Lawn Work with Calculus D: By: Anand Veeraswamy Pd. 7.

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Presentation transcript:

Lawn Work with Calculus D: By: Anand Veeraswamy Pd. 7

The Meaning of Calculus In the most simplistic form, calculus can be defined as either the rate of change or accumulation. The names of these types of calculus is known as differential and integral calculus respectively. When these two types of calculus are used together, they are known as the fundamental theorems of calculus. Two major mathematical scholars known as Isaac Newton and Gottfried Leibniz accidentally stumbled upon similar creations of the same mathtype and commonly accused each other of plagiarism. However, as a result of their work, we have the commonly taken math course of calculus.

Definitions: The Average Value Theorem:

Free Response Question (Question #1, 2014) Grass clippings are placed in a bin, where they decompose. For the amount of grass clippings remaining in the bin is modeled by, where A(t) is measured in pounds and t is measured in days. (a) Find the average rate of change of A(t) over the interval. Indicate units of measure. (b) Find the value of A′(15). Using correct units, interpret the meaning of the value in the context of the problem. (c) Find the time t for which the amount of grass clippings in the bin is equal to the average amount of grass clippings in the bin over the interval (d) For t > 30, L(t), the linear approximation to A at t = 30, is a better model for the amount of grass clippings remaining in the bin. Use L(t) to predict the time at which there will be 0.5 pound of grass clippings remaining in the bin. Show the work that leads to your answer.

(a) Find the average rate of change of A(t) over the interval. Indicate units of measure.

(b) Find the value of A′(15). Using correct units, interpret the meaning of the value in the context of the problem. The amount of grass clippings in the bin is decreasing at a rate of (or 0.163) lbs/day at time t=15 days.

(c) Find the time t for which the amount of grass clippings in the bin is equal to the average amount of grass clippings in the bin over the interval

(d) For t > 30, L(t), the linear approximation to A at t = 30, is a better model for the amount of grass clippings remaining in the bin. Use L(t) to predict the time at which there will be 0.5 pound of grass clippings remaining in the bin. Show the work that leads to your answer.

Additional Graphs:

References AP College Board Virtual TI-83 emulator Wolfram Alpha