Presentation on theme: "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."— Presentation transcript:
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 1 st 20 minutes?
When the coffee was 1 st 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. This will be the average change in temperature.
Find the instantaneous rate of change of the temperature of the coffee at 5 min.? The slope of the tangent line gives the instantaneous rate of change.
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.