Mathematics 180  Midterm Exam #1  February 11, 2002

MATHEMATICS 180 – Section 005

NAME

DATE

1. 2. 3. 4. TOTAL 1.(20 pts.) Let f(x) = 2x^{3}. Consider the point P = (1, 2) on the graph of f(x).

a) Find the equation of the line L tangent to f(x) at P.

b) Find points on the graph of f(x) at which the tangent line has slope m = 6. Write down the equations of the tangent lines at these points.

2. (30 pts.) Compute the following a) y(x) = (x + 1)^{3} − \frac{1}{x^{2}} , y′(x) =

b)

\frac{d}{dx}( ^{3} \sqrt{x^{2}} · ^{3} \sqrt{x^{4}}) =

c) y(x) = (x^{2} + 1)^{4}, y′(x) =

d) y(x) = \frac{4}{\sqrt{x^{2}+2}}, y′(x) =

e) \frac{d}{dx}(2x^{2} − 3) |_{x=5}=

f) f(x) = a^{2}x^{2} + 2b^{3}x + c^{4}.

\frac{d}{dx}f(x) =

\frac{d^{2}}{dx^{2}} f(x) =

3. (30 pts.) Compute the following a) f(P) = (P + 1)^{5}.

\frac{d^{2}}{dP^{2}} f(P) =

b)

\frac{d^{2}}{dx^{2}} (x^{3} − x − 1) |_{x=2}=

c) f(x) = (\sqrt{x} + 1)^{3/2}. f′(x) |_{x=4}=

d) f(x) = x^{10}.

\frac{d^{3}}{dx^{3}} f(x) =

e)

lim_{x→4} \frac{x^{2} − 16}{4 + x} =

f)

lim_{x→2} \sqrt[3]{x^{3} + 19} =

4.(20 pts.) Let f(x) = \frac{3}{x}.

a) Compute f(1 + h) − f(1),

f(1 + h) − f(1) =

b) Compute the difference quotient \frac{f(1+h)−f(1)}{h} and simplify,

\frac{f(1 + h) − f(1)}{h} =

c) Compute the limit in b) when h → 0, that is

f′(1) = lim_{h→0} \frac{f(1 + h) − f(1)}{h} =