Name:
Practice Final.
CLOSED book. Remember that points are given for the steps that you perform,
not just the answer. Also, full points are given only for precise answers so use your
calculator carefully.
p
1. (a) (5 marks) Dierentiate f (x) = (2xx+,1)x2 with respect to x.
p
(b) (5 marks) Dierentiate g() = ( + 7) sin( 3 ) with respect to .
2. (a) (5 marks) Calculate
Z
3
, sin Z
d.
(b) (5 marks) Calculate (t3 + 4) (t + 1)
1
dt.
3. (a) (2 marks) Write down the limit denition of the derivative f (x) of a
function f .
0
(b) (8 marks) Find the derivative of f (x) = (x + 2)3 by using the limit
denition.
4. A ball is thrown straight up so that its height (in feet) after t seconds is
s(t) = ,16t2 + 32t + 6.
(a) (5 marks) What is the ball's velocity when it hits the ground?
(b) (5 marks) What is the ball's maximum height?
2
5. Let f be a continuous function and let its derivative f have the graph shown:
0
y
(0; 1) bc
b
(,3; 0)
b
b
(,2; 0)
(2; 0)
x
c
b
(a) Where is f increasing/decreasing ?
(b) Where is f concave up/down ?
(c) Assume that f (0) = 0. Draw the graph of y = f (x).
y
x
6. Answer 'true' (T) or 'false' (F) by circling the appropriate letter.
T/F
\The equation xy + x3 = y=2 can be written in the form y = f (x)
for some function f (x)."
T/F
\The slope of the tangent line to the curve y = x3 + 2 is dierent
at every point of the curve."
T/F
\If f (c) = 0 then f has an inection point at (c; f (c))."
00
Z 2
Z 2
\
j sin(x)jdx =
j cos(x)jdx."
0
0
p
T/F
Z 3 \The area of the region bounded by y = x, y = 0 and x = 9 is
9 , y2 dy".
T/F
0
3
7. (10 marks) A woman on a dock is pulling on a rope fastened to the bow of
a small boat. If the woman's hands are 10 feet higher than the point where
the rope is attached to the boat and if she is retrieving the rope at a rate of
2 feet per second, how fast is the boat approaching the dock when 20 feet of
rope are still out ?
4
8. (10 marks) Find the area of the region bounded by the curves y = x2 , 3x
and y = ,x2 .
9. Evaluate the following integrals:
Z 2
(a) (5 marks) (x2 +x 4+x 2+ 1)2
1
(b) (5 marks)
Z
p
0
5
dx:
p3
p + 3) dt.
( 3 t2 + 3)2
t cos( t2
5
10. An icecream cone is full of jello icecream (with weight density = 32 pounds
per cubic inch). If the icecream cone is 8 inches tall and has radius 2 inches,
nd the work done in sucking all of the icecream over the top of the cone.
[Hint: you should introduce coordinates so that the point of the cone is at
the origin.]
6