Practice problems for these solutions can be found at: Definite Integrals Practice Problems for AP Calculus
Part A The use of a calculator is not allowed.
Note that f (x )= cos x – x2 is an even function. Thus you could have written and obtained the same result.
Part B Calculators are allowed.
Thus at each of the points at (1, 1) and (1, –5) the graph has a horizontal tangent.
Enter [Solve] (9x^{ 2} – 8x – 27=0, x ) and obtain x =3 or x = – 1.
Thus at each of the points (3, –2) and (–1, –2), the graph has a vertical tangent. See Figure 11.93.
Step 1. See Figure 11.94. Let P =x + y where P is the length of the pipe and x and y are as shown. The minimum value of P is the maximum length of the pipe to be able to turn in the corner. By similar triangles,
Step 2.Find the minimum value of P. Enter Use the [Minimum] function of the calculator and obtain the minimum point (9.306, 22.388).
Step 3.Verify with the First Derivative Test. Enter y 2=(y^{1}(x ), x ) and observe. (See Figure 11.95.)
Step 4.Check endpoints. The domain of x is (6,∞). Since x =9.306 is the only relative extremum, it is The absolute minimum. Thus the maximum length of the pipe is 22.388 feet.

The function f has a relative minimum at x = – 5 and x =3, and f has a relative maximum at x = – 1 and x =7.
The function f is concave upward on intervals (–6, –3) and (1, 5).
 See Figure 11.91.
 See Figure 11.92.
 A change of concavity occurs at x = – 3, x =1, and x =5.
 (Calculator)
Ask a Question
Have questions about this article or topic? AskPopular Articles
 Kindergarten Sight Words List
 First Grade Sight Words List
 10 Fun Activities for Children with Autism
 Signs Your Child Might Have Asperger's Syndrome
 Theories of Learning
 A Teacher's Guide to Differentiating Instruction
 Child Development Theories
 Social Cognitive Theory
 Curriculum Definition
 Why is Play Important? Social and Emotional Development, Physical Development, Creative Development