Lamb’s Final Exam Review for Math 1910
1. A ladder 20 feet long is leaning against a wall. If the foot of the ladder is being pulled away at a rate
of 2 feet per second how fast is the top of the ladder sliding down the wall
A) when the foot is 6
feet from the wall? B) when the top is 12 feet above ground?
2. Two straight roads intersect at right angles. Car A is on one road moving toward the intersection at
a speed of 60 mph. Car B is on the other road moving from the intersection at a speed of 40 mph.
When Car A is 2 miles from the intersection and Car B is 4 miles from the intersection, how fast is the
distance between the cars changing?
3. The total cost of producing x units of a commodity per week is C  x   200  5 x  .2 x 2
A) find
the marginal cost when production is 100 units.
B) Use the marginal cost to approximate the cost of producing the 101 st unit.
C) Find the exact cost of producing the 101st unit.
4. An oil spill from a ruptured tanker spreads in a circular pattern whose radius increases at a constant
rate of 2 ft/s. How fast is the area of the spill increasing when the radius of the spill is 40 ft.
5. How can a function fail to be differentiable at a point? Either no limit exists at that point or the
point is a sharp edge such as an absolute value function where no tangent line could exist.
6. Suppose a baseball is tossed straight upward and that its height (in feet) as a function of time (in
seconds) is given by the formula h  t   150t  16t 2
A)
B)
C)
D)
E)
find the instantaneous velocity and acceleration of the baseball at time t.
What is the maximum height attained by the ball?
What is the average velocity of the ball during the time interval from t=1 to t=4?
How long does it take before the ball lands?
At what time is the height of the ball 112 feet?
7. Find all local maximum and minimum values and absolute max and min values for the function
f  x 
1
x  cos x on the interval 0, 2  .
2
dy
if 5x 2 y  3 y  x3  2
dx
3x  2
9. Given y 
. Find y'
2
2
 x  4
8. Find
10. You want to build a rectangular fence for your dog in the back yard. If you can use the back of the
house for one of the sides of the fence and you have 30 feet of fence available what is the
maximum area possible for the fence?
11. Find the x-coordinate of the point of inflection of the function f ( x)  2 x3  x 2  x  1
12. Let f ( x) 
3
x ln x . Find the interval on which f is increasing.
13. Find the interval on which f ( x)  2 xe x is concave upward.
cos x
14. Find the derivative of the function f(x) =
x
1
3
15. The formula for the volume of a cone is V   r 2 h . Find the rate of change of the volume of the
cone if the radius is changing at a rate of 3 inches per minute and the height is 4 times the radius
when the radius of the cone is 8 inches.
dy
by implicit differentiation.
dx
17. f ( x)  ( x  1)2 ( x  2), find f '(2).
16. If x3  y 2  6, find
sin x
x 0 2 x
18. Find lim
19. A particle moves along a straight line with equation of motion s  t 3  2t 2 . Find the value at
which the acceleration is equal to zero.
20. A particle moves on the x-axis with velocity ( )
at time . At
the position of the particle is 2. What is the - coordinate of the position at
, the - coordinate of
?
21. Find the equation of the line tangent to f ( x)  x 2  5x  1 at the point (4,-3).
22. Find lim
x 2
x2  x  6
x2  5x  6
THERE WILL ALSO BE 3 LIMIT PROBLEM BASED ON INTERPRETING A GRAPH