APPM 1350 Final Exam (150 pts) Spring 2011
INSTRUCTIONS: Books, notes, and electronic devices are not permitted. Write (1) your name,
(2) 1350/FINAL, (3) instructor’s name and (4) SPRING 2011 on the front of your bluebook. Also
make a scoring table with room for 8 problems and a total score. Work all problems. Start each
problem on a new page. Box your answers. A correct answer with incorrect or no supporting work
may receive no credit, while an incorrect answer with relevant work may receive partial credit.
— SHOW ALL WORK —
1. (24 pts - 6pts ea) Assume yis a function of x, find y0given:
(a) y=1
ln(x)(b) xey2= tan−1(ex)−y(c) y= (cos x)x(d) y=Zex
e
(t)ln(t)dt
2. (24 pts - 6pts ea) Evaluate the integrals:
(a) Zln(x2e√x)
xdx (b) Zdx
x3/2+x1/2(c) Z4
0
x
√1+2xdx (d) Z1
0
ln(sinh(x) + cosh(x)) dx
3. (24 pts - 6pts ea) Find the limits
(a) lim
x→∞(1 −2x)1/x (b) lim
x→−∞
x
√1 + x2(c) lim
x→∞
1
2 tan−1(x)−π(d) lim
x→e
1
x−e
4. (a) (5 pts) Find the linearization of f(x) = ln(1 −x) at a= 0.
(b) (5 pts) Use your linearization from part (a) to approximate ln(0.99).
5. (a) (5 pts) Consider a cylinder with a circular base and a fixed height of 7 inches. Suppose the radius
of the base is r. If V is the volume of the cylinder, find the rate of change of volume in terms of the
rate of change of the radius rwith respect to time. (Note, V= 7πr2)
(b) (10 pts) Suppose the radius of the cylinder mentioned in part (a) is measured to be 0.8 inches
with an error in measurement of 0.01 inches, use differentials to estimate the percentage error in
calculating the volume of the cylinder.
6. The velocity function of a particle moving along a straight line is given by v(t) = t2+ 3t−4 m/s
with initial position s(0) = 106 m.
(a) (5 pts) Find the position of the particle at any time t
(b) (5 pts) Find the acceleration of the particle at any time t
(c) (10 pts) Find the total distance travelled by the particle during the first 3 seconds.
7. (10 pts) Use the Mean Value Theorem to prove that if a < b then sin(b)−sin(a)≤b−a.
8. Given that f(x) = x2
2+ln[(x−2)2]
2,f0(x) = x+1
(x−2), and f00(x) = 1 −1
(x−2)2
(a) (2 pts) State the domain of f(x).
(b) (4 pts) Does f(x) have any vertical asymptotes? Justify your answer with a limit.
(c) (4 pts) Does f(x) have any horizontal asymptotes? Justify your answer with a limit.
(d) (4 pts) On what interval (or intervals) is f(x)increasing and/or decreasing ? Justify your answer.
(e) (4 pts) On what interval (or intervals) is f(x)concave up and/or concave down? Justify your answer.
(f) (2 pts) Find all local maximum or minimum values of f(x).
(g) (3 pts) Sketch the graph of f(x) = x2
2+ln[(x−2)2]
2. (Clearly label and sketch your graph.)
END