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More Applications of Derivatives Practice Problems for AP Calculus

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By — McGraw-Hill Professional
Updated on Oct 24, 2011

Review the following concepts if needed:

Part A—The use of a calculator is not allowed.

1. Find the linear approximation of f (x) = (1+ x)1/4 at x = 0 and use the equation to approximate f (0.1).
2. Find the approximate value of using linear approximation.
3. Find the approximate value of cos 46° using linear approximation.
4. Find the point on the graph of y = |x3| such that the tangent at the point is parallel to the line y – 12x = 3.
5. Write an equation of the normal to the graph of y = ex at x = ln 2.
6. If the line y – 2x = b is tangent to the graph y = – x2 + 4, find the value of b.
7. If the position function of a particle is s(t) = – 3t2 + 4, find the velocity and position of particle when its acceleration is 0.
8. The graph in Figure 9.6-1 represents the distance in feet covered by a moving particle in t seconds. Draw a sketch of the corresponding velocity function.
9. The position function of a moving particle is shown in Figure 9.6-2. For which value(s) of t(t1, t2, t3) is:
1. the particle moving to the left?
2. the acceleration negative?
3. the particle moving to the right and slowing down?
10. The velocity function of a particle is shown in Figure 9.6-3.
1. When does the particle reverse direction?
2. When is the acceleration 0?
3. When is the speed the greatest?
11. A ball is dropped from the top of a 640-foot building. The position function of the ball is s(t) = – 16t2 + 640, where t is measured in seconds and s(t) is in feet. Find:
1. The position of the ball after 4 seconds.
2. The instantaneous velocity of the ball at t = 4.
3. The average velocity for the first 4 seconds.
4. When the ball will hit the ground.
5. The speed of the ball when it hits the ground.
12. The graph of the position function of a moving particle is shown in Figure 9.6-4.
1. What is the particle's position at t =5?
2. When is the particle moving to the left?
3. When is the particle standing still?
4. When does the particle have the greatest speed?

Part B—Calculators are allowed.

13. The position function of a particle moving on a line is s(t) = t3 – 3t2 + 1, t ≥ 0 where t is measured in seconds and s in meters. Describe the motion of the particle.
14. Find the linear approximation of f (x) = sin x at x = π. Use the equation to find the approximate value of f.
15. Find the linear approximation of f (x) = ln (1+ x) at x = 2.
16. Find the coordinates of each point on the graph of y2 = 4 – 4x2 at which the tangent line is vertical. Write an equation of each vertical tangent.
17. Find the value(s) of x at which the graphs of y = ln x and y = x2 + 3 have parallel tangents.
18. The position functions of two moving particles are s1(t) = ln t and s2(t) = sin t and the domain of both functions is 1 ≤ t ≤ 8. Find the values of t such that the velocities of the two particles are the same.
19. The position function of a moving particle on a line is s(t) = sin(t) for 0 ≤ t ≤ 2π. Describe the motion of the particle.
20. A coin is dropped from the top of a tower and hits the ground 10.2 seconds later. The position function is given as s(t) = – 16t2v0t + s0, where s is measured in feet, t in seconds and v0 is the initial velocity and s0 is the initial position. Find the approximate height of the building to the nearest foot.
21. Find the equation of the tangent line to the curve defined by x = cos t – 1, y = sin t + t at the point where x = .
22. An object moves on a path defined by x = e2t + t and y = 1+et. Find the speed of the object and its acceleration vector with t = 2.
23. Find the slope of the tangent line to the curve r = 3 sin 4θ at θ = .
24. The position of an object is given by . Find the velocity and acceleration vectors, and determine when the magnitude of the acceleration is equal to 2.
25. Find the tangent vector to the path defined by r = at the point where t = 4.

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