Math 141H.1, Honors Calculus II

Midterm Exam 1

Stephen G. Simpson January 23, 2002

1. Let a function f be defined by

f(x) = log_{10} \sqrt{x^{4} − 16} .

On which intervals is f one-to-one? For each of these intervals, find the inverse function of f.

2. If log_{5} 100 = a, find log_{100} 5.

3. Solve the initial value problem

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

4. Find the derivative of log_{2}(sec^{−1} x).

5. Simplify the expression

ln \sqrt{x} + ln(x\sqrt{x}) + ln (\frac{5}{x^{2}} ) + log_{2}(32x) .

For which values of x is this expression defined?

6. Find the indefinite integral

∫ \frac{dx}{\sqrt{16 + 6x − x^{2}}} .

7. Evaluate the definite integral

∫_{−3\sqrt{2}}^{−2\sqrt{3}} \frac{dx}{x\sqrt{x^{2} − 9}} .

8. If f(x) = x^{5} + 2x − 8 and g = f^{−1}, find g′(−8).