REVIEW, MIDTERM #2: MATH 1B

Problem 1. The equation

e^{x}y′ = x + y

is:

(a) Linear;

(b) Separable;

(c) Both linear and separable;

(d) Neither linear nor separable.

Problem 2. Evaluate the limit:

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

REVIEW, MIDTERM #2: MATH 1B

Problem 3. Consider the differential equation

y′ = −x^{2}(x^{2} + y^{2}).

Which of the following statements is not true? (a) The solutions to this equation are decreasing functions.

(b) Every solution has a critical point.

(c) If y(x) is a solution to this equation, then so is cy(x).

(d) There are no constant solutions to this equation.

Problem 4. Compute a series expansion for

∫ \sqrt{1 + x^{3}} dx.

Given an estimate for the integral

∫_{0}^{1} \sqrt{1 + x^{3}} dx

with an error bounded by 1/160.

REVIEW, MIDTERM #2: MATH 1B

Problem 5. If f(x) = ln(1 − x^{2}), what is the value of f^{(2n)}(0)? (a) 1/2n (b) (2n)!/n (c) −1/n

(d) 0

(e) −(2n)!/n (f) 1/(2n)!

Problem 6. Determine if the series

∑_{n=2}^{∞} \frac{1}{n ln n(ln ln n)^{2}}

is convergent or divergent.

REVIEW, MIDTERM #2: MATH 1B

Problem 7. The interval of convergence of the series

f(x) = ∑_{n=1}^{∞} \frac{(x − 6)^{n}}{3^{n}(n + 2)}

is:

(a) |x − 6| < 3

(b) |x| < 3

(c) [3, 9]

(d) 3 ≤ x < 9

(e) The series converges for all x.

Problem 8. Solve the initial value problem

(x^{2} + 1)\frac{dy}{dx} = 2xy + 4x^{2} + 4, y(0) = 0.

REVIEW, MIDTERM #2: MATH 1B

Problem 9. Which of the following statements is true?

(a) If ∑_{n} c_{n}2^{n} converges, then ∑_{n} c_{n}(−3)^{n} converges.

(b) If ∑_{n} c_{n}2^{n} converges, then ∑_{n} c_{n}(−2)^{n} converges.

(c) If ∑_{n} c_{n}(x − 3)^{n} diverges when x = 2, then it diverges when x = 5.

(d) None of the above statements are true.

Problem 10. A tank contains 100 L of water with a salt concentration of 0.01 kg/L. Brine that contains 0.02 kg/L of salt enters the tank at 5 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after t minutes?

REVIEW, MIDTERM #2: MATH 1B

Problem 11. Calculate the Maclaurin series for

f(x) = \frac{x^{2}}{(1 + x)^{2}} .

Problem 12. Determine if the series

∑_{n=2}^{∞} \frac{(−1)^{n}n}{2n^{3} − n − 1}

is convergent or divergent.

REVIEW, MIDTERM #2: MATH 1B

Problem 13. Which of the following statements is true? (a) If ∑_{n} a_{n} and ∑_{n} b_{n} are divergent, then so is ∑_{n}(a_{n} + b_{n}).

(b) If a_{n} > 0 and lim_{n→∞}(a_{n+1}/a_{n}) < 1, then lim_{n→∞} a_{n} = 0.

(c) If a_{n} > 0 and ∑_{n} a_{n} is divergent, then ∑_{n} \sqrt{a_{n}} is convergent.

(d) If a_{n}, b_{n} > 0, then ∑_{n}(−1)^{n}(a_{n} + b_{n}) = ∑_{n}(−1)^{n}a_{n} + ∑_{n}(−1)^{n}b_{n}.

(e) Mathematics is not a rewarding discipline of study.

Problem 14. Determine if the series

∑_{n=1}^{∞} \frac{2^{n}}{n\sqrt{n^{n}}}

is convergent or divergent.

REVIEW, MIDTERM #2: MATH 1B

Problem 15. The series

∑_{n=1}^{∞}(−1)^{n} \frac{sin n}{n}

is:

(a) Convergent by the alternating series test.

(b) Divergent by the test for divergence.

(c) Convergent by the integral test.

(d) Convergent by the comparison test.

(e) None of the above.

Problem 16. Find the Taylor series for

f(x) = e^{2x}

at x = 1. What is its radius of convergence? Compute the Taylor polynomial T_{3}(x) for f(x) at x = 1.

REVIEW, MIDTERM #2: MATH 1B

Problem 17. Find the orthogonal trajectory of the family of curves given by

y = \frac{k}{1 + x^{2}} .

Problem 18. Determine if the series

∑_{n=1}^{∞} ln (\frac{1 + 1/n}{(n + 2)/(n + 1)})

is convergent or divergent. If it is convergent, find its sum.