   Chapter 7.P, Problem 10P

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# Suppose that f is a positive function such that f′ is continuous.(a) How is the graph of y = f ( x ) sin n x related to the graph of y = f ( x ) ? What happens as n → ∞ ?(b) Make a guess as to the value of the limit lim n → ∞ ∫ 0 1 f ( x ) sin n x   d x based on graphs of the integrand.(c) Using integration by parts, confirm the guess that you made in part (b). [Use the fact that, since f′ is continuous, there is a constant M such that | f ′ ( x ) | ≤ M for 0 ≤ x ≤ 1 .]

To determine

a) To Find: how the graphs of y=f(x) and y=f(x)sinnx are related and what happens as n

Explanation

Calculation: For convenience we took  two cases by taking the function f(x)=ex and by taking f(x)=x

The graphs of the function y=ex and y=exsinnx are given below

Similarly the graphs of the function y=x and y=xsinnx are given below

To determine

b) To Find :limn01f(x)sinnxdx

To determine

c) To Find :limn01f(x)sinnxdx using integration by parts and compare with the answer in part (b)

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