Calculus Notes 3.4: Rates of Change in the Natural and Social Sciences. Start up: This section discusses many different kinds of examples. What is the.

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Presentation transcript:

Calculus Notes 3.4: Rates of Change in the Natural and Social Sciences. Start up: This section discusses many different kinds of examples. What is the main idea underlying them all? A particle moves along the y-axis so that its position at time t is given by . For what value of t is the velocity of the particle zero? Answers: All of the examples involve expressing quantities as an average rate of change, and then using the idea of the derivative to compute an instantaneous rate of change. Derivative:

Calculus Notes 3.4: Rates of Change in the Natural and Social Sciences. Example 1: Suppose C(x) is the total cost that a company incurs in producing x units of a certain commodity. The function C is called the cost function. The instantaneous rate of change of cost with respect to the number of items produced, is called the marginal cost by economists. What is the actual cost of producing the 501st item?

Calculus Notes 3.4: Rates of Change in the Natural and Social Sciences. Example 2: A particle moves according to a law of motion s=f(t), t≥0, where t is measured in seconds and s in feet. a. Find the position at t=1, t=2, t=3, and t=6. b. Find the velocity at time t. c. Find the velocity at t=2 and t=4. d. When is the particle at rest?

Calculus Notes 3.4: Rates of Change in the Natural and Social Sciences. Example 2: A particle moves according to a law of motion s=f(t), t≥0, where t is measured in seconds and s in feet. e. When is the particle moving forward (that is, in the positive direction)? f. Find the total distance traveled on the intervals [0,1], [0,2], [0,3], [0,6] g. When is the particle speeding up? Slowing down? Speeding up: Slowing down: PS 3.4 pg.166 #2, 3, 8, 12, 15, 18, 20, 22, 29, 30 (10)