AP Calculus BC 2008



AP Calculus BC 2011 Form B Name_________________________

Parametric, Arc Length, Logistic, Work

1. Given [pic] and [pic]. Write the Cartesian form by eliminating the parameter.

2. [pic]

Find the slope of the tangent line and concavity at (6, 306).

3. Sketch the graph below. Indicate direction of motion.

[pic] for -2≤t≤ 3. [pic]

4. Multiple choice. A curve C is defined by the parametric equations [pic] and [pic]. Which of the following is an equation of the line tangent to the graph of C at the point (2, 27)?

(A) [pic] (B) y = 27(x -2)+27 (C) x = 3 (D) [pic] (E) y = 37

5. Multiple choice: For 0 ≤t≤[pic], an object travels along an elliptical path given by the parametric equations x = 4cost and y = 3 sint . At the point where t = [pic], the object leaves the path and travels along the line tangent to the path at that point. What is the slope of the line on which the object travels?

[pic]

6. [pic]

Find the points at which the tangent to the curve is

(a)horizontal

(b) vertical

7. Multiple choice. The position of a particle moving in the xy-plane is given by the parametric equations

[pic] and [pic]

For what values of t is the particle at rest?

(A) 0 only (B) 2 only (C) 3 only (D) 0 and 3 only (E) 0, 2 and 3

8. Given the logistic differential equation [pic].

a. What is the carrying capacity?

b. When is the growth rate for the logistic curve the greatest?

c. When is the logistic curve increasing?

d. What are the equilibrium solutions?

e. What is the [pic]?

f. What is the [pic]?

9. The number of bears that can be supported in a given National Park is a solution of the differential equation [pic]. Fifteen bears were brought into the park when it opened.

a) Find the solution for the logistic differential equation.

b) How many days will it take for the bear population to reach 50?

10. Multiple choice. The length of a curve from x =1 to x = 4 is given by [pic]. If the curve contains the point (1, 7), which of the following could an equation for this curve?

(A) [pic] (B) [pic] (C) [pic] (D) [pic] (E) [pic]

11. Multiple choice. The number of moose in a national park is modeled by the function M that satisfies the logistic differential equation [pic], where

t is the time in years and M(0)=25. What is [pic]?

(A) 0 (B) 50 (C) 100 (D) 125 (E) 200

12. Multiple choice. A particle moves in the xy-plane so that its position at any time t is given by x(t) = [pic] and y(t) = sin(4t). What is the speed of the particle when t = 3?

(A) 6.884 (B) 47.393 (C) 3.062 (D) 9.016 (E) 2.909

13. Suppose that a bee follows the trajectory [pic] but lands on a wall at time t = 10. Show the calculus used to set up the following and use your calculator to evaluate.

a. At what time is the bee flying vertically?

b. How far has the bee flown from t = 0 to t = 10.

c. What was the bee’s speed at t =[pic]?

14. An object moving along a curve in the xy-plane is at position (x(t), y(t)) at time t with

[pic] and [pic] for t ≥ 0.

At time t = 0, the object is at position (−3,−4).

(a) Find the speed of the object at time t = 4. (Show what you set up on your calculator and solve).

(b) Find the total distance traveled by the object over the time interval 0≤t≤4. (Show what you set up on your calculator and solve).

(c) Find x(4). (Show what you set up on your calculator and solve).

(d) For t >0, there is a point on the curve where the line tangent to the curve has slope 2. At what time t is the object at this point? Show your work.

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