Application of Derivatives

Topics in this chapter

• Derivative-Based Rate of Change
• Related Rates and Motion
• Increasing and Decreasing Functions
• Local and Absolute Extrema
• Tangents, Normals, and Geometry
• Differentials and Linear Approximation
• Derivative-Based Optimization

Cheat sheet

Practice questions

1. Question 1Q-AA174F

   For the function f(x) = x³ − 3x², on which intervals is f increasing?

   1. A.(−∞, 0) and (2, ∞)
   2. B.(0, 2)
   3. C.(−∞, 2)
   4. D.(0, ∞)
   See the answer and solution
2. Question 2Q-C610AB

   The radius of a circular balloon is increasing at the rate of 0.5 cm/s. At what rate is its area increasing when the radius is 4 cm?

   1. A.2π cm²/s
   2. B.4π cm²/s
   3. C.8π cm²/s
   4. D.16π cm²/s
   See the answer and solution
3. Question 3Q-E59343

   For f(x) = x³ − 6x² + 9x + 1, which statement about its local extrema is correct?

   1. A.There is a local maximum at x = 3 and a local minimum at x = 1.
   2. B.There is a local maximum of 5 at x = 1 and a local minimum of 1 at x = 3.
   3. C.There are no local extrema because f′(x) is never zero.
   4. D.There is a local maximum of 1 at x = 3 and a local minimum of 5 at x = 1.
   See the answer and solution
4. Question 4Q-C48718

   Two positive numbers have sum 20. Find the maximum possible value of their product.

   See the answer and solution
5. Question 5Q-F80671

   For the function f(x) = x³ − 12x + 5, the intervals on which f is increasing are:

   1. A.(-∞, -2) and (2, ∞)
   2. B.(-2, 2)
   3. C.(-∞, 2)
   4. D.(-2, ∞)
   See the answer and solution