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Lesson 3-2

Lesson 3-2. Difference Quotients and One Definition of Derivatives. Remember the Automatic Door Opener problem from our very first lesson. We first discussed finding the average speed. Graphically, the Average Rate of Change is represented by the secant line . Recall.

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Lesson 3-2

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  1. Lesson 3-2 Difference Quotients and One Definition of Derivatives

  2. Remember the Automatic Door Opener problem from our very first lesson. We first discussed finding the average speed. Graphically, the Average Rate of Change is represented by the secant line.

  3. Recall The instantaneous rate of change of the dependent variable (d) is the limit of the average rates as the time interval gets closer to zero. Graphically, the Instantaneous Rate of Change is represented by the tangent line. This limiting value is called the derivative of the dependent variable with respect to the independent variable.

  4. Calculating rates of door speed Recall The limit of these calculations is a derivative... As we close in on a change in t = 0 the instantaneous rate doesn't change much.

  5. Definition of Derivative(at x = c form) There is a relationship, yet a difference, between the original function f(x), and the function “derived” from it f(c), which indicates its rate of change.

  6. Graphically

  7. Example of finding the derivative using the limit of the difference quotient. Find the value of the derivative at x = -1.

  8. So, f ’(-1) = 1 Let’scheckitout graphically! Window: x(-5,5) xscl = 1 y(-5,5) yscl = 1 Trace to x = -1, observe y =

  9. Take a look at its table: Press TBLSET tblStart…………………..-1.004 Δtbl……………………….0.0001 Be sure Independent…….Auto Press TABLE

  10. Graphic Check Window: x(-5,5) xscl = 1 y(-5,15) yscl = 1 Point-Slope Form of an equation What's the slope of the tangent line?

  11. Example of finding the derivative using the limit of the difference quotient. Free Fall on JupiterThe equation for free fall at the surface of Jupiter is with t in seconds. Assume a rock is dropped from the top of a 500 meter cliff. Find the speed of the rock at t = 2 sec. So, its falling at 45.76 m/s

  12. Example of finding the derivative using the limit of the difference quotient. Volume of a SphereWhat is the rate of change of the volume of a sphere with respect to the radius when the radius is r = 2 in? Ω

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