Trajectories in Animation, Mathematics, and Physics

Week 9 Calculus Reading Assignment
Stewart 4.2, 4.3, 4.6

Stewart 4.2: Maximum and Minimum Values

Stewart 4.3: Derivatives and the Shapes of Curves

Stewart 4.6: Optimization Problems

notes:

We conclude our study of calculus for the quarter with more applications of derivatives to determine the maximum and minimum values of functions, the shapes of graphs of functions, and optimization problems. There is some new (or updated) terminology, but not very many new concepts. Reviewing section 2.8 when studying section 4.3 is recommended. As with our study of related rates, our main goal is to use the tools of calculus to solve interesting pure and applied mathematics problems.

key terms/ideas:

• absolute minimum, absolute maximum, extreme values (extrema)
• local minimum, local maximum
• Extreme Value Theorem
• Fermat’s Theorem
• critical number
• closed interval method to find extrema
• Mean Value Theorem
• increasing functions, decreasing functions, increasing/decreasing test
• first derivative test
• concave upward, concave downward, concavity test
• second derivative test
• optimization
• problem solving strategies (p. 299)

key questions:

1. What are extreme values (extrema) of a function? What is the absolute maximum of a function? The absolute minimum? Local maximum? Local minimum? What is the distinction between absolute and local maximum and minimum?
2. What is the Extreme Value Theorem?
3. What are the critical numbers of a function?
4. How do you find extreme values of a function?
5. How do you determine whether a function is increasing or decreasing on an interval? How do you determine the intervals where a function is increasing or decreasing?
6. How do you determine whether a function is concave up or concave down on an interval? How do you determine the intervals where a function is concave up or down?
7. How do you determine points of inflection of a function?
8. What is optimization? How does it connect to finding extreme values of a function?
9. Looking at the various examples in section 4.6, where is the calculus? What else do you need to use besides calculus to solve these problems?
10. How does the problem solving strategy outlined on p. 299 compare to the problem-solving strategies you’ve used in calculus, physics, animation, and other subjects you’ve studied? What is similar to the creative process used in art? in science?