MAT 131 Calculus I ~~ First Midterm Examination

March 9, 1994

This is an 90-minute test. Work all questions. You may use a programmable, graphing calculator. As usual, no credit for unexplained work!

1. Three functions of x are tabulated below (the values are rounded off to three decimal places). One is linear (of the form mx+b), one is exponential (of the form Ce^(kx)) and one is quadratic (of the form ax^2+bx+c). Which is which? Explain your answers.


x       f(x)       g(x)        h(x)
1.1    6.400      7.050       6.340
1.2    5.328      5.875       5.423
1.3    4.335      4.896       4.506
1.4    3.362      4.080       3.589
1.5    2.409      3.400       2.672


2. The function f(x)= x^3 + 0.2 x is invertible; let the function g be its inverse. Your calculator can generate a graph of f with range -1.5 <= x <= 1.5, -1 <= y <= 1 a. Use this to sketch a graph of g(x), for -1 <= x <= 1.

b. Use your calculator to determine the value of g(0.55) to 2 decimal places.

3. A cup of hot coffee left in a cool room will get colder. If the room temperature is 20 degrees C, and the cup starts off at 100 degrees C, Newton's Law of Cooling implies that the temperature H of the coffee, in degrees Centigrade, as a function of time t in minutes will be

H(t) = 20 + 80 e^{-kt}

for some k > 0.

a. Suppose that after 12 minutes the coffee has cooled to 60 degrees C. Use this information to calculate k.

b. How long will it take the coffee to cool to 40 degrees C? Show your work!

c. Sketch the graph of H(t). Pay careful attention to the initial value H(0), the limiting value as t --> \infty, the slope and the concavity of your graph.

4. The position of an object falling vertically through air was recorded every 1/10 second; part of the record is shown in this table:



time (seconds    13.1       13.2       13.3     13.4     13.5
height(cm.)     529.67     423.10     307.83   183.95    51.54



a. Estimate the instantaneous velocity of this object at 13.4 seconds. Show your work.

b. Estimate the acceleration of this object at 13.4 seconds. Show your work.

Alternate versions:
1. Permute f,g,h.
2. 0.45, 0.35
3. 14,16
4. 13.2, 13.3.