SOLUTION MTH 142 Practice Problems for Exam 2 - Spring 2000

Last changed: March 30 (a more complete answer No.15 was added since March 12)

NOTE: The last line of answer 10 was modified. Also, please correct the following images:
problems 4,5,6,7,8,9, the Riemann sums should begin at j = 0
problem 12 b, change a0 = Pi/2

1. a) Take  in  to get

2. This gives the answer in less time than actually calculating the coefficients one by one (which is ok too).

b) Replace the function by the first few terms of the series to get

3. a) Direct calculation gives the polynomial

4. b)

From the plot above we see that the gap between f ( x ) and p ( x ) is the largest at the endpoint x = 1.2.
Therefore the maximum error in the interval (0.7,1.2) is E = f ( 1.2 ) - P (1.2 ) = 0.20911

5. a) The radius of convergence is given by

6. b) The interval

7. a) By taking sections perpendicular to the axis of rotation, we get washers''. At the tickmark  the washer has inner radius , outer radius , and thickness .

8. The Riemann sum that approximates the volumen is

The volume is obtained by taking limit as . We have,

9. a) By taking sections perpendicular to the axis of rotation, we get disks''. At the tickmark  the radius  and thickness . The Riemann sum that approximates the volumen is

10. b) The volume is obtained by taking limit as . We have,

11. a)

12.

b)

13. a)

14.

b)

c)

15. A cross-section of the cone (shown in the figure) is bounded by the lines  and .

16. Introduce tick marks in the -axis. The slab  at height  is a disk with radius  and thickness , so its volume is , and its weight is . The work involved in raising the slab a distance of  to the top of the cone is

The total work is approximated by

The exact work is given by

17. a) A sketch of the dam is shown in the figure.

18. Note that the equation of the right hand, non-horizontal side is . Introduce tick marks , in the y axis. At height , the slab has area , and the pressure at this height is . Therefore the force on the slab is

The total force is approximated by

The exact value of the total force is obtained by taking the limit of the sum as :

19. The following table is helpful:
20.  day amount before 24hrs are up Jan 2 Jan 3 Jan 4 Jan 20

Then, the amount right before noon on January 20th is

21. a) The Taylor polynomial of degree 2 of f(x) is

22. so the absolute value of the error in approximating f(0.5) by P(0.5) is

b) The function

is decreasing on the interval (0,0.5), so it attains its maximum in that interval at t=0. The maximum is

The error bound when approximating f(0.5) by P(0.5) is

23. a)

24. b) we have that

For details, we calculate  step by step below.

Therefore, the Fourier polynomial we seek is .

25. Solution 1: Note that  behaves like  when  is large, so we suspect that the series diverges. The following inequalities are clearly valid:

26. The term in the center simplifies to . Since  diverges, so does . This is our original series with only the first term changed, so our original series  also diverges.

Solution 2: We use the integral test. Set  for . Now  Therefore, the series diverges too.

27. Solution 1: Note that . Therefore the series diverges.

28. Solution 2: Using comparison, we have

Since  is divergent, so is our original series.

29. Solution 1:

Solution 2: Since the series is of the alternating (sign) type. To apply Leibnitz test we must first check two things. The first is,

and this is clearly true as each new term on the right hand side is obtained from the previous one upon multiplication by which is a number less than 1. Also, we must check that

One can see that the above statement is true once the limit is rewritten in this form:

(note that is less than 1, so raising this to a large power produces a small number). Because the two conditions to apply the test are satisfied, we have that, by Leibnitz test, the series is convergent.