Calculus I Focus Questions, Quiz #1

These questions are meant to help you synthesize the main calculus concepts covered so far.  Every answer goes back, ultimately, to the first problem set you did (the 9 simple problems where you were to describe a picture).  You might review this set and see how the concepts developed there can be refined to answer the questions posed here.

1.  In terms of two simple examples involving money, pay rate and time, explain the difference between the meaning of the slope of a trapezoid and the area of a trapezoid. Be sure to explain your results in terms of units and also in terms of meanings.

In terms of two simple examples involving velocity, distance and time, explain the difference between a situation where you obtain a meaningful result by subtracting two quantities and dividing by a time interval and a situation where you obtain a meaningful result by averaging two quantities and multiplying by a time interval. Be sure to explain your results in terms of units and also in terms of meanings.

2.  In terms of two simple examples involving velocity, distance and time, explain the difference between the meaning of the slope of a trapezoid and the area of a trapezoid. Be sure to explain your results in terms of units and also in terms of meanings.

In terms of two simple examples involving money, pay rate and time, explain the difference between a situation where you obtain a meaningful result by subtracting two quantities and dividing by a time interval and a situation where you obtain a meaningful result by averaging two quantities and multiplying by a time interval. Be sure to explain your results in terms of units and also in terms of meanings.

3.  In terms of two simple examples involving depth, rate of depth change and time, explain the difference between the meaning of the slope of a trapezoid and the area of a trapezoid. Be sure to explain your results in terms of units and also in terms of meanings.

In terms of two simple examples involving temperature, rate of temperature change and time, explain the difference between a situation where you obtain a meaningful result by subtracting two quantities and dividing by a time interval and a situation where you obtain a meaningful result by averaging two quantities and multiplying by a time interval. Be sure to explain your results in terms of units and also in terms of meanings.

4.  In terms of two simple examples involving temperature, rate of temperature change and time, explain the difference between the meaning of the slope of a trapezoid and the area of a trapezoid. Be sure to explain your results in terms of units and also in terms of meanings.

In terms of two simple examples involving depth, rate of depth change and time, explain the difference between a situation where you obtain a meaningful result by subtracting two quantities and dividing by a time interval and a situation where you obtain a meaningful result by averaging two quantities and multiplying by a time interval. Be sure to explain your results in terms of units and also in terms of meanings.

5.  Explain how y ' (t) and y ' (t0) can be used to estimate y(t+Dt). Use a labeled picture as well as a diagram.

6.  Explain in terms of rates and amounts why and how, given the values of a derivative function y ' (t) at two nearby t values, we can estimate the change in its antiderivative y(t) between these t values by averaging two values of y ' (t) and multiplying by the corresponding time interval.

7.  Explain in terms of rates and amounts why and how, given two values of a function y(t), we can estimate the value of its derivative function y ' (t) in the vicinity of the two corresponding t values by subtracting the two y values and dividing result by the corresponding time interval.

8.  Explain in terms of rates and amounts why and how, given two nearby values of a derivative function y ' (t), we can estimate the change in its antiderivative y(t) between the corresponding t values by calculating the area of a trapezoid on the graph of y ' (t) vs. t.

9.  Explain in terms of rates and amounts why and how, given two values of a function y(t), we can estimate the value of its derivative function y ' (t) in the vicinity of the two corresponding t values by calculating the slope of a trapezoid on a graph of y(t) vs. t.

10.  Explain the difference between a situation in which you would do each of the following; if there is no such situation for a given item tell why:

11.   Explain how to get a tangent-line approximation to the function y(t) at the t = t0 point. Use a labeled sketch in your explanation.

12.  Explain why the expression [ y(t0+Dt) - y(t0) ] / Dt represents the average rate of change of y with respect to t, between t0 and t+Dt, and why the limiting value of this expression, as Dt -> 0, represents the instantaneous rate of change of y with respect to t at t = t0.