21-121  IM/Econ Calculus I  Fall 2000

Review Problems for Test 2

1. Find f′(x) for each of the following.

(a) f(x) = \sqrt{x^{4} + 1}

(b) f(x) = 3 sin(x^{2} + 1) + 2e^{−x}

(c) f(x) = x^{3}ln(1 + x^{4})

(d) f(x) = cos^{−1}(\sqrt{x})

(e) f(x) = \frac{e^{sin x}}{1 + e^{cos x}}

(f) f(x) = 2ln(e^{x} + e^{−x})

(g) f(x) = sin^{−1}(2x) + tan^{−1}(3x)

2. Solve for x.

(a) e^{3x} = 6e^{−2x}

(b) lnx + ln(x − 1) = ln6

3. Find the tangent line to the curve xy^{4} + y + x^{3} = 2 at the point (1,1)

4. Find f′(x), f′′(x), f′′′(x) for f(x) = \sqrt{1 − 3x}.

5. Find \frac{dy}{dx} for each of the following.

(a) y = (1 + x^{2})^{cos x}

(b) y = (1 + \sqrt{x})^{lnx}

6. Find \frac{d^{2}y}{dx^{2}} if y^{3} + xy = 2.

7. Suppose that the demand for a manufacturer’s product satisfies

p^{3} + pq + 10p = 2000

where p is the price per unit and q is the number of units.

(a) Use implicit differentiation to find \frac{dp}{dq}.

(b) Find the marginal revenue when p = 10, q = 90.

8. Suppose that f and F are differentiable functions satisfying F (x) > 0 for all x; f′(x) = F (x) and F′(x) = f(x) for all x; (F (x))^{2} − (f(x))^{2} = 1 for all x, and f is one to one. Let g denote the inverse function of f, i.e. y = g(x) if and only if x = f(y). Find an explicit formula for g′(x).

9. A CMU business student and a CMU engineering student are talking with each other at the same spot on campus. At the instant when the conversation ends, the business students starts walking north at 3 meters per second and the engineering student starts walking east at 2 meters per second. How fast is the distance between the students changing at the instant precisely 30 seconds after the conversation has ended.

10. A spherical balloon is being inflated at a constant rate of 16 ft^{3} per minute. If

R denotes the radius of the balloon, find \frac{d^{2}R}{dt^{2}} at the instant when R = 2 ft. (Here t is time.)

11. Assume that f is differentiable and satisfies f(6) = 11 and f′(6) = 3. Find an approximate value for f(5.9).

12. Find the absolute maximum and the absolute minimum of f(x) = x^{2/3} − \frac{2}{3}x + 1 on [−1, 1.5].