Definitions: A critical point of a function is any point where the derivative is zero, or is not differentiable. Statements: The average value of the function on the interval is The average slope (or rate of change) of the function … Continue reading

# Tag Archives: calculus

The volume of a cone can be approximated by carving the cone into slices, and approximating each slice with a cylinder. To then set it up as a Riemann sum, it helps to orient the cone so that we can … Continue reading

The volume of a pyramid can be approximated by carving the pyramid into slices, and approximating each slice with a rectangular cuboid. To then set it up as a Riemann sum, it helps to orient the pyramid so that we … Continue reading

The graph of the function is below. Which of the follow four numbers could be an estimate of ? (a) -98.35 (b) 71.84 (c) 100.12 (d) 93.47 Solution: Area is not negative, since graph is above the -axis. It completely … Continue reading

Suppose that we use subintervals to approximate . Without computing the Riemann sums, find the difference between the right-and left-hand Riemann sums. Solution: If the intervals are assumed to be the same width, then the last 499 terms of the … Continue reading

Without computing the integral, decide if is positive or negative, and explain your decision. Solution: We know that (in blue) is positive but decreasing. On the other hand, (in green) is positive on , and then repeats itself (but negative) … Continue reading

For practicing understanding the implications of lefthand vs. righthand (or lower bound vs. upper bound integrals), it's nice to be able to graph a function with a grid suited to the task. I plotted a portion of using Mathematica. f[x_] … Continue reading

There are lots of bells and whistles available in Mathematica, but sometimes I like something really plain. For example, I wanted my students to find slope analytically given two points. On graph paper with a full grid, they could count … Continue reading

In physics and calculus, we study radioactive decay processes. So as a lab, we could play around with some radioactive isotopes! For a variety of reasons, I settled for a simulation. That gives us the luxury of being able to … Continue reading

Most of the functions we look at in high school math are continuous everywhere they are defined. But not always. Graphing piecewise functions with software requires defining the function on distinct intervals. Grapher (on the Mac) provides a nice facility … Continue reading

Mathematica, WolframAlpha, and Grapher provide tools for solving differential equations. But you don't get to see what's happening under the hood. Differential equations often don't have a tidy closed form solution, and a numerical method is needed. Euler's method requires … Continue reading

If you are manually graphing slope fields, here are some scalable graphics: Slope field, grid Slope field, grid You can plot slope fields with a solution to the differential equation in Grapher (Mac only). If you haven't used Grapher, it … Continue reading

We make lots of graphs. Sometimes I embed the graphs in the handout. But other times, such as for creating graphs that will be cut out and attached to a group chart, a separate piece of graph paper is better. … Continue reading

Written solutions are available at the AP Exam question bank. Whiteboard screencasts for these problems: 2007 AP Free-response Form B, problem 1 2007 AP Free-response Form B, problem 2 2007 AP Free-response Form B, problem 3 If you would like … Continue reading

We use the second edition of this reformed calculus text for Calculus AB at Ypsilanti High School. Solutions to problems p 176, #6 p 176, #10 … Continue reading