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# Rates of Change Practice Questions

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Updated on Oct 1, 2011

To review these concepts, go to Rates of Change Study Guide.

## Rates of Change Practice Questions

For each of the following four graphs, give the rate that a slope represents.

1.

2.

3.

4.

1. The height of a tree after t years is feet when t ≥ 1. How fast is the tree growing after 5 years?
2. The level of a river t days after a heavy rainstorm is L(t) = – t2 + 8t + 26 feet. How fast is the river's level changing after 7 days?
3. When a company makes and sells x cars, its . How fast is its profit changing when the company makes 50 cars? Should the company make more cars?
4. When a container is made x inches wide, it costs dollars to make. How is the cost changing when x = 3 inches? Would it be cheaper to increase or decrease the width?
5. An electron in a particle accelerator is s(t) = t3 + 2t2 + 10t meters from the start after t seconds. Where is it after 3 seconds? How fast is it moving then? How fast is it accelerating then?
6. A brick is dropped from 64 feet off the ground. What is its position function? What is its velocity function? What is its acceleration? When will it hit the ground? How fast will it be traveling then?
7. A bullet is fired upward at 800 feet per second from the ground. How high is it when it stops rising and starts to fall?
8. A rock is thrown 10 feet per second down a 1,000-foot cliff. How far has it gone down in the first 4 seconds? How fast is it traveling then?

Differentiate the following practice problems.

1. y = 4x5 + 10cos(x) + 3
2. g(x) = 8x + 3 – cos(x)
3. h(x) = cos(x) + cos(5)
4. Find the equation of the tangent line to

Differentiate the following.

1. f(x) = 1 + x + x2 + x3 + ex
2. g(t) = 12ln(t) + t2 – + 4
3. y = cos(x) – 10ex + 8x
4. h(x) = √x – 8ln(x)
5. k(u) = 3x + 5ex + 11
6. Find the second derivative of f(x) = ex + ln(x).
7. Find the 100th derivative of g(x) = 3ex.
8. What is the slope of the tangent line to f(x) = ln(x) at x = 10?

### Solution

1. pay rate in dollars per hour
2. fuel economy in miles per gallon
3. baby's growth rate in pounds per month
4. sunflower's growth rate in inches per week
5. increasing by 1 foot per year
6. decreasing by 6 feet per day
7. The profit is increasing by \$3,750 per car, so the company would increase its profits if it made more cars.
8. C'(3) = 4.8 – ≈ 2.13, so the cost would increase by about \$2.13 if the width were increased by an inch. This indicates that the cost to make the container would be cheaper if x were decreased from 3.
9. After 3 seconds, it is at s(3) = 75 meters from the start. At that moment, it is traveling at v(3) = 49 meters per second and accelerating at a(3) = 22 meters per second per second.
10. The position function is s(t) = –16t2 + 64, the velocity function is v(t) = –32t, and the acceleration is a constant a(t) = – 32. It will hit the ground when t = 2 seconds and be traveling at v( 2) = – 64 feet per second (downward) at that instant.
11. The position function is s(t) = –16t2 + 800t and the velocity function is v(t) = – 32t + 800. The bullet will stop in the air when the velocity is zero. This happens at t = 25 seconds, when the bullet is s( 25) = 10,000 feet in the air.
12. The position function is s(t) = –16t2 – 10t + 1,000, so after 4 seconds, it is s(4) = 704 feet off the ground. It has therefore fallen 296 feet by this moment. It is traveling v(4) = – 138 feet per second (downward) at this moment.
13.   = 20x4 – 10sin(x)
14.  f '(t) = 3cos(t) –
15. g'(x) = 8 + sin(x)
16. r'(θ) = cos(θ) – sin(θ)
17. h'(x) = –sin(x) because cos(5) is a constant
18. Because f () = sin () + cos () = 1 + 0 = 1, the point is (, 1). The slope is f ' () = – 1, so the equation is y = – (x) + 1 = –x + + 1.
19. f '(x) = 1 + 2x + 3x2 + ex
20. g'(t) = + 2t
21.   = –sin(x) – 10ex + 8
22. h'(x) =
23. k'(u) = + 5 ex
24. f'(x) = ex + so f "(x) = ex
25. g(100)(x) = 3 ex
26. f'(10) =

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