Calculus Answers

Let 𝛼 be increasing and 𝑓 ∈ 𝑅(𝛼) on [𝑎, 𝑏]. What condition can we impose on 𝑓

so that the given equality holds?

𝑏𝑏

|∫ 𝑓(𝑥) 𝑑𝛼(𝑥)| = ∫ |𝑓(𝑥)| 𝑑𝛼(𝑥) 𝑎𝑎

Assume that 𝛼 is increasing on [𝑎, 𝑏]. Note that if 𝑓 ∈ 𝑅(𝛼) on [𝑎, 𝑏], then 𝑓2 ∈ 𝑅(𝛼) on [𝑎, 𝑏]. Use this statement to prove that if 𝑓, 𝑔 ∈ 𝑅(𝛼) on [𝑎, 𝑏], then 𝑓𝑔 ∈ 𝑅(𝛼) on[𝑎,𝑏].

1. Consider the graph of the function y = sin x + cos x. Describe its overall shape.

• Is it periodic?
• How do you know?

2. Using a graphing calculator or other graphing device, estimate the x- and y-values of the maximum point for the graph (the first such point where x > 0). It may be helpful to express the x-value as a multiple of π.

3. Now consider other graphs of the form y = A sin x + B cos x for various values of A and B.

• Sketch the graph when A = 2 and B = 1, and, find the x - and y-values for the maximum point. (Remember to express the x-value as a multiple of π, if possible.)
• Has it moved?

4. Repeat and sketch the graph for A = 1, B = 2.

• Is there any relationship to what you found in part (2)?

5. Explain what you have discovered from completing this activity using details and examples.