MTH 142 Practice Problems for Exam 2 - Spring 2000

Last changed: March 12

Sections 8.1--8.3, 9.1--9.5, pp. 481--487, and the Series Handout.

Note: the exam will have fewer questions!
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  1. a) Obtain the first three nonzero terms of the Taylor series of $f(x) = e^{-x^2}$ about $a=0$.

  2. b) Use part (a) to calculate
    \begin{displaymath}\lim_{x \rightarrow 0} \frac{-1+e^{-x^2}}{x^2}\end{displaymath}
  3. a) Obtain $P_3(x)$ = the Taylor polynomial of order 3 of $\tan x$ about $a=\pi/4$

  4. b) Use the plots of $f(x)$ and $P_3(x)$ to obtain an approximate value of the maximum error $\vert f(x)-P_3(x)\vert$ for $0.7 \leq x \leq 1.2$.

  5. a) Calculate the radius of convergence of the series $\displaystyle \sum_{n=0}^\infty n 3^n (x - 2)^n$.

  6. b) Sketch in the number line an open interval of points $x$ for which the series converges.

  7. A solid $S$ is produced by revolving about the x-axis the region $R$ in the plane bounded by $y=1$$y = 2x^2$, and $x=0$.

  8. a) Write down a Riemann sum that approximates the volume of the solid.

    b) Find the exact volume of the resulting solid.

  9. Answer questions (a) and (b) of the previous problem, only now the solid is produced by revolving $R$ about the $y$-axis.
  10. An oil slick has the shape of a circle with radius 7,000 feet. After measurements were taken, it has been determined that the density of the oil at "r" feet from the center of the circle is given by the formula



    \begin{displaymath}d(r) = \frac{0.004}{1+r^2} \quad \mbox{Kg/ft}^2\end{displaymath}

    a) Write down a Riemann sum that approximates the total mass of the oil.

    b) Use part (a) to express the mass of the oil as an integral.

  12. A 1 m. long rod has (linear) density $d(x) = 2.0 + 0.015 x \ \mbox{gr/m} $, where $x$ is the distance from one end.

  13. a) Obtain a Riemann sum that approximates the total mass of the rod.

    b) Calculate the total mass of the rod as an integral.

    c) Calculate the center of mass of the rod.

  14. $20$ feet tall water tank has the shape of an inverted cone (i.e, the vertex is at the bottom) with circular top with radius 10 feet and height 20 feet. Water weighs $62.4 \ \mbox{lbs/ft}^3$.

  15. a)Write down a Riemann sum that approximates the work required to take the water out of the tank from the top.

    b) obtain the exact work in part (a) by calculating a suitable integral.

  16. A dam has the shape of a trapezoid, with horizontal parallel sides measuring 30 ft. (bottom) and 60 ft (top). The height of the dam is 30 ft., and one vertical side is perpendicular to both base and top. The dam has water up to the top on one side. (Water weighs 62.4 $lbs/ft^3$.)

  17. a) Write down a Riemann sum that approximates the total force excerted by the water on the dam.

    b) Obtain an integral that gives the total force of the water on the dam.

  18. A certain amount of fresh water shrimp is placed in a tank together with 2 lbs. of food, at 12:00 p.m. on January 1st. An additional 2 lbs. of food are added to the tank every 24 hours. After every 24 hours, $ 85 \% $ of the food either decomposes or is eaten.

  19. How much food is in the tank right before 12:00 p.m. on January 20 th? Give details, and explain how you arrived at your answer.

  20. If $P_2(x)$ is the order 2 Taylor polynomial approximating the function $f(x) = \frac{1}{2+x}$ near $x=0$, estimate the error in the approximation $P_2 (0.5) \approx f(0.5)$ by two methods:

  21. a) Using the plots of $P_2(x)$ and $f(x)$ directly.

    b) Using the inequality studied in class for error estimation when approximating a function by a Taylor polynomial.

  22. a) Plot the $2 \pi$ periodic function $f(x)$ given by $ f(x) = \vert x\vert,$ for $ - \pi < x \leq \pi$. For the plot, use the region $- 3 \pi \leq x \leq 3 \pi$, and $- 3 \pi \leq y \leq 3 \pi$.

  23. b) Calculate the Fourier polynomial of order 2 of $f(x)$.

    Determine if the series given below converges or diverges
    (Note: each one can be treated in more than one way!)

  24. $\displaystyle \sum_{n=0}^\infty \frac{n^2}{1 + n^3}$
  25. $\displaystyle \sum_{n=1}^\infty 1 + \frac{1}{2^n}$
  26. $\displaystyle \sum_{n=3}^{\infty} \frac{(-2)^{n+1}}{{\pi}^n}$

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