Math Workshop Spring 2015
Problem Set I (Algebra and Calculus)
Due date: 5:00pm, January 19, 2015 (Monday)
Turn in to Instructor – ASM 2091
E-mail: [email protected]
Note: If I’m not in the office when you come by to turn in Problem Set I (38 subproblems or credits), please slide your solutions under my office door and send an e-mail
to me for this matter. My reply e-mail serves as the confirmation of receiving your
homework assignment. No late report will be accepted except you present the original
documents for medical or family emergency.
Grading:
(# of problems with correctness or credits in parts I and II)/(# of problems or credits
available in parts I and II)
Qualification for earning an extra-credit of 3% on the semester grade in my finance
classes:
The above grading ≥ 80%
The number of credits for this problem set = 38 (credits or sub-problems)
1. (2 credits) The graphs of
and
are given.
a) For what values of is
?
b) Find the values of
and
.
2. (1 credit) Find an equation of the tangent line to the curve
.
at the point
3. (1 credit) A rectangle has perimeter 18 m. Express the area of the rectangle as a
function A(l) of the length l of one of its sides.
4. (1 credit)
If
f
and g are
, find
.
continuous
functions
with
and
1
5. (1 credit) The point P(4, 2) lies on the curve
. If Q is the point
, use
your calculator to find the slope of the secant line PQ (correct to six decimal places) for
the value of
.
6. (1 credit) Find the domain of the function
.
7. (2 credits) If an equation of the tangent line to the curve
is
find
and
.
at the point where
8. (1 credit) The top of a ladder slides down a vertical wall at a rate of 0.15 m/s. At the
moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall
at a rate of 0.2 m/s. How long is the ladder?
9. (2 credits) Find the first and the second derivatives of the function
10. (1 credit) Find
in terms of
.
.
11. (1 credit) Suppose that
Find
12. (1 credit) Find the tangent line to the ellipse
at the point
13. (1 credit) Estimate the absolute minimum value of the function
decimal places on the interval
.
to two
14. (1 credit) The average cost of producing x units of a commodity is given by the
equation
. Find the marginal cost at a production level of 1,255 units
(up to 2 decimal places).
15. (1 credit) Find the number c that satisfies the conclusion of the Mean Value Theorem
on the given interval.
.
16. (1 credit) Use Newton's method with the specified initial approximation
to find
, the third approximation to the root of the given equation. (Give your answer to four
decimal places.)
17. (1 credit) Find a cubic function
that has a local maximum
value of 112 at 1 and a local minimum value of –1,184 at 7.
18. (1 credit) The sum of two positive numbers is 36. What is the smallest possible value
of the sum of their squares?
2
19. (1 credit) The manager of a 138-unit apartment complex knows from experience that
all units will be occupied if the rent is $860 per month. A market survey suggests that, on
the average, one additional unit will remain vacant for each $10 increase in rent. What
rent should the manager charge to maximize revenue?
20. (1 credit) Use Newton's method to approximate the indicated root of
in
the interval [1, 2], correct to six decimal places. Use
as the initial approximation.
21. (1 credit) Evaluate the Riemann sum for
,
with four
subintervals, taking the sample points to be right endpoints (up to 2 decimal places).
22. (1 credit) Evaluate the integral.
dy. (OK, the integrand is 16y-y2 )
23. (1 credit) Evaluate the integral by making the given substitution.
,
.
24. (1 credit) Find the solution of the equation correct to four decimal places.
.
25. (1 credit) Suppose g is the inverse function of a differentiable function f and
. If
26. (1 credit) If
27. (1 credit) Evaluate the integral.
28. (1 credit) Evaluate the integral.
. Select the correct answer.
a.
b.
c.
d.
e.
29. (1 credit) Find the Maclaurin series for f.
3
30. (1 credit) Find the Taylor series for
that f has a power series expansion.
centered at the given value of a. Assume
.
31. (1 credit) Find the Taylor polynomial
for the function
at the number a = 1.
. Select the correct answer.
a.
b.
c.
d.
e.
32. (1 credit) Find all the second partial derivatives of
.
33. (1 credit) Use Lagrange multipliers to find the maximum of f subject to the given
constraint(s).
34. (1 credit) Use Lagrange multipliers to find the maximum value of the function
subject to the given constraint.
. Select the correct
answer.
a.
b.
c.
d.
e.
35. (1 credit) Use Lagrange multipliers to find the dimensions (X, Y, Z) of the
rectangular box with largest volume if the total surface area is given as 54
.
4