Section 2.4b. The “Do Now” Find the slope of the given curve at x = a. Slope:

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Presentation transcript:

Section 2.4b

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?

From physics, with an object in free-fall on Earth, the position function is 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 instantaneous rate of change of its position with respect to time at time t :

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:

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: The rock’s speed at the instant t = 1: ft/sec Average speed of the rock over the interval between t = 1 and t = 1 + h sec:

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

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:

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: The volume is changing at a rate of 16 in per inch of radius Rate of change of this function: 3

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

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???

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

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

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