 Rate of change and tangent lines

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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.

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.?

A tangent line in geometry is a line that touches a circle in exactly one point. This is not always the same in calculus. Think of it as a line that goes in the direction of the curve if you zoom really close to the curve. When you zoom in close enough to any curve it will appear to be a straight line. This line is the same as the tangent line at that point.

Click Here to see an example of tangent lines

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

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.

Lets find the instantaneous rate of change at x=3
That is find the slope of the tangent line at x=3

In the last example I drew in the tangent line and found some points on it to find the slope. This time we will find it a little more exact.

The point (3,0) is on the graph A generic point on the graph is
Plug 3 into the equation The point (3,0) is on the graph A generic point on the graph is

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

What happens when we slide the generic point closer to the point (3,0)?
Click on when to see

You can graph this on your calculator to find the limit.
If we 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 This is the instantaneous rate of change at x=3, the slope of the tangent at x=3 and the derivative at x=3

Now lets find the derivative another way.
Graph the original function and not the slope of the secant like we did previously

Lets Review Find the average rate of change over the given interval.

Find the derivative at x = 3 using the limit then check with the calculator.
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