Math 1300 Fall 2012 Exam #3 Solutions 1. Multiple Choice. (a ...

(c) Show that the function f(x) = √ x satisfies the conclusion of the Mean Value Theorem
on [0, 4] by finding the value c.
Solution: The slope of the secant line is
√ √
f(b) − f(a) 4 − 0
=
b − a 4 − 0
On the other hand we have f ′ (x) = 1
2 x−1/2 , so that
= 2
4
f ′ f(b) − f(a)
(c) =
b − a
1
2 c−1/2 = 1
2
c −1/2 = 1
c = 1.
= 1
2 .
6. Suppose you want to use 24 feet of fencing to enclose three separate square-shaped regions,
one which has side length x and the other two which each have side length y.
(a) Draw a picture of the situation, with labels.
Solution: I had in mind Bizarro Mickey Mouse.
(b) Find a formula for x in terms of y.
Solution: The total perimeter must be 4x + 4y + 4y = 24, which means x = 6 − 2y.
(c) What is the largest y can be? What is the smallest it can be?
Solution: The largest y could be is if x is (close to) 0, which means y = 3. The smallest
y can be is 0.
(d) Find a formula for the total area A in terms of y.
Solution:
A = x 2 + 2y 2 = 2y 2 + (6 − 2y) 2 .
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