REVIEW, FINAL: MATH 1B

Problem 1. Consider the differential equation

y′′ + y = sin^{2} x.

Which of the following statements is true? (a) Every solution y(x) to this equation is concave up.

(b) 2 cos x + 3 sin x is a solution.

(c) A particular solution can be found using A sin^{2} x.

(d) If y_{1}(x) and y_{2}(x) are solutions to the equation, then so is y_{1}(x) + y_{2}(x).

(e) None of the above.

Problem 2. Use series to compute

∫ \frac{e^{x} − 1}{x} dx.

What is its radius of convergence?

REVIEW, FINAL: MATH 1B

Problem 3. Which of the following functions does not have an elementary antiderivative?

(a) \frac{e^{tan^{−1} x}}{1 + x^{2}}
(b) \sqrt{x^{3} + 1}
(c) x^{5}e^{x}
(d) \frac{cos(ln x)}{x}
(e) tan^{3} x.

Problem 4. Consider the differential equation

y′′ + xy′ − 2y = 0.

Suppose that y(x) = ∑_{n=0}^{∞} c_{n}x^{n} is a solution to this equation satisfying y(0) = 0, y′(0) = 1. Compute the fourth Taylor polynomial for y(x) at x = 0.

REVIEW, FINAL: MATH 1B

Problem 5. Consider the antiderivative of a rational function:

F (x) = ∫_{a}^{x} \frac{P (t)}{Q(t)} dt.

Which of the following is true?

(a) F (x) is a rational function.

(b) F (x) is a combination of rational functions and logarithms.

(c) F (x) is a combination of rational functions, logarithms, and inverse trigonometric functions.

(d) F (x) may fail to be an elementary function.

(e) lim_{x→∞} F (x) = ∞.

Problem 6. Use series to estimate the arc length of y = (2/5)x^{5/2} for 0 ≤ x ≤ 1 with an error < 0.02.

REVIEW, FINAL: MATH 1B

Problem 7. Let f(x) be a continuous function on the interval [a, b]. Which of the following is correct?

(a) If f(x) is increasing, then the midpoint approximation gives an upper bound for ∫_{a}^{b} f(x) dx.

(b) If f(x) is increasing, then the trapezoidal rule gives an upper bound for ∫_{a}^{b} f(x) dx.

(c) If f(x) is decreasing, then the left endpoint approximation gives an upper bound for ∫_{a}^{b} f(x) dx.

(d) If f(x) is concave up, then the right endpoint approximation gives a lower bound for ∫_{a}^{b} f(x) dx.

(e) None of the above.

Problem 8. Solve the differential equation y′ = xe^{x}y(y + 1).

REVIEW, FINAL: MATH 1B

Problem 9. Which of the following is a solution to z^{4} = −1 + \sqrt{3}i? (a) \sqrt[4]{2}e^{5iπ/6}

(b) \sqrt[4]{2}e^{−2iπ/3}
(c) \sqrt[4]{1/8}(−1 + \sqrt{3}i)
(d) \sqrt[4]{2}(−\sqrt{3} + i).

Problem 10. Evaluate

∫_{−1}^{1} \frac{1}{\sqrt{4x^{2} − x^{4}}} dx.

REVIEW, FINAL: MATH 1B

Problem 11. For which value of m do we have

∫_{0}^{π} cos^{2} mx sin mx dx = 2/3?

(a) 0

(b) 2

(c) −1

(d) 1

(e) None of the above.

Problem 12. The punch at a party is kept in a 10 L bowl and is initially half-full of cranberry juice. Some unruly partygoers decide to spike the punch with PCP and are able to slip a conconction with a toxicity of 5 mg/L into the punch at a rate of 1 L/min. The punch is being drunk at a rate of 1/2 L/min. When the bowl is full, how toxic is the punch?

REVIEW, FINAL: MATH 1B

Problem 13. Which of the following statements is incorrect?

(a) If a_{n}, b_{n} > 0, ∑_{n} a_{n} converges and the sequence {b_{n}/a_{n}} converges, then ∑_{n} b_{n} converges.

(b) If a_{n} > 0 for all n and a_{n} → 0, then ∑_{n}(−1)^{n}a_{n} converges.

(c) If a_{n} > 0 for all n and ∑_{n} a_{n} converges, then ∑_{n}(−1)^{n}a_{n} converges.

(d) If ∑_{n} a_{n} converges then ∑_{n} a_{n}/2^{n} converges.

(e) If a_{n} > 0 and ∑_{n} a_{n} converges then ∑_{n}(−1)^{n}a_{n}^{2} converges absolutely.

Problem 14. Determine if the series

∑_{n=1}^{∞}(−1)^{n}( \sqrt[n]{n} − 1)^{n}

is convergent or divergent.