Mathematics for Management -- Supplementary Electronic Materials

The Average Value of a Function

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1. If the average value of a function \(f(x)\) on the interval \([a,b]\) is \(10\), then \(\int^b_a \, f(x) \, \textrm{d} x\) is

\( \frac{10}{b-a} \) \( \frac{f(a)+f(b)}{10} \) \( \frac{b-a}{10} \) \( 10 (b - a) \)

2. The average value of the function \(f(x) = (x-1)^2\) on the interval \([1,5]\) is

\( -\frac{16}{3} \) \( \frac{16}{3} \) \( \frac{64}{3} \) \( \frac{66}{3} \)

3. The average value of the function \(f(x) = \ln^2(x)\) on the interval \([2,4]\) is approximately

\( 1.204 \) \( 2.159 \) \( 2.408 \) \( 8.636 \)

4. The average value of the function \(f(x) = {\rm{e}}^{4 x^2}\) on the interval \([-\frac{1}{4}, \frac{1}{4}]\) is approximately

\( 0.272 \) \( 0.545 \) \( 1.090 \) \( 2.180 \)

5. The average value of the function \(f(x) = \sqrt{x+1}\) on the interval \([3,8]\) is

\( \frac{35}{15} \) \( \frac{28}{15} \) \( \frac{\sqrt{12}-1}{5} \) \( \frac{38}{15} \)

6. The average value of a function \(f(x)\) is \(-10\) on the interval \([-4,-2]\) and \(2\) on the interval \([-2,0]\) respectively. What is the average value of \(g(x)\) on the interval \([-4,0]\)?

\( -8 \) \(-4 \) \( -2 \) \( 4 \)

7. The average value of the function \(f(x) = \frac{-15}{(1+x)^2}\) on the interval \([1,6]\) is

\( \frac{-7}{14} \) \( \frac{-15}{14} \) \( \frac{-15}{7} \) \( \frac{-7}{3} \)

8. The average value of the function \(f(x) = \cos(x)\) on the interval \([1,5]\) is approximately

\( -0.990 \) \( -0.128 \) \( -0.450 \) \( 0.412 \)

9. The average value of the function \(f(x) = |x|\) on the interval \([-1,1]\) is

\( -\frac{1}{2} \) \( 0 \) \( \frac{1}{2} \) \( 1 \)

10. Let \(f(x)\) be a twice differentiable function such that \(f(4) = 10\) and \(f(6) = 7\). Which of the following must be true for the function \(f(x)\) on the interval \(4 \leq x \leq 6\)?
I) The average value of \(f(x)\) is \(\frac{17}{2}\).
II) The average rate of change of \(f(x)\) is \(-\frac{3}{2}\).
III) The average value of \(f'(x)\) is \(-\frac{3}{2}\).

I) and II) only I) and III) only II) and III) only none of these

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Solution: 1d; 2b; 3a; 4a; 5d; 6d; 7b; 8c; 9c; 10b

Here, a, b, c, d indicate the 1st, 2nd, 3rd, and 4th answer choice, respectively, for the numbered questions.