AP Calculus Testbank (Chater 1) (Mr. Surowski) Part I. Multile-Choice Questions 1. The grah in the xy-lane reresented by x = 3 sin t and y = cost is (A) a circle (B) an ellise (C) a hyerbola (D) a arabola (E) a line. If a article moves in the xy-lane so that at time t > its osition vector is (e t ;e t3 ), then its velocity vector at time t = 3 is (A) (ln 6; ln( 7)) (B) (ln 9; ln( 7)) (C) (e 9 ; e 7 ) (D) (6e 9 ; 7e 7 ) (E) (9e 9 ; 7e 7 ) 3. A article moves along a ath described by x = cos 3 t and y = sin 3 t. The distance that the article travels along the ath from t = to t = ß is (A).75 (B) 1.5 (C) (D) 3:5 (E) :75 4. The area enclosed by the olar equation r = 4 + cos, for»» ß, is (A) (B) 9ß (C) 18ß (D) 33ß (E) 33ß 4 5. If, for t > ; x = t and y = cos t, then dy dx = (A) cos t (B) sin t (C) sin t (D) sin t (E) cos t
6. Find the area inside one loo of the curve r = sin. (A) ß 16 (B) ß 8 (C) ß 4 (D) ß (E) ß 7. Find the length of the arc of the curve defined by x = 1 t and y = 1 9 (6t +9)3,fromt = to t =. (A) 8 (B) 1 (C) 1 (D) 14 (E) 16 8. If f is a vector-valued function defined by f (t) = (sin t; sin t), then f (t) = (A) ( 4 sin t; cost) (B) ( sin t; cos t) (C) (4 sin t; cos t) (D) (4 sin t; cos t) (E) ( cos t; 4sint) 9. A solid is formed by revolving the region bounded by the x-axis, the lines x = and x = 1 and the arametric curve x = sin t; y = 1+cos (t) about the x-axis. Write down the integral which will comute the volume of the solid. (A) ß (B) ß (C) ß Z ß Z ß Z ß (1 + cos 4 (t)) (1+cos (t)) (sin t +(1+cos (t)) ) (D) ß (E) ß Z ß Z ß (1 + cos (t)) cos t(1 + cos (t))
1. The length of the curve determined by x = 3t and y = t from t = to t = 9 is (A) (B) (C) (D) (E) Z 9 Z 16 Z 16 Z 3 Z 9 9t +4t 4 9 16t 9+16t 9 16t 9+16t. 11. The length of the curve determined by x = t 3 and y = t 3 from t = to t = 1 is (A) 5 7 (B) 5 (C) 3 (D) 5 (E) 3 1. The area of the region inside the olar curve r = 4 sin but outside the olar curve r = is given by (A) (B) 1 (C) 1 (D) 1 (E) 1 Z 3ß=4 ß=4 Z 3ß=4 ß=4 Z 3ß=4 ß=4 Z 3ß=4 ß=4 Z 3ß=4 ß=4 (4 sin 1) d (4 sin (4 sin ) d ) d (16 sin 8) d (4 sin 1) d.
13. A article moves on a lane curve so that at any time t > its osition is defined by the arametric equations x(t) = 3t 7 and y(t) = 4t +1 3t (A) 6; 1 1. The acceleration vector of the article at t = is (B) (C) (E) 17; 17 6 1; 47 1 1; 33 1 14. The acceleration of a article is described by the arametric equation x (t) = t 4 + t and y (t) = 1. If the velocity vector of the 3t article when t = is (4; ln ), what is the velocity vector of the article when t = 1? 5 (A) 4 ; 1 3 3 (B) 1 ; ln 4 3 3 (C) 1 ; ln 3 5 (D) 4 ; 3 ln (E) 3 1 ; 1 3 ln 6; 17 6
Part II. Free-Resonse Questions 1. A moving article has osition (x(t); y(t)) at time t. The osition of the article at time t = 1 is (; 6), and the velocity vector at any 1t time t > is given by ; +. 1t 1 (a) Find the acceleration vector at time t = 3. (b) Find the osition of the article at time t = 3. (c) For what time t > does the line tangent to the ath of the article at (x(t);y(t)) have a sloe of 8? (d) The article aroaches a line as t! 1. Find the sloe of this line. Show that work that leads to your conclusion.. The figure above shows the grahs of the line x = 5 y and the 3 curve C is given by x = 1+y. Let S be the region bounded by the two grahs and the x-axis. The line and the curve intersect at oint P. (a) Curve C is a art of the curve x y = 1. Show that x y = 1 can be written as the olar equation r 1 = cos. sin (b) Use the olar equation given above to set u an integral exression with resect to the olar angle that reresents the area of S.
3. The figure above shows the ath traveled by a roller coaster car over the time interval» t» 18 seconds. The osition of the car at time t can be modeled arametrically by x(t) = 1t + 4 sin t y(t) = ( t)(1 cos t) where x and y are measured in meters. The derivatives of these functions are given by x (t) = 1 + 4 cos t y (t) = ( t) sin t + cos t 1 (a) Find the sloe of the ath at time t =. Show the comutations that lead to your answer. (b) Find the acceleration vector of the car at the time when the car s horizontal osition is x = 14. (c) Find the time t at which the car is at its maximum height, and find the seed, in m/sec, of the car at this time. (d) For < t < 18, there are two times at which at the car is at ground level (y = ). Find these two times and write an exression that gives the average seed, in m/sec, of the car between these two times.
4. A article moves in the xy-lane so that its osition at any time t, for ß t ß, is given by x(t) =sin 3t and y(t) = t. (a) Sketch the ath of the article in the xy-lane rovided. Indicate the direction of motion along the ath. y 6 ff 1 1 - x (b) Find the range of x(t) and the range of y(t). (c) Find the smallest ositive value of t for which the x-coordinate of the article is a local maximum. What is the seed of the article at this time?? (d) Is the distance traveled by the article from t = ß to ß greater than 5ß. Justify your result. 5. A moving article has osition (x(t); y(t)) at time t. The osition of the article at time t = 1 is (7; ), and the velocity vector at any t time t > is given by ; 4+. 3t 3 (a) Find the osition of the article at t = 3. (b) Will the line tangent to the ath of the article at (x(t)); y(t)) ever a sloe of zero? If so, when? If not, why not?
6. A article starts at oint A on the ositive x-axis at time t = and travels along the curve from A to B to C to D, as shown above. The coordinates of the article s osition (x(t); y(t)) are differentiable functions of t, where x (t) = dx = 9 cos ßt 6 sin ß t +1 ; y (t) = dy is not exlicitly given. At time t = 9, the article reaches its final osition at oint D on the ositive x-axis. (a) At oint C, is dy a reason for each answer. ositive? At oint C, is dx ositive? Give (b) The sloe of the curve is undefined at oint B. At what time t is the article at oint B? (c) The line tangent to the curve at the oint (x(8);y(8)) has equation y = 5 x. Find the velocity vector and the seed of the 9 article at this oint. (d) How far aart are oints A and D, the initial and final ositions, resectively, of the article?
7. The figure above shows the grahs of the circles x + y = and (x 1) + y = 1: The grahs intersect at the oints (1; 1) and (1; 1). Let R be the region in the first quadrant bounded by the two circles. (a) Write the olar equations of the two circles. (b) Write down an integral in terms of the olar angle which comutes the area of R. 8. A article moves in the xy-lane so that the osition of the article at any time t is given by x(t) =e 3t + e 7t and y(t) =3e 3t e t : (a) Find the velocity vector of the article in terms of t, and find the seed of the article at time t =. (b) Find dy dx dy in terms of t, and find lim. t!1 dx (c) Find each value t at which the line tangent to the ath of the article is horizontal, or exlain why none exists. (d) Find each value t at which the line tangent to the ath of the article is vertical, or exlain why none exists.
9. An object moving along a curve in the xy-lane has osition (x(t); y(t)) at time t with dx = 3 + cos t. The derivative dy dx (1; 8). is not exlicitly given. At time t =, the object is at osition (a) Find the x-coordinate of the object at time t = 4. (b) At time t =, the value of dy is 7. White an equation for the line tangent to the curve at the oint (x(); y()) (c) Find the seed of the object at time t =. (d) For t 3, the line tangent to the curve at (x(t); y(t)) has a sloe of t +1. Find the acceleration vector of the object at time t = 4. 1. A article moving along a curve in the lane has osition (x(t); y(t)) at time t, where dx = t 4 +9 and dy = et +5e t for all real values of t. At time t =, the article is at the oint (4; 1). (a) Find the seed of the article and its acceleration vector at time t =. (b) Find an equation of the line tangent to the ath of the article at time t =. (c) Find the total distance traveled by the article over the time interval» t» 3. (d) Find the x-coordinate of the osition of the article at time t = 3.
11. The curve above is drawn in the xy-lane and is described by the equation in olar coordinates r = + sin( );»» ß; where r is measured in meters and is measured in radians. The derivative of r with resect to is given by dr d = 1+cos( ). (a) Find the area bounded by the curve and the x-axis. (b) Find the angle that corresonds to the oint on the curve with x-coordinate. (c) For ß 3 < < ß 3 ; dr is negative. What does this fact say d about r? What does this fact say about the curve? (d) Find the value of in the interval»» ß that corresonds to the oint on the curve in the first quadrant with greatest distance from the origin. Justify your answer.