Math 165 - Assignment #5
Name:
Due 7/23/2013
SHOW YOUR WORK!
This assignment covers sections 4 - 6, 8,9 from chapter 3. Make sure you justify your answer clearly! Simply writing
the final result does not mean you receive full credit. You must also show you understand the procedure.
1.- Your dream of becoming a Compsognathus breeder has finally come true. You are constructing a set of rectangular
pens in which to breed your vicious friends. The overall area you are working with is 120 square feet, and you
want to divide the area up into six pens of equal size as shown below.
The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. Find the dimensions of the pen
as to minimize the cost of the fencing.
2.- A wire of length 100 centimeters is cut into two pieces; one is bent to form a square, and the other is bent to form
an equilateral triangle. Where should the cut be if the sum of the two areas is to be a minimum? What about a
maximum? (Hint: Allow the possibility of no cut being made.)
3.- Make an analysis similar to that done in class (or that of examples 4 and 5 on section 3.5) and sketch the graph
2
+x−6
of f (x) = x x−1
.
4.- Sketch the graph of a function f that has the following properties
(i) f is everywhere continuous;
(ii) f (3) = 1;
(iii) f 0 (x) < 0 for x < 3, f 0 (x) > 0 for x > 3, f 00 (x) < 0 for x 6= 3
5.- A function is defined and a closed interval is give. Decide whether the Mean Value Theorem applies to the given
function on the given interval. If it does, find a possible value of c; if not, state the reason.
(i) f (x) =
x
x−2
on the interval [−1, 1].
(ii) f (x) = x1/3 on the interval [−100, 100].
(iii) f (x) = tan(x) on the interval [0, 2π].
(iv) f (x) = JxK on the interval [−2, 2].
6.- Show that if f is the quadratic function defined by f (x) = αx2 + βx + γ, α 6= 0, then the number c of the Mean
Value Theorem is always the midpoint of the given interval [a, b]. (Hint: (a2 − b2 ) = (a + b)(a − b).)
7.- Find the general antiderivative F (x) + C for each of the following:
(a) f (x) = 2x3 + πx2 − 1.
(b) f (x) = x
q
3
x
− 5x3 .
(c) f (x) = sin(x) − sec(x) tan(x).
(d) f (x) = −2 sin(x) cos(x)
p
1 + cos2 (x)
8.- Solve the solution to differential equation
dy
dx
= y 5 subject to y(0) = 1.