Problem Set 2
Math 1090-001 (Spring 2001)
Due: Friday Feb. 2, 2001
1
Instructions
1. Solve all the following problems and submit your solutions on separate sheets of paper. In particular, do not write
your solutions on the margin of this problem sheet.
2. Your solutions must be neat and legible. Clearly indicate the flow of logic in your solutions.
3. Staple your solution set if you have used more than one sheet.
Problems
1. Textbook problems: 1.5.44, 1.5.46, 1.5.47, and 1.5.50.
2. Textbook problems: 1.6.14, 1.6.26, 1.6.38, 1.6.48, 1.6.49, 1.6.50, 1.6.56.
3. a)
b)
What does the p-intercept represent for a supply function S(p)? For a demand function D(p)?
Should the p-intercept of a supply function S(p) be positive, negative or zero? How about that of a demand
function D(p)? Explain your answer.
4. Sketch each of the following quadratic functions, clearly indicating its vertex, axis of symmetry and y-intercept.
Find its maximum or minimum.
a)
b)
c)
d)
f (x) = 3x2 − 8x − 3.
f (x) = −2x2 − 6x + 1.
f (x) = (2x − 1)(3x + 2).
f (x) = 2x(5 − 3x).
5. Solve the following quadratic equations. You have to decide which method to use.
a)
b)
c)
x2 − 6x − 7 = 0.
p
x2 + (3)x − 1 = 0.
(2x + 3)(x − 2) = (3x − 2)(x + 3) + x + 1.
6. The following three diagrams show three possible graphs of the the function f (x) = ax 2 + bx + c. In each case,
determine the signs of a, c, and b2 − 4ac.
6
6
-
6
-
-
7. Show that the equation x2 − 6cx + 12c2 has no real roots for any real non-zero number c.
Hint: Check its discriminant.
8. Show that among all rectangles having a certain fixed perimeter, the one with the largest area is a square.
• Let the perimeter be 2k, the length of the rectangle be x. What is its width y in terms of k and x?
• If the area of the rectangle is A, what is A in terms of x and y.
• Using the previous two parts, write A as a function of x alone (i.e. eliminate y from the expression for A. However, the answer to this
question will involve the constant k.)
• You should have found that A is a quadratic function of x. Explain why this quadratic function has a maximum rather than a minimum.
Find this maximum value and at what value of x it is attained.
• Find the width y corresponding to the above maximum.
• Explain why the “largest-area” rectangle is a square.