MATH 1220 Calculus II Midterm Examination 2 2022

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Calculus II (MATH 1220)

Midterm Examination 2

2022


Part I: Answer each of the following questions by encircling your choice  [each 3

points].

1. The definition of an infinite sum involves having the terms of the sum go to zero “fast enough”.

T       F

2. Any polynomial with real coefficients can be factored into a product of linear polynomials with real coefficients.                                   T       F

3. If řn“(8)k an  converges for some large k, then so will řn“(8)1 an .                                    T    F

4. řn“(8)1 n´n  converges.                                    T       F

5. If the sequence an  diverges and the sequence bn  diverges, then the sequence an ` bn  diverges.

T       F

6. If the sequence an  converges, then limn8 pan ´ an`1q“ 0.                                    T       F

7. If fpxq is a positive, continuous and decreasing function for n ě 1 and if an  “ fpnq for all n ě 1, then Σn“(8)1 an  “ş1(8) fpxq dx.                                      T       F

8. If an  ą 0 and an  Ñ 0, the series Σn“(8)1 an  converges.                                   T       F

9. If an    ą  0 for n  “  1, 3, 5, ¨ ¨ ¨  and an    ă  0 for n  “  2, 4, 6, ¨ ¨ ¨ ,  then Σn“(8)1 an   converges.

T       F

10. If an  ą 0 and an  ă en , then Σn“(8)1 an  converges.                                   T       F


Part II: Give the most simplified answer for each of the following questions.  [each 5 points]

11. Integrate

(a) ş  dt

(b) ş4(3) dt

(c) ş  dt

12.  (a) Estimate the minimum number of subintervals needed to approximate the integral

5x4  dx

using Simpson’s Rule with an error of magnitude less than 104 .

(b) A town wants to drain and fill a small polluted swamp.

 

The swamp averages 5 ft deep. About how many cubic yards of dirt will it take to fill the area after the swamp is drained?

13.  (a) Find the lim an  if an  “ n sin  .

Part  III:  Solve  each  of the  following  questions  by  showing  your  steps  in  detail neatly [each 10 points].

14. Use integration by rational fractions to compute ż

15. a. Use the trapezoidal rule to estimate ş0(1) x2 dx using four subintervals.

b. Use the midpoint rule to estimate ş0(1) x2 dx using four subintervals. Compare the result with the actual value of this integral.

16. In the following geometric series

8

ÿ ´ ¯n

n“0

write out the first few terms of the series to find a and r, and find the sum of the series.  Then express the inequality in terms of x and find the values of x for which the inequality holds and the series converges.

17. For what values of a, if any, does the series

8

ÿ ˜  ´ ¸

 

(1) Use a series representation of sin 3x to   nd values of r and s for which

x(l)im!0     +  + s  = 0:

(2) Find a parametrization for the line through the point (a; b) having slope m.



















2022-06-17