21-121  IM/Econ Calculus I  Fall 2000

Review Problems for Test 1

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

(a) lim_{x→2} \frac{x^{3} + x − 2}{x − 1} (b) lim_{x→1} \frac{x^{2} − 3x + 2}{x^{3} − 1}
(c) lim_{x→3^{+}} \frac{|3 − x|}{9 − x^{2}}  (d) lim_{x→3} \frac{|3 − x|}{9 − x^{2}}

2. Consider the function f defined by

f(x) = \{ \begin{matrix}−1 & x < 0 \\ x^{2} & 0 ≤ x < 1 \\ 2 & x = 1 \\ x & 1 < x < 2 \\ 3 − x & 2 ≤ x \end{matrix}

(a) Sketch the graph of y = f(x).

(b) Evaluate lim_{x→1}f(x) or explain why it does not exist.

(c) Where is f continuous?

3. Use the definition of derivative to find f′(x) for each of the following:

(a) f(x) = 1 + 3x − x^{2} (b) f(x) = \frac{2}{x + 1}

(c) f(x) = \sqrt{x − 1}

4. Show that the equation

x^{10} + \frac{1}{\sqrt{x + 3}} − 10x + 2 = 0

has at least one (real) root.

5. Find the tangent line to the curve at the point P.

(a) y = \frac{x}{x^{2} + 1} P = (1, \frac{1}{2}).

(b) y = x^{4} − 3x^{2} + 5 P = (−1, 3)

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

(a) f(x) = (x^{3} − \frac{1}{x}) (e^{3x} − 2e^{5x})

(b) f(x) = \frac{4}{x^{5} − 2x + 7}

(c) f(x) = \sqrt{x}e^{−x}

(d) f(x) = (x^{2} + 1)^{3} (x^{2} − 1)

(e) f(x) = \sqrt{x} + \frac{1}{x} + \frac{1}{x^{2}}

7. Suppose that the demand for a monopolists product is given by p = 2 + \frac{200}{\sqrt{q}}, where p is the price per unit and q is the number of units, and that the average cost function is given by \overline{c} = 3 + \frac{100}{q}.

(a) Find the marginal revenue function.

(b) Find the marginal cost function.

(c) Find an expression for the profit P as a function of q.

(d) Find all values of q at which \frac{dP}{dq} = 0.

8. Suppose that the demand for a monopolists product is given by p = F (q) where F is a function satisfying F (100) = 3 and F′(100) = −.01. Find the marginal revenue when q = 100.

9. Find a line that passes through the point (1, \frac{11}{4}) and is tangent to y = x^{2} +2x somewhere.

10. Evaluate the following limit or explain why it does not exist:

lim_{x→0^{+}} (\frac{1}{\sqrt{x^{2} + x}} − \frac{1}{\sqrt{x}}) .

11. Assume that f : (0, ∞) → ℝ satisfies f(s) − f(t) = f(\frac{s}{t}) for all s, t > 0 and lim_{h→0} \frac{f(1 + h)}{h} = 3. Show that f is differentiable on (0, ∞) and find a formula for f′(x).