is the area of the circular base of the cone increasing? c) At this moment, how fast is the height of the pile increasing? __________________________________________________________________ 4) A person is running around an elliptical track. The equation of the track is 4x ! + 25y ! = 200. a) When the person is at the point (5,2), her x-­‐coordinate is increasing at 6 units per minute. Describe how her y-­‐coordinate is changing. b) Can she run in such a way that her x-­‐coordinate changes at a constant rate? Explain. c) The inside of the track is heavily wooded, and she cannot see through the woods. There is a bear standing outside the woods at the point (8,0). When she is at the point (5,2), can she see the bear? Explain.