REVIEW, FINAL: MATH 1B ADDITIONAL PROBLEMS

Problem 15. The integral

∫_{a}^{b} f(x) dx

was approximated using the Trapezoidal rule and n = 10. Using the error bound it was found that |E_{T} | ≤ 1. For what value of n will |E_{T} | ≤ 10^{−6}?

(a) 999

(b) 10074

(c) 1053

(d) 60

(e) The answer cannot be determined from the information given.

Problem 16. Find the area of the region bounded by the curve

y = sin^{−1} x

and y = 0, x = 1/2.

REVIEW, FINAL: MATH 1B ADDITIONAL PROBLEMS

Problem 17. Let z_{1}, z_{2} be the solutions to the equation z^{2} − z + 7 = 0. What is the value of z_{1} + z_{2} + 7/(z_{1}z_{2})?

(a) 0

(b) 2

(c) 1 + i

(d) −1

(e) None of the above.

Problem 18. Solve the differential equation

y′′ − 2y′ + y = x.

REVIEW, FINAL: MATH 1B ADDITIONAL PROBLEMS

Problem 19. Determine if the series

∑_{n=3}^{∞} \frac{1}{n(ln n)^{3}}

is convergent or divergent.

Problem 20. Use Euler’s method with step size 1/2 to estimate y(1) where y(x) is the solution to the intial-value problem y′ = x + 2y^{2}, y(0) = 0.

REVIEW, FINAL: MATH 1B ADDITIONAL PROBLEMS

Problem 21. What is the value of

∑_{n=1}^{∞} \frac{(−1)^{n}π^{2n}}{2^{2n+1}(2n + 1)!}?

(a) 0

(b) −1

(c) 1/π − π/2 (d) 1/π + π/2 (e) (2 − π)/2π.

Problem 22. Evaluate the limit

lim_{x→0} \frac{(sin 2x − 2x)^{2}}{x^{2}(e^{x} − 1)^{3}} .

REVIEW, FINAL: MATH 1B ADDITIONAL PROBLEMS

Problem 23. Consider the series

∑_{n=0}^{∞}(−1)^{n} \frac{n − 1}{n} sin^{2} n.

Which of the following statements is true? (a) The series is absolutely convergent by the integral test.

(b) The series is convergent by the alternating series test.

(c) The series is divergent by the test for divergence.

(d) The series is convergent by the comparison test.

(e) None of the above.

Problem 24. Evaluate

∫ \frac{dx}{\sqrt{x^{2} − 2x}}.

REVIEW, FINAL: MATH 1B ADDITIONAL PROBLEMS

Problem 25. Consider the sequence defined by

a_{n} = \frac{ne^{1/n}}{3n − 1}.

What is

lim_{n→∞} a_{n}?

(a) 0

(b) ∞

(c) 1/3

(d) 1

(e) The sequence does not have a limit.

Problem 26. Evaluate

∫_{0}^{π} sec x dx.

REVIEW, FINAL: MATH 1B ADDITIONAL PROBLEMS

Problem 27. Find the area of the surface obtained by rotating the parabola y = x^{2} from x = 0 to x = 1 around the y-axis.

Problem 28. Find the particular solution of

y′ + y = x + e^{x}

satisfying y(0) = 0.

REVIEW, FINAL: MATH 1B ADDITIONAL PROBLEMS

Problem 29. Find the twelfth derivative of (x + 1)^{3}e^{x} at x = −1.

Problem 30. Determine if the series

∑_{n=1}^{∞} \frac{cos(1/n)}{n}

is convergent or divergent.