Let F(x) = a x (t) dt for the specified function and interval [a, b].

Let F(x) = ∫_a^xƒ(t) dt for the specified function ƒ and interval [a, b]. Use a CAS to perform the following steps and answer the questions posed.

a. Plot the functions ƒ and F together over [a, b].

b. Solve the equation F′(x) = 0. What can you see to be true about the graphs of ƒ and F at points where F′(x) = 0? Is your observation borne out by Part 1 of the Fundamental Theorem coupled with information provided by the first derivative? Explain your answer.

c. Over what intervals (approximately) is the function F increasing and decreasing? What is true about ƒ over those intervals?

d. Calculate the derivative ƒ′ and plot it together with F. What can you see to be true about the graph of F at points where ƒ′(x) = 0? Is your observation borne out by Part 1 of the Fundamental Theorem?

Explain your answer.

Transcribed Image Text:

f(x) = x cos TX, COS [0, 2π]