Chapter 4 AP Review Questions

You may use a calculator for the first two questions only.

1. The wind chill is the temperature, in degrees Fahrenheit (°F), a human feels based on the air temperature, in degree Fahrenheit, and the wind velocity v, in miles per hour (mph). If the air temperature is 32°F, then the wind chill is given by W(v) = 55.6 – 22.1v0.16 and is valid for 5 ≤ v ≤ 60.

a. Find W′(20). Using correct units, explain the meaning of W′(20) in terms of the wind chill.

b. Find the average rate of change of W over the interval 5 ≤ v ≤ 60. Find the value of v at which the instantaneous rate of change of W is equal to the average rate of change of W over the interval 5 ≤ v ≤ 60

c. Over the time interval 0 ≤ t ≤ 4 hours, the air temperature is a constant 32°F. At time t = 0, the wind velocity is v = 20 mph. If the wind velocity increases at a constant rate of 5 mph per hour, what is the rate of change of the wind chill with respect to time at t = 3 hours? Indicate units of measure.

2. The wave pool at DorneyPark is 15ft deep at the deep end, and zero feet deep at the shallow end. It is 35 ft wide and 75ft in length as shown in the diagram below. As the pool is being filled for the season water flows in at a rate of 2.5 cubic feet per second.

a) If the pool is initially empty, how long will it

take to fill the pool? Include units of measure.

b) When the water is 4 ft deep, how fast is the depth of

the water increasing? Include units of measure.

c)At what rate is the area of the surface of the water increasing when the

water is 4ft deep? Include units of measure.

3. The twice-differentiable function f is defined for all real numbers and satisfies the following conditions:

f(0) = 2, f′(0) = -4, and f″(0) = 3

  1. The function g is given by g(x) = eax + f(x) for all real numbers, where a is a constant. Find g ′(0) and g″(0) in terms of a. show the work that leads to your answers.
  2. The function h is given by h(x) = cos(kx)f(x) for all real numbers, where k is a constant. Find h′(x) and write an equation for the line tangent to the graph of h at x = 0.

4. Let f be a twice-differentiable function such that f(2)=5 and f(5) = 2. Let g be the function given by g(x) = f(f(x)).

  1. Explain why there must be a value c for 2 < c < 5 such that f ′(c)= -1.
  2. Show that g′(2) = g′(5). Use this result to explain why there must be a value k for 2 < k < 5 such that g″(x) = 0
  3. Show that if f ″(x) = 0 for all x, then the graph of g does not have a point of inflection.
  4. Let h(x) = f(x) – x. Explain why there must be a value r for 2 < r < 5 such that h(r) = 0.