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Modeling with rates of change

Modeling with rates of change. Section 2.4a. The “Do Now”. Find the slope of the given curve at x = a. Slope:. The “Do Now”. Find the slope of the given curve at x = a. Bonus Question: What is the equation of the tangent to this curve at x = –2?.

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Modeling with rates of change

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  1. Modeling with rates of change Section 2.4a

  2. The “Do Now” Find the slope of the given curve at x = a. Slope:

  3. The “Do Now” Find the slope of the given curve at x = a. Bonus Question: What is the equation of the tangent to this curve at x = –2?

  4. From physics, with an object in free-fall on Earth, the position functionis given as (y is feet fallen after t seconds) A body’s average speed along a given axis for a given period of time is the average rate of change of this position function. Its instantaneous speed at any time t is the instantaneousrate of change of its position with respect to time at time t :

  5. Guided Practice Find the speed of a falling rock (acted on by Earth’s gravity only) at t = 1 sec. Position function of the rock: Average speed of the rock over the interval between t = 1 and t = 1 + h sec:

  6. Guided Practice Find the speed of a falling rock (acted on by Earth’s gravity only) at t = 1 sec. Position function of the rock: Average speed of the rock over the interval between t = 1 and t = 1 + h sec: The rock’s speed at the instant t = 1: ft/sec

  7. Guided Practice At t sec after lift-off, the height of a rocket is ft. How fast is the rocket climbing after 10 sec? This is the position function for the rocket… we seek the instantaneous rate of changeof this function at t = 10… Let Speed: The rocket’s speed at t = 10 sec is 60 ft/sec

  8. Guided Practice What is the rate of change of the volume of a sphere with respect to the radius when the radius is r = 2 in.? We need a function for the volume with respect to the radius: Rate of change of this function:

  9. Guided Practice What is the rate of change of the volume of a sphere with respect to the radius when the radius is r = 2 in.? We need a function for the volume with respect to the radius: Rate of change of this function: The volume is changing at a rate of 16 in per inch of radius 3

  10. Guided Practice At what point is the tangent to the given function horizontal? First, find the slope of the tangent at x = a:

  11. Guided Practice At what point is the tangent to the given function horizontal? First, find the slope of the tangent at x = a: The tangent at x = a is horizontal where the slope is zero: The tangent line is horizontal at the point: Can we support this answer graphically???

  12. Guided Practice Find the equations of all lines tangent to that passthrough the point (1, 12). First, sketch a graph of this situation. Slope of the curve at x = a:

  13. Guided Practice Find the equations of all lines tangent to that passthrough the point (1, 12). Our points: Slope of the tangent through these points: Equation for slope:

  14. Guided Practice Find the equations of all lines tangent to that passthrough the point (1, 12). At , the slope is Point-slope form (with point (1, 12)): The two tangent lines At , the slope is Point-slope form (with point (1, 12)):

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