Assignment 1
• Answer the questions in separate sheets of paper, and CLEARLY
indicate your name and student ID on the TOP of the FIRST
PAGE.
• Submit your assignment to the MAT1010 dropbox on the sixth floor
of Cheng Dao Building, before 6PM of Tuesday, February 28.
• Explain your answer clearly, either in mathematical symbols or in
words.
1. Recall that if
L
,
1 + Ce−kt
where L, C and k are positive constants, then P is called a logistic function
of t.
P = f (t) =
(a) Show that P is an increasing function of t.
(b) Show that this function has only one inflection point, which occurs
when P = L/2.
二阶导数为0
2. Consider the function f (x) = x3 −3x2 −9x−5. Find the global maximum
and global minimum of f defined on the closed interval [−4, 4].
3. A car speeds up at a constant rate from 10 to 70 mph (miles per hour)
over a period of half an hour. Its fuel efficiency (in miles per gallon) increases
with speed; values are shown in the following table. Using these numbers,
make an upper estimate of the quantity of fuel used during the half hour.
1
4. For q ≥ 0, the marginal cost function of a product for a manufacturer is
given by
C 0 (q) = 0.006q 2 − q + 56 (dollars per unit).
The fixed cost for the product is 20000 dollars.
(a) Find the cost function C(q).
Z
49
(b) Fill-in-the-blank: the number
C 0 (q)dq represents the
48
Your answer in the blank should contain AT MOST six words.
5. Lily is the monopolist of a certain product. The price she sets for her
product is based on the demand curve of the product, which is shown below.
(a) Let B(q) be the average revenue function. That is, B(q) = R(q)/q,
where R(q) is the revenue function of the product. Suppose we draw
the graph of B(q) on the same axes below. What is the relation between this graph and the demand curve? Explain.
(b) Lily uses the demand curve to find a “special value”, following the
procedure below. She starts with a quantity q0 and considers the
corresponding p-value p0 on the demand curve. From the point S =
(0, p0 ), she draws a line l that is parallel to the line tangent to the
demand curve at the point with q = q0 . The line l will hit the vertical
line q = q0 at a point T = (q0 , p∗ ), and p∗ is the “special value”.
What is the meaning of this “special value” p∗ ? Justify your answer
mathematically.
2
.
6. Compute the following integrals.
Z
2x3
(a)
dx.
x4 + 1
Z
4
(b)
x7 ex dx.
Z
(c)
π/2
cos2 x dx.
0
3