a.) Given the function f(x)=x−2cosx,−2≤x≤0, use a graph to estimate the absolute maximum and minimum values.
b.) Use calculus to find the exact maximum and minimum values.
a.)
Based from the graph, the absolute minimum value is approximately f(−0.5)≈−2.20 and the absolute maximum value is approximately f(−2)=−1.20
b.) To find the exact value, we take the derivative of the function.
f′(x)=ddx(x)−2ddx(cosx)f′(x)=1+2sinx
when f′(x)=0
0=1+2sinx2sinx=−1x=sin−1[−12]x=−π6
We have either maximum or minimum at x=−π6
when x=−π6
f(−π6)=−π6−2(cos(−π6))f(−π6)=−2.2556
Evaluating the function of its interval [−2,0].
When x=0,
f(0)=0−2cos(0)f(0)=−2
When x=−2,
f(−2)=−2−2cos(−2)f(−2)=−1.1677
Therefore, the absolute minimum is exactly at f(−π6)=−2.2556 and the absolute maximum is exactly at f(−2)=−1.1677.
Thursday, May 18, 2017
Single Variable Calculus, Chapter 4, 4.1, Section 4.1, Problem 62
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