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!

(Click here to see a solution)

- a) Obtain the first three nonzero terms of the Taylor series of about .
- a) Obtain = the Taylor polynomial of order 3 of about
- a) Calculate the radius of convergence of the series .
- A solid is produced by revolving about the x-axis the region in the plane bounded by , , and .
- Answer questions (a) and (b) of the previous problem, only now the solid is produced by revolving about the -axis.
- 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
- A 1 m. long rod has (linear) density , where is the distance from one end.
- A 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 .
- 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 .)
- 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, of the food either decomposes or is eaten.
- If is the order 2 Taylor polynomial approximating the function near , estimate the error in the approximation by two methods:
- a) Plot the periodic function given by for . For the plot, use the region , and .

b) Use part (a) to calculate

b) Use the plots of
and
to obtain an approximate value of the maximum error
for .

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

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

b) Find the exact volume of the resulting solid.

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.

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.

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.

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.

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.

a) Using the plots of
and
directly.

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

b) Calculate the Fourier polynomial of order 2 of .

**Determine if the series given below converges or diverges**

(Note: each one can be treated in more than one way!)