Practice exam 2.
Name:
CLOSED book, calculators allowed. Remember that points are given for the
steps that you perform, not just the answer.
1. Evil Farmer Brown has 120 feet of fence with which he plans to enclose a
rectangular prison for his children along one side of his 200 foot barn. He has
four children, and he plans to make a single prison for each child, as shown.
Barn
(a) What are the dimensions of the prison that will maximize its area ?
(b) What is the maximum area of each of the four prisons?
2. Answer 'true' (T) or 'false' (F) by circling the appropriate letter.
T/F
T/F
T/F
T/F
\A quadratic function has no points of inection."
1
\The function f (x) = x 3 has no critical points."
\The function f (x) = cos x has innitely many critical points."
\If x = c is a critical point of the function f then f (c) = 0."
0
1
3. (a) Find
dy
dx
by implicitly dierentiating the following:
2
2
x3
y3
2y = 2
2
(b) Find the equation of the tangent line to the curve x 3
the point (1; 1).
4. Sketch the graph of a function
the stated conditions:
2
y3
2y = 2 at
on the interval [ 2; 7] which satises all of
f
( 2) = 4; f (1) = 1 f (3) = 3 f (7) = 5
f (x) < 0 for
2 < x < 1, f (x) > 0 for 1 < x < 3 and 3 < x < 7.
f (x) < 0 for 2 < x < 3, f (x) > 0 for
2 < x < 2 and 3 < x < 7.
f
0
00
0
00
Draw a large picture so that it's easy to distinguish between concave up and
concave down.
2
5. Sketch the graph of y = f (x) = x4
2x3 as follows:
(a) Find f (x) and establish a sign diagram for f (x).
0
0
(b) Find f (x) and establish a sign diagram for f (x).
00
00
(c) (3 marks) Plot all of the critical points and potential points of inection
on a graph, then use your sign diagrams to help you get the concavity
and slope of f correct.
3