Math 142 (Sections 1&2)  University of South Carolina
D. Meade  Fall 1995

Calculus I Review Problems

1. Find the indicated limit or explain why it does not exist.

(a) lim_{u→1} \frac{u^{2} − 1}{u + 1}
(b) lim_{u→−1} \frac{u^{2} − 1}{u + 1}
(c) lim_{x→0} \frac{tan x}{sin(2x)}
(d) lim_{x→0} \frac{sin(x)}{x}
(e) lim_{x→∞} \frac{sin(x)}{x}
(f) lim_{x→∞} \frac{3x^{2} − 2x + 7}{2x^{2} + 5x + 9}

2. State the definition of f′, the derivative of a function f. Describe the derivative of a function in geometric terms.

3. Compute the derivative of each of the following functions.

(a) f(x) = x^{3} − 3x^{2} + x^{−2}

(b) y = sin(t^{2}) − sin^{2}(t)

(c) g(u) = \frac{3u − 5}{(u^{2} − 1)^{2}}

(d) L(x) = ∫_{1}^{x} \frac{1}{t} dt

Hint: Use the Fundamental Theorem of Calculus.

4. A page of a book is to contain 27 square inches of print. If the margins at the top, bottom, and one side are 2 inches and the margin at the other side is 1 inch, what size page would use the least paper?

5. Compute each definite or indefinite integral.

(a) ∫_{0}^{1} x^{3} − 3x^{2} + 3\sqrt{x} dx

(b) ∫ \frac{y^{2} − 1}{(y^{3} − 3y)^{2}} dy

(c) ∫_{0}^{π/2} cos^{4} x sin x dx

6. Find the area of the region bounded by y = x − x^{2} and the x-axis.

7. Find the length of the arc of the curve y = \frac{x^{3}}{3} + \frac{1}{4x} from x = 1 to x = 3.