2The 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
3This 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?
4When the coffee was 1st made the temperature was After 20 minutes the temperature was
5The line connecting these two points is a secant line.
6Find the slope of this line Find the slope of this line. This will be the average change in temperature.
7The 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.?
8A 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.
14Find the slope of the secant The average rate of change from [-3,2] is 3.
15Lets find the instantaneous rate of change at x=3 That is find the slope of the tangent line at x=3
16In 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.
17The point (3,0) is on the graph A generic point on the graph is Plug 3 into the equationThe point (3,0) is on the graphA generic point on the graph is
18Find the slope of the secant line between the point (3,0) and the generic point
19What happens when we slide the generic point closer to the point (3,0)? Click on when to see
20You 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.
21You 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
22Now lets find the derivative another way. Graph the original function and not the slope of the secant like we did previously