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!n$t@nt@n30u$ @nd @v3r@g3 R@t3$ 0f Ch@ng3

!n$t@nt@n30u$ @nd @v3r@g3 R@t3$ 0f Ch@ng3. @l3x J0n3$ @nd m@ry b33ch @k@: A + @13X & M@1!c!0u$ M@rY. http://bedesblog.files.wordpress.com/2009/04/pennies1.jpg. @v3r@g3 R@t3 0f Ch@ng3. Definition: rate of change of the dependent variable on an interval of the independent variable.

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!n$t@nt@n30u$ @nd @v3r@g3 R@t3$ 0f Ch@ng3

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  1. !n$t@nt@n30u$ @nd @v3r@g3 R@t3$ 0f Ch@ng3 @l3x J0n3$ @nd m@ry b33ch @k@: A + @13X & M@1!c!0u$ M@rY http://bedesblog.files.wordpress.com/2009/04/pennies1.jpg

  2. @v3r@g3 R@t3 0f Ch@ng3 • Definition: rate of change of the dependent variable on an interval of the independent variable For example, in the graph to the left, the average rate of change of f(x) from x = 0 to x = 1 is the change in f(x) divided by the change in x.

  3. @v3r@g3 R@t3 0f Ch@ng3 • For a car moving forward, the position of the car as a function of time can be graphed. The average rate of change is the change in distance divided by the change in time. • This represents the average speed of the car. http://s.chakpak.com/se_images/136705_-1_564_none/cool-car-wallpaper.jpg

  4. Gr@ph!c@1 R3pp!n’ The average rate of change can be represented graphically by drawing a secant line (a line between two different points-shown in green) and calculating its slope. Calculate the slope using: where a and b are the endpoints of the interval Substitute in values found on graph to get: The average change from x = 0 to x = 1 for this graph is 4

  5. @1g3br@!c R3pp!n’ • Use the function to evaluate the slope of the secant line from x = 0 to x = 1 by calculating the slope between the two points • f(x) = x3+2x2+x+1 • f(0) = 03+2(02)+0+1 = 1 • f(1) = 13+2(12)+1+1 = 5 • Calculate the slope using these values: Picture from http://images.buycostumes.com/mgen/merchandiser/27253.jpg

  6. Num3r!c@1 R3pp!n’ • For the function: f(x) = x3+2x2+x+1 a table of values is given below: Use the equation for slope to evaluate the slope from x = 0 to x = 1 using the values given in the table:

  7. !n$t@nt@n30u$ R@t3 0f Ch@ng3 • Definition: the rate of change of the dependent variable at a specific value of the independent variable For example, for the graph to the left, the instantaneous rate of change at x = 0 is the exact slope of the graph at x = 0. This value cannot be determined using only the function, but there are numerous ways to approximate this value (and later you will learn how to calculate this exactly).

  8. !n$t@nt@n30u$ R@t3 0f Ch@ng3 • For a car moving forward, the instantaneous rate of change of its position over time is the exact speed of the car at a specific time. This is impossible to calculate exactly using the position-time graph, but can be approximated. However, for a velocity-time graph for the car, the speed would be the y-value at a point, but the instantaneous rate of change of velocity would be the acceleration. http://bigpicture.typepad.com/writing/images/peugeot_908_rc.jpg

  9. Gr@ph!c@1 R3pp!n’ The instantaneous rate of change at x = 0 can be represented graphically by drawing a tangent line (a straight line that touches the curve only at one point) at a given point and calculating its slope. Calculate the slope for two estimated points on the tangent line using the equation: Substitute in values found on graph for the points (-1, 0) and (1, 2) to get: The approximated instantaneous rate of change at x = 0 for this graph is 1

  10. Gr@ph!c@1 R3pp!n’ • The instantaneous rate of change can be approximated from the average rate of change by using the slope of the secant line as the change in x goes to 0 (giving you the tangent line mentioned before).

  11. @1g3br@!c R3pp!n’ • Use the method for algebraically calculating average rate of change. Do this multiple times with smaller and smaller intervals around the given point to estimate the instantaneous rate of change.

  12. @1g3br@!c R3pp!n’ • f(x) = x3+2x2+x+1 • x = 0 to x = 0.01: • x = 0 to x = 0.001: • As the x-interval around x = 0 gets smaller, the values of the average rate of change go from 1.0201 to 1.002001, so they approach 1. • This is the limit of the average rate of change as the interval of x-values decreases to 0.

  13. Num3r!c@1 R3pp!n’ • For a mass oscillating on a spring, the distance (in ft) of the mass from the ground is recorded each second. A sample of the data is shown below: Approximate the instantaneous rate of change of the mass’s position at t = 4 s. Use the equation for slope to evaluate the slope around t = 4 using the values given in the table (this will give you a rough estimate of the instantaneous rate of change): The instantaneous rate of change at t = 4 s is approximately -2 ms-1 (the negative signifies that the distance between the mass and the ground is decreasing). *Note that this is also the average rate of change from t = 2 s to t = 4 s

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