18.01 PROBLEM SET 2

Hand in to Room 2-108 by 1:00 pm Friday, February 23.

The due date and time are absolute. Late homeworks are accepted until the solutions are posted, but the score will be multiplied by 1/2. Everything included here may be used later. You do not need to

memorize the proofs, but you need to know the facts.

1. 84/3, 19, 23, 33, 35, 37, 41, 42. 217/25. X.8/3.

(10 points)

2. Graph the function

\frac{(x − 1)^{2}}{(x − 2)^{2}(x^{2} − 1)}.

Find the domain, asymptotes, limit at infinity (and negative infinity), discontinuities, and zeroes of the function.

(5 points)

3. The hyperbolic functions are defined by the following formulas:

sinh x = \frac{e^{x} − e^{−x}}{2} cosh x = \frac{e^{x} + e^{−x}}{2} .

Prove

2 sinh x cosh x = sinh 2x
cosh^{2} x + sinh^{2} x = cosh 2x
cosh^{2} x − sinh^{2} x = 1
\frac{d}{dx} sinh x = cosh x
\frac{d}{dx} cosh x = sinh x.

Sketch the graphs of sinh x, cosh x, and sinh^{−1} x. Compute \frac{d}{dx} sinh^{−1} x. (10 points)

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