5 Steps to a 5 AP Calculus BC 2019

Applications of Derivatives 155

8.2 Applied Maximum and Minimum Problems

Main Concepts:General Procedure for Solving Applied Maximum and Minimum

Problems, Distance Problem, Area and Volume Problem, Business

Problems

STRATEGY

General Procedure for Solving Applied Maximum

and Minimum Problems

Steps:

1. Read the problem carefully, and if appropriate, draw a diagram.


2. Determine what is given and what is to be found and represent these quantities by
   mathematical symbols.


3. Write an equation that is a function of the variable representing the quantity to be
   maximized or minimized.


4. If the equation involves other variables, reduce the equation to a single variable that
   represents the quantity to be maximized or minimized.


5. Determine the appropriate interval for the equation (i.e., the appropriate domain for
   the function) based on the information given in the problem.


6. Differentiate to obtain the first derivative and to find critical numbers.


7. Apply the First Derivative Test or the Second Derivative Test by finding the second
   derivative.


8. Check the function values at the end points of the interval.


9. Write the answer(s) to the problem and, if given, indicate the units of measure.

Distance Problem

Find the shortest distance between the pointA(19, 0) and the parabolay=x^2 − 2 x+1.

Solution:

Step 1: Draw a diagram. (See Figure 8.2-1.)

Figure 8.2-1

Step 2: LetP(x,y) be the point on the parabola and letZrepresent the distance between

pointsP(x,y) andA(19, 0).