6.5E: Exercises for Section 6.5

For 1-6, calculate the average value of the following functions on the specified interval. Sketch the graph. Find the value of \(c\) such that \(f_\text{ave}=f(c)\). Round the answer to the nearest hundredth.

1. \(f(x)=1+2x\); \([0,3]\)

Answer

\(f_{ave}=4;c=\frac{3}{2}\)

2. \(f(x)=x^4\); \([-2,4]\)

Answer

\(f_{ave}=\frac{176}{5};c \approx 2.44\)

3. \(f(x)=\cos(x)\); \([0,\frac{\pi}{2}]\)

Answer

\(f_{ave}=\frac{2}{\pi};c\approx 0.88\)

4. \(f(x)=6(x-3)^2\); \([0,6]\)

Answer

\(f_{ave}=18;c\approx 1.27,4.73\)

5. \(f(x)=\ln(x)\); \([1,e^3]\)

Answer

\(f_{ave}=\frac{1+2e^3}{e^3-1};c\approx 8.65\)

6. \(f(x)=e^x\); \([-4,-1]\)

Answer

\(f_{ave}=\frac{e^3-1}{3e^4};c\approx -2.15\)

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