SAMPLE MIDTERM 2 MAT 142 Midterm will be Thursday, Nov 15, 2001

1. Place the letter corresponding to the correct answer in the box next to each question.

(i) If we divide 3x + 2 into 3x^{2} − 7x, the remainder term is (a) 0 (b) −1 (c) 1 (d) 6 (e) 2 (f) none of these.

(ii) What is the limit of the sequence given by a_{n} = n^{1/n}? (a) 0 (b) 1/2 (c) 1 (d) e (e) e^{2} (f) none of these.

(iii) Suppose \frac{x−1}{(x+1)^{3}} = \frac{A}{x+1} + \frac{B}{(x+1)^{2}} + \frac{C}{(x+1)^{3}} . Then C = (a) −2 (b) −1 (c) 0 (d) 1 (e) 2 (f) none of these.

(iv) Which of the following improper integrals converges? (a) ∫_{0}^{1} \frac{1}{x}dx (b) ∫_{0}^{1} x^{−2}dx (c) ∫_{0}^{1} x^{−1/2}dx (d) ∫_{1}^{∞} \frac{1}{x}dx (e) ∫_{1}^{∞} x^{−1/2}dx (f) none of these.

(v) Suppose \frac{x^{2}+1}{(x−1)(x−2)(x−3)} = \frac{A}{x−1} + \frac{B}{x−2} + \frac{C}{x−3}. Then A = (a) −2 (b) −1 (c) 0 (d) 1 (e) 2 (f) none of these.

(vi) Which of the following improper integrals diverges? (a) ∫_{0}^{∞} e^{−x}dx (b) ∫_{0}^{∞}(1 + x^{2})^{−1}dx (c) ∫_{1}^{∞} \frac{x}{1+x^{2}} dx (d) ∫_{1}^{∞} \frac{x}{1+x^{3}} dx (e) ∫_{0}^{∞} \frac{sin x}{1+x^{2}} dx (f) none of these.

(vii) If we use the substitution x = tan θ, the integral ∫ \frac{dx}{\sqrt{4+x^{2}}} becomes (a) ∫ sec θdθ (b) ∫ tan θdθ (c) ∫ sin^{2} θdθ (d) ∫ sec θ tan θdθ (e) ∫ sec^{2} θdθ (f) none of these.

(viii) Suppose f is positive and decreasing and ∫_{0}^{∞} f(x)dx converges. Then which of the following must be true?

(a) ∫_{0}^{∞} f^{2}(x)dx must converge (b) ∫_{0}^{∞} f^{2}(x)dx must diverge (c) lim_{x→∞} f(x) > 0 (d) lim_{x→∞} f(x)/x = 1 (e) ∫_{0}^{∞} xf(x)dx must diverge (f) none of these.

(ix) What is the limit of the sequence \sqrt{n}/\sqrt{n + 2}? (a) 0 (b) 1/2 (c) 1/\sqrt{2} (d) 1 (e) 2 (f) none of these.

(x) Evaluate the series 1 − 2x + 4x^{2} − 8x^{3} + 16x^{4} − 32x^{5} + … where it converges. (a) \frac{1}{1−x} (b) \frac{1}{1−2x} (c) \frac{1}{1+2x} (d) \frac{1}{1+x^{2}} (e) \frac{2}{1−x} (f) none of these.

2. Find each of the following integrals.

(i) ∫ x^{2} sin xdx (ii) ∫ \frac{x}{(x−1)(x−2)}dx

(iii) ∫ \frac{dx}{x^{2}\sqrt{x^{2}+1}}

(iv) ∫ \frac{1−x}{\sqrt{1−x^{2}}} dx

(v) ∫ \frac{1}{1+sin x}dx (Hint: multiply and divide by 1 − sin x)

3. Determine whether each of the following infinite series converges or diverges (you do not need to find the limit if it converges) and explain why.

(i) ∑_{n=1}^{∞} \frac{n^{2}}{3+n^{3}}

(ii) ∑_{n=1}^{∞} n^{3}3^{−n} (iii) ∑_{n=1}^{∞} n^{−n} (iv) ∑_{n=1}^{∞}(\sqrt{n + 1} − \sqrt{n})

4. Evaluate each of the following limits.

(i) lim_{x→0} \frac{1−cos x}{x^{2}}

(ii) lim_{x→0^{+}} x ln x (iii) lim_{x→1}(x^{2} − 2x + 1)^{x−1}

(iv) lim_{x→∞} \frac{3x−5}{2x^{2}−x+2}

(v) lim_{x→∞}(1 + \frac{1}{x})^{x}.

5. For what values of p does the series ∑_{n=1}^{∞} \frac{n^{p}}{\sqrt{n^{3}+1}} converge? Justify your answer.

6. Find the sum of the infinite series ∑_{n=1}^{∞} \frac{2n+1}{n^{2}(n+1)^{2}} . Justify your answer.

7. Prove that if {a_{n}} is a bounded sequence then it has a subsequence which is monotone.