Short Calculus M112 Sample Exam I

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  1. The height of an object $(y)$ thrown upward as a function of time $(t)$ is given in the table below:

    MATH

    a) Find the average velocity between $t=2$ and $t=6$.

    b) Give your best estimate of the instantaneous velocity at $t=4$.

  2. Estimate the slope of the tangent to the graph of $y=0.87^{x}$ at $x=2$. Use at least three secant lines to come up with your estimate.$\qquad $

  3. Use the definition of the derivative to find $f~^{\prime }(2)$ given that $f(x)=3x^{2}-x$.

  4. One of the functions below is $f(x)$, one is $f~^{\prime }(x)$ and one is MATH. Label each. On what intervals is $f(x)$ increasing? Decreasing? Concave up? Concave down?


    sampleExamISpring08__16.png

  5. Find the equation of the line that goes through the points (1,2) and (4,0).

  6. A population is declining in size exponentially. One year there were 2 thousand in the population and a year later there were 1.4 thousand. Find the equation of the exponential function that represents this population, using $t=0$ as the initial year. What are the annual and continuous rates of decline?

  7. The distance (s, in feet) of a mouse from a cat as a function of time (t, in seconds) is given by the function $s=f(t).$ Explain what each of the following means in words, using correct units: $f(0)=10$, $f~^{\prime }(1)=2$, $f~^{\prime }(4)=-3$, $f(7)=0$.

  8. Suppose that score on a test in directly proportional to the square root of the time spent studying for the test. Also suppose that one hour of study results in a score of 65. How much studying does it take to get 100? Sketch a graph of score versus study time.

  9. A boy leaves for school, but when he is about 1/4 of the way to school, he realizes that he forgot his homework. He turns around and goes back home, gets his homework, and then runs as fast as he can to school. Assume that school is a mile away. Sketch a graph of his distance from home versus time. Also sketch a graph of his velocity versus time.

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