Math 220 Calculus Review Problems

Based on the text: Hughes-Hallet, Gleason, et al, Calculus, Wiley.

1. Find the derivatives for the following functions:

(a) f(x) = (3x^{2} + π)(e^{x} − 4)

(b) f(x) = \frac{x}{x^{3} + 4}

(c) g(t) = ln(t + \sqrt{t^{2} − 1})

2. Find \frac{dy}{dx}:

(a) 2y^{2} + xy = x^{2} + 3

(b) \sqrt{x + y} + \sqrt{xy} = 6

(c) x sin y + cos 2y = cos y

(d) y = ∫_{0}^{x^{2}} sec t dt

3. Integrate (remember to check your answers)

(a) ∫_{1}^{4}(\sqrt{t} − \frac{2}{\sqrt{t}})dt

(b) ∫ \frac{x + 1}{x^{2} + 2x} dx

(c) ∫ x sin x dx

(d) ∫ x^{2}e^{x}dx

4. A object is dropped from a height of 64 ft. Find the height of the object above the ground, h(t), and the velocity, v(t), as functions of time. Also find the time the object hits and the velocity on contact. Hint: use the fact that the rate of change of velocity is equal to the constant acceleration due to gravity, i.e.

\frac{dv}{dt} = −g, g = 32 ft/sec^{2}

5. Find the function y(x) whose derivative satisfies:

(a) \frac{dy}{dx} = x^{3} + sin x

(b) \frac{dy}{dx} = 2y

6. If y(x) = ∫_{x}^{0} e^{t} dt, find y′(x).