## MAT 131 Calculus I Final Examination

December 16, 1992

This is a 3-hour test. Work all questions. You may use a graphing calculator. Write your name and section number in your bluebook''.

1. Sketch the following graphs.
a. Plot the distance from the top of the hill as a function of time. The sled started down the hill, going faster and faster until it hit a wall.
b. Plot the distance from your dog to your front door as a function of time. As you walked home with your dog on a leash, he ran in circles around you, wrapping the leash around your body (so the circles got smaller and smaller). Finally you picked him up and carried him home.

2. a. Calculate the derivative of f(x) = x^2 cos(x).
b. Calculate the derivative of f(x) = 1/[ e^{sin(2x)}].
c. Calculate the slope dy/dx of the curve x^3 + xy + y^4 = 3 at the point x= 1 , y= 1 .
d. Calculate an anti-derivative for f(x) = 2x^2 - 5.
e. Calculate an anti-derivative for f(x) =sin(2x).

3. The height h(t) of a marble falling through thick syrup is given at various times t in the following table:

t (sec)          0.0      0.2      0.4      0.6      0.8
h (inches)        12     11.5     10.8      9.9      8.8


Estimate the velocity h' (in inches/sec) at t= 0.4 .

4. a. Give the equation of the line tangent to the graph of f(x) = cos(x^2) when x = 2. (Note: x is in radians!)
b. Calculate the approximation to f(2.1) given by the tangent line approximation at x = 2.

5. If the sum of two non-negative numbers is 12, what is the maximum value of one times the square of the other?

6. A jet touches down on the deck of an aircraft carrier at t=0 and immediately starts decelerating. At touchdown it has 200 feet to go before the end of the runway. The table below gives the jet's velocity when it was tested on a longer runway. Assuming these are the same velocities when it attempts to land on the aircraft carrier, does the jet come to a stop before it reaches the end of the runway, or does it go plunging off into the water? Explain in detail.

t (sec)       0.0   0.5   1.0   1.5   2.0   2.5   3.0
v (feet/sec)  235   178   114    67    33     13    0


7. You want to approximate

\int_0^1 e^{x^2}dx

                   /1
|     2
|    x
|   e    dx
|
0/

using a left-hand sum. Suppose you do not know the true value. How many (equal) subdivisions do you need to get within .05 of the true value? Note that the function to be integrated is monotonic increasing on the interval given.

8. Given that f(0)=0, and that the derivative f '(x) is given by
f '(x)= cos(x^2)- (sin x)^2 sketch the graph of f(x) for
0 < = x < = 3, showing on your graph the coordinates of critical points and inflection points. Hint: use your calculator to produce the graph of f ', and use that information to determine where f is increasing, where it is decreasing and what the concavity of the graph of f will be.