Good things to know for the AP calculus test


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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

Volume of cones


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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

Volume of pyramids


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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

HHG p159 num23


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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

HHG p159 num22


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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

HHG p159 num21


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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

Riemann sum graphics


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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

Graphing slope fields


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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