2.3: Using Derivatives to Describe Families of Functions
- Page ID
- 111980
Overview Video Section 2.3
Supplemental Videos
Working with Families of Functions (part 1):
Working with Families of Functions (part 2):
Homework Exercises 2.3
1. For some positive constant C, a patient's temperature change, T, due to a dose, D, of a drug is given by \(T=(\frac{C}{2}-\frac{D}{3})D^2\).
a. In order to find the dosage that maximizes the temperature change, first find the derivative of T. Keep in mind that D is the variable, and C is a constant.
b. Find the critical numbers of T .
c. Test the critical numbers on a first derivative sign chart. What value of D maximizes T?
d. The sensitivity of the body to the drug is defined as dT/dD. To find the dosage that maximizes sensitivity, find the 2nd derivative of T, determine the critical numbers, and test with the 2nd derivative sign chart. What value of D maximizes sensitivity?
2. The graph below is of the derivative of g(x).
a. Make a first derivative sign chart to determine where g is increasing and decreasing.
b. Sketch an approximate graph of the second derivative of g.
c. Make a second derivative sign chart to determine where g is concave up and concave down.
d. Where does the graph of g have inflection points?
e. Where are the global (absolute) maxima and minima of g on [-2, 2]?
f. Sketch an approximate graph of g.
3. Consider the one-parameter family of functions given by \(p(x)=x^3-ax^2\), where \(a>0\).
a. Sketch a plot of a typical member of the family, using the fact that each is a cubic polynomial with a repeated zero at x =0 and another zero at x = a. (You may use a graphing utility like DESMOS with a slider for a.)
b. Find all critical numbers of p.
c. Compute \(p''\) and find all values for which \(p''(x)=0\). Hence, construct a second derivative sign chart for p.
d. Describe how the location of the critical numbers and the inflection point of p change as a changes. That is, if the value of a is increased, what happens to the critical numbers and inflection point?