MTH 142 Practice Problems for Exam 1     Last updated 9/25/02


This is a selection of sample problems. The actual test will have fewer problems.


Part I. No Calculators


You may not have a calculator while this part of the test is in your posession. When you are through with this part, hand it in. Then you may work on the rest of the test using a calculator.


\begin{displaymath}
\begin{array}{lll}
\displaystyle
(1.) \
\int \frac{6 t + 1}...
...
\displaystyle
(14.) \ \int \sqrt{3 - x^{2}}\ dx &
\end{array}\end{displaymath}


SHORT ANSWERS FOR PART I
  1. Use partial fractions. ANS: $ \frac{-5}{1 + t} + \log (t) - \log (1 + t)+C$
  2. Substitution $w = \sqrt{x}$. ANS: $ 2\, e^{{\sqrt{x}}}+C$
  3. By parts. ANS: $ - \frac{7}{4} + \frac{e^2}{4}$
  4. Substitution $w = 3 \sin{x} + 1$. ANS: $ \frac{2}{9} \, {\left(1 + 3\, \sin (x)\right)^{3/2}} $
  5. Long division. ANS: $\frac{3}{4}\, x + \frac{1}{4}\, x^2 + \frac{9}{8}\, \log (-3 + 2\, x) +C$
  6. Except for a constant factor, the numerator is the derivative of the denominator.. ANS: $ \frac{1}{2}\,\log (-9 + 8\, x + x^2) +C$
  7. Complete the square. ANS: $ \frac{-4}{5}\, \arctan (\frac{1 + 2\, x}{5}) +C$
  8. Substitution $w = e^{x}+2$. ANS: $ -\cos (2 + e^x) +C$
  9. Long division. ANS: $2 - \frac{3\, \pi }{4} + 3\, \arctan (\frac{1}{3})$
  10. Substitution $w = \sqrt{x}$, $w^{2}=x$ and $2 w dw = dx$. ANS: $2\, {\sqrt{x}} -2\, \log (1 + {\sqrt{x}}) +C $
  11. Write $\ln (2 t) = 1 \cdot \ln (2t)$, use integration by parts.. ANS: $-t + t\, \log (2\, t) $
  12. Use guess and check. Note that $a$ and $b$ are constants. ANS: $- \frac{a}{b} \, \cos (b\, t) +C$
  13. Trigon. substitution $x = 2 \sin (x)$, $dx = 2 \cos (x)$. ANS: $ \frac{-3}{2}\, x\, {\sqrt{16 - x^2}} + 24\, \arcsin (\frac{x}{4})+C$
  14. Trigon. substitution $x = \sqrt{3} \sin (x)$, $dx = \sqrt{3} \cos (x)$. ANS: $\frac{1}{2} \, x\, {\sqrt{3 - x^2}} + \frac{3}{2}\, \arcsin (\frac{x}{{\sqrt{3}}}) +C$

Part II. Calculators are allowed


15.
Use the midpoint rule with $n=3$ to approximate $\int_{0}^4 \frac{1}{1+x} dx$. Do the calculations by hand, write down details.
16.
The following table gives values of a function $f$, whose concavity does not change in the interval [0,2]:


x 0.0 .25 .50 .75 1.00 1.25 1.50 1.75 2.00
f(x) 0.0 1.492 2.08 2.48 2.75 2.92 3.00 2.98 2.86


We want to estimate $\int_0^2 f(x) dx $.

a) Find the midpoint estimate with 4 subdivisions, MID(4).

b) You can easily calculate that LEFT(4) = 3.915 and RIGHT(4) = 5.345. Find the trapezoid and Simpson's estimates TRAP(4) and SIMP(4).

17.
The numerical approximation of an integral $\int_a^b f(x) dx$ with LEFT(n),RIGHT(n),TRAP(n), and MID(n), produced the numbers shown below (not necessarily in the same order):

\begin{displaymath}
6.725,\quad 6.745,\quad 6.762, \quad 6.800
\end{displaymath}

It is known that the function $f(x)$ is decreasing and concave up. Choose a suitable method for each number. Give a reason for your choice.

18.
The exact value of an integral $\int_a^b f(x) dx$ is $2.50$. It is also known that LEFT(5) = 2.532431 and MID(5) = 2.502215.

a) What is the error in each case.

b) If only one decimal is correct in both figures, how many decimals you predict will be correct when using MID(50) to calculate the integral? Explain.

Say which of the following integrals are improper. For those that are improper, determine if they are convergent or divergent.

\begin{displaymath}
\displaystyle
(19.) \
\int_1^\infty \frac{3}{\sqrt{2 + x}}\...
...quad
(22.) \
\int_{-\infty}^\infty e^{3t}\ dt
\displaystyle
\end{displaymath}

Use the comparison test to determine which of the following improper integrals converge.

\begin{displaymath}
\displaystyle
(23.) \
\int_1^\infty \frac{x}{\sqrt{1 + x^6}...
...splaystyle
(24.) \ \int_2^\infty \frac{ t^2 + 1}{t^2 - 1}\ dt
\end{displaymath}