Mathematics 210-2, MIDTERM II, February 26, 2002
1. (15 points) Differentiate the following functions. Do not simplify the answer!
(a) f(x)=3xe−x.
(b) f(x)=x3
−
2
x10
.
(c) f(x)=ln
x−1
x.
2. (5 points) Find the limit: lim
x→∞
3x2
−5x+6
5x2−4.
3. (10 points) Find an equation of the tangent line to the graph of f(x)=√2x−3
at the point (6,3).
4. (10 points) Find the absolute maximum and minimum values of the function
f(x)=x3
−12xover the interval [0,3].
5. (10 points) Find (x3+3x−
1
x)dx.
6. (10 points) Find the particular solution of the differential equation y=x1/2
−x−1/2
given that y=1/3 when x=4.
7. (10 points) Suppose that $1000 is invested in a savings account in which interest
is compounded continuously at 2.7% per year; that is the balance P(t) grows at the rate
P(t)=.027 ·P(t), where tis in years and Pis in dollars. Find P(t) and determine when
the investment will triple itself. (Give the answer using logarithms, do not calculate as a
decimal.)
8. (30 points) Given f(x)= 2
1−x2.Find the following:
(a) Intercepts.
(b) Asymptotes.
(c) f(x), intervals of increasing or decreasing, and relative extrema.
(d) f(x), intervals of concavity (up or down), and inflection points.
(e) Sketch the graph of y=f(x).