Rate of change and tangent lines

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Presentation transcript:

Rate of change and tangent lines

The average rate of change of a function over an interval is the amount of change divided by the length of the interval. On a graph this is equal to the slope of a secant line

This graph shows the temperature of a cup of coffee over a 30 minute period. What is the average rate the coffee cools during the 1st 20 minutes?

When the coffee was 1st made the temperature was After 20 minutes the temperature was

The line connecting these two points is a secant line.

Find the slope of this line Find the slope of this line. This will be the average change in temperature.

Find the average rate of change of Over the interval

Find the slope of the secant Find the average rate of change of Over the interval Find the slope of the secant

Find the slope of the secant The average rate of change from [-3,2] is 3.

The slope of the tangent line gives the instantaneous rate of change. Find the instantaneous rate of change of the temperature of the coffee at 5 min.?

The slope of the tangent line at x=5 will give the instantaneous rate of change

The slope of the tangent line will give the instantaneous rate of change

Click Here to see an example of tangent lines

Lets find the instantaneous rate of change at x=3 That is find the slope of the tangent line at x=3 We are going to write an expression for the slope of the secant from the point where x=3 to a second point. We will then look at what happens to the slope of the secant as the second point moves closer to (3,0). Demonstration

Plug 3 into the equation Find the slope of the secant line between the point (3,0) and the generic point

Find the slope of the secant line between the point (3,0) and the generic point

You can graph this on your calculator to find the limit. Take the limit of the slope of the secant line as the x value of the generic point gets closer to 3. You can graph this on your calculator to find the limit.

You can graph this on your calculator to find the limit. Take the limit of the slope of the secant line as the x value of the generic point gets closer to 3. You can graph this on your calculator to find the limit.

You can graph this on your calculator to find the limit. Take the limit of the slope of the secant line as the x value of the generic point gets closer to 3. You can graph this on your calculator to find the limit.

You can graph this on your calculator to find the limit. Take the limit of the slope of the secant line as the x value of the generic point gets closer to 3. You can graph this on your calculator to find the limit.

You can graph this on your calculator to find the limit. Take the limit of the slope of the secant line as the x value of the generic point gets closer to 3. You can graph this on your calculator to find the limit.

You can graph this on your calculator to find the limit. Take the limit of the slope of the secant line as the x value of the generic point gets closer to 3. You can graph this on your calculator to find the limit.

You can graph this on your calculator to find the limit. Take the limit of the slope of the secant line as the x value of the generic point gets closer to 3. You can graph this on your calculator to find the limit.

You should have gotten -12 You should have gotten -12. This is the instantaneous rate of change at x=3, the slope of the tangent at x=3 and the derivative at x=3

Find the derivative at x = 3 using the limit of the slope of the secant. 7

Let’s Review Find the average rate of change over the given interval.

Find the derivative at x = -1 using the limit of the slope of the secant. -(-1-3) 4

A ball is thrown straight up from a rooftop 160 feet high with an initial velocity of 48 feet per second. The function Describes the balls height above the ground, s(t), in feet, t seconds after it is thrown. The ball misses the rooftop on its way down and eventually strikes the ground. What is the average velocity of the ball from 0 to 4 seconds? What is the instantaneous velocity at 3 seconds?

Worksheet

Find the equation of the tangent line at x=2 First find the derivative at x=2 which is the slope of the tangent line.

Find the equation of the tangent line at x=2

Find the equation of the tangent line at x=0 First find the derivative at x=0 which is the slope of the tangent line.

Find the equation of the tangent line at x=0

Wkst II