SAMPLE FINAL MAT 142 FINAL is Thursday, May 11, 2000, 12-3pm Psych A-137

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

(i) The sequence \{a_{n}\} = \{1 + (−1)^{n} \frac{1}{n} converges to (a) 0 (b) −1 (c) 1 (d) \frac{1}{2} (e) −\frac{1}{2} (f) it diverges

(ii) The sequence \{a_{n}\} = \{(−1)^{n}(1 − \frac{1}{n})\} has least upper bound equal to (a) −1 (b) 0 (c) 1 (d) 2 (e) \frac{1}{2} (f) it has no upper bound

(iii) Define a sequence by a_{0} = 1, a_{n} = \frac{3}{2}a_{n−1}. Then the sequence converges to (a) 0 (b) 1 (c) 2 (d) 4 (e) \frac{3}{2} (f) the sequence diverges

(iv) The infinite series ∑_{n=0}^{∞} 3^{−n} converges to (a) 0 (b) \frac{1}{3} (c) \frac{2}{3} (d) \frac{3}{4} (e) 1 (f) none of these

(v) The infinite series ∑_{n=0}^{∞} \frac{2^{n}}{n!} converges to (a) 0 (b) 1 (c) e (d) e^{2} (e) 2 (f) none of these

(vi) What is the inverse function of y = \sqrt{1 − x^{2}} on (0, 1)? (a) x^{2} − 1 (b) \sqrt{1 − x^{2}} (c) x^{2} + 1 (d) \sqrt{1 + x^{2}} (e) \sqrt{1 − x} (f) none of these

(vii) ∫_{0}^{2} \frac{2x}{x^{2}−5}dx = (a) ln 2 (b) ln 5 (c) − ln 5 (d) − ln 2 (e) 0 (f) none of these

(viii) \frac{d}{dx}2^{x^{2}} = (a) 2^{x^{2}} (b) 2^{x^{2}}2x (c) 2^{x^{2}} ln 2 (d) 2^{x^{2}}2x ln 2 (e) 2^{x^{2}}x^{2} ln 2 (f) none of these

(ix) Find the limit lim_{x→0^{+}}(1 + x)^{1/x}.

(a) 0 (b) 1/e (c) 1 (d) e (e) ∞ (f) none of these

(x) What is \frac{d}{dx} sin^{−1}(x),|x| < 1?

(a) x/\sqrt{1 + x^{2}} (b) 1/\sqrt{1 + x^{2}} (c) 1/\sqrt{1 − x^{2}} (d) −1/\sqrt{1 − x^{2}} (e) 1/(|x|\sqrt{x^{2} − 1}) (f) none of these 2. Evaluate each of the following integrals. You may use the table of integrals at the end of

the book.

(i) ∫ sin^{3}(x)dx

(ii) ∫ \frac{dx}{1+sin 3x}

(iii) ∫ \sqrt{x^{2} − 1}dx

(iv) ∫ \frac{dx}{\sqrt{4+x^{2}}}

(v) ∫ \frac{\sqrt{x+2}}{x}dx

3. Put a “C” in the box if the series converges and a “D” if it diverges.

(i) ∑_{n=1}^{∞} \frac{sin n}{n^{3}}
(ii) ∑_{n=1}^{∞} \frac{1}{1+n^{2}}

(iii) ∑_{n=1}^{∞}(−1)^{n}n^{1/2}

(iv) ∑_{n=1}^{∞} \frac{n^{4}}{2^{n}}

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

4. Write out the Maclaurin series for sin x and tan x up to the third power. Use these series to

evaluate

lim_{x→0} \frac{sin x − tan x}{x^{3}}.

5. Prove that

1 + \frac{1}{2} · 2 + \frac{1}{4} · 3 + … + \frac{n}{2^{n−1}} + … = 4.

(Hint: differentiate the equation \frac{1}{1−x} = 1 + ∑_{n=1}^{∞} x^{n} and plug in an appropriate value.)

6. Give the definition of the hyperbolic functions sinh x and cosh x. Using these definitions, show sinh 2x = 2 sinh x cosh x.