Solved: In Exercises 85–88, let for the specified function
Chapter 5, Problem 87CE(choose chapter or problem)
In Exercises 85-88, let \(F(x)=\int_{a}^{x} f(t) d t\) for the specified function \(f\) and interval \([a, b]\). Use a CAS to perform the following steps and answer the questions posed.
a. Plot the functions \(f\) and \(F\) together over .
b. Solve the equation \(F^{\prime}(x)=0\). What can you see to be true about the graphs of \(f\) and \(F\) at points where \(F^{\prime}(x)=0\)? 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 \(f\) over those intervals?
d. Calculate the derivative \(f^{\prime}\) and plot it together with \(F\). What can you see to be true about the graph of \(F\) at points where \(F^{\prime}(x)=0\)? Is your observation borne out by Part 1 of the Fundamental Theorem? Explain your answer.
\(f(x)=\sin 2 x \cos \frac{x}{3},[0,2 \pi]\)
Equation Transcription:
Text Transcription:
F(x)=integral _a ^x f(t) dt
f
[a, b]
f
F
[a, b]
F’(x)=0
F
f
f
F
F
f prime (x)=0
f(x)=sin 2x cos x/3, [0, 2pi]
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