Mathematics for Management -- Supplementary Electronic Materials

Quiz: Definite Integration by Parts

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1. Assume \(f(x)\) is a function with continuous first and second derivatives, and furthermore assume that \(f(0) = 1\), \(f(2) = 3\), \(f'(0) = 5\), and \(f'(2) = 7\). What is the value of \(\int^2_0 \, 3x f''(x) \, \textrm{d} x\)?

\( 32 \) \( 36 \) \( 16 \) \( 8 \)

2. Use integration by parts to evaluate the integral \(\int^1_0 \, x f''(x) \, \textrm{d} x\), given the values of \(f(x)\), \(f'(x)\), and \(f''(x)\) at \(x=0\) and \(x=1\) below \[ \begin{array}{c || c c c} x & f(x) & f'(x) & f''(x) \\ \hline 0 & 3 & 2 & -1 \\ 1 & 2 & 4 & 1 \end{array} \]

\( 2 \) \( 3 \) \( 5 \) \( 4 \)

3. Determine the value of \(\int^2_1 \, \ln(x) \, \textrm{d} x\).

\( 2 \ln(2) - 1 \) \( 2 \ln(2) + 1 \) \( 2 \ln(2) - 3 \) none of these

4. Consider evaluating the integral \(\int^{2}_0 \, x \sin(x+1) \, \textrm{d} x\). Which of the below would be the most appropriate selection and set-up of \(u\), \(\textrm{d} u\), \(v\) and \(\textrm{d} u\) if you were to attempt to evaluate the integral using integration by parts?

\( \left\{ \begin{array}{l c l} u \, = \, x & & \textrm{d} v \, = \, \sin(x+1) \textrm{d} x \\[1mm] \textrm{d} u \, = \, \textrm{d} x & & v \, = \, -\cos(x+1) \end{array} \right. \) \( \left\{ \begin{array}{l c l} u \, = \, x & & \textrm{d} v \, = \, \sin(x+1) \textrm{d} x \\[1mm] \textrm{d} u \, = \, \frac{1}{2} x^2 \textrm{d} x & & v \, = \, \cos(x+1) \end{array} \right. \) \( \left\{ \begin{array}{l c l} u \, = \, \sin(x+1) & & \textrm{d} v \, = \, x \textrm{d} x \\[1mm] \textrm{d} u \, = \, \cos(x+1) \textrm{d} x & & v \, = \, \frac{1}{2} x^2 \end{array} \right. \) \( \left\{ \begin{array}{l c l} u \, = \, \sin(x+1) & & \textrm{d} v \, = \, x \textrm{d} x \\[1mm] \textrm{d} u \, = \, -\cos(x+1) \textrm{d} x & & v \, = \, 1 \end{array} \right. \)

5. Determine the value of \(\int^{\pi/2}_0 \, x^3 \sin(x) \, \textrm{d} x\).

\( \frac{3}{4} \pi^2 - 3 \pi + 6 \) \( \frac{3}{4} \pi^2 + 3 \pi - 6 \) \( \frac{3}{4} \pi^2 + 6 \) \( \frac{3}{4} \pi^2 - 6 \)

6. Determine the value of \(\int^{\pi}_0 \, x^3 \sin(x) \, \textrm{d} x\).

\( \pi^3 - 6 \pi \) \( -\pi^3 - 6 \pi \) \( \pi^3 + 6 \pi \) \( \pi^3 + 6 \pi \)

7. Determine the value of \(\int^1_0 \, x {\rm{e}}^x \, \textrm{d} x\).

\( 0 \) \( 1 \) \( \rm{e} \) \( \frac{1}{\rm{e}} \)

8. Determine the value of \(\int^{2.2}_{0.2} \, x {\rm{e}}^x \, \textrm{d} x\).

\( 7.8036 \) \( 11.807 \) \( 14.034 \) \( 19.611 \)

9. Determine the value of \(\int^{\rm{e}}_1 \, x^2 \ln(x) \, \textrm{d} x\).

\( \frac{2}{9} {\rm{e}}^3 - \frac{1}{9} \) \( \frac{1}{3} {\rm{e}}^3 + \frac{1}{9} \) \( \frac{1}{3} {\rm{e}}^3 - \frac{1}{9} \) \( \frac{2}{9} {\rm{e}}^3 + \frac{1}{9} \)

10. Suppose the interest rate is constantly \(5\%\) and the income stream is given by the function \(A(t) = 1000 + 50t\). What is the approximate present value of this income stream over the next \(10\) years rounded to the nearest integer?

\( 7681 \) \( 7869 \) \( 7983 \) \( 8145 \)

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

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