MTH 142 Practice Problems for Exam 2 - Fall 1999

Sections 8.1-8.3, 9.1-9.4, and Series Handout.

1. a) Obtain the Taylor polynomial of order 2 of f(x) = 1+e-x2 about a = 0.

b) Use part (a) to get an approximation to ò00.2 f(x) dx.

2. a) Obtain P3(x) = the Taylor polynomial of order 3 of tanx about a = p/4

b) Use the plots of f(x) and P3(x) to obtain an approximate value of the maximum error |f(x)-P3(x)| for 0.7 £ x £ 0.8.

3. Consider the series ån = 0¥ n 3n (x - 2)n.

a) Calculate the radius of convergence

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

4. a) Use the Taylor series of sin(x) to obtain the first four nonzero terms of the Taylor series of sin( -3 x2 ) for x near a = 0.

b) Use part (a) to obtain the first four nonzero terms of the Taylor series of x3 sin( -3 x2 ) for x near a = 0.

5. The region R in the plane bounded by y = 1, y = 2x2, and x = 0 is revolved about the x-axis.

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

b) Find the exact volume of the resulting solid.

6. A straight road joins the center of Town A to a railroad crossing that is 2000 meters away. The number of people living along the road is determined by the density function d(x) = 0.5 e- 0.0001 x people/m, where x is the distance in meters to the center of town.

a) Write down a Riemann sum that approximates the total number of people living along the road.

b) Obtain an approximate value for the total number of people living along the road by calculating an integral.

7. A 1 m. long rod has (linear) density d(x) = 2.0 + 0.015 x  gr/m where x is the distance from one end.

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.

8. A water tank has the shape of an inverted cone, with circular top with radius 10 feet and height 20 feet.

a)Write down a Riemann integral 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.

9. 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/ft3.)

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.

10. 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.

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.

11. Study the convergence of the series given below (Note: each one can be treated in more than one way!)

a) ån = 0¥ [(n2)/( 1 + n3)]

b) ån = 1¥ 1 + [1/( n2)]

c) ån = 3¥ [((-2)n+1)/( pn)]

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