Determining Rates of Change from an Equation Recall that we were able to determine the average rate of change of a quantity by calculating the slope of the secant line joining two points on the curve. We were also able to estimate the instantaneous rate of change in four ways: Drawing a tangent line and then using two points on this tangent line, calculate the slope of the tangent line. Estimate the slope of the tangent line by calculating the slope of the secant line using a small preceding interval and the given table of values. Estimate the slope of the tangent line by calculating the slope of the secant line using a small following interval and the given table of values. Estimate the slope of the tangent line by calculating the slope of the secant line using a small centered interval and the given table of values.
Using an Equation We can do all of these things again today but without having the graph or the table of values. Instead we will use a formula for our calculations. This formula gives us more flexibility because we can calculate the y-value (Temperature) for any x-value (time).
Temperature Example revisited
Example 2 continued The formula for the Temperature-time graph is given by: This is written using function notation. T(t) is read as T as a function of t, or T of t. Later we will see examples such as f(x)=3x. This is read as f of x equals 3x. To find the value of the function when x is 4, we write f(4)=3(4). We say f of 4 =12. You just substitute 4 for the x. In this case y=f(x).
Average rate of change What was the average rate of change of the temperature with respect to time from t=0s to=20s ? How did we do this yesterday? The average rate of increase in temperature is 2.5 degrees per second.
Estimating instantaneous rate of change Estimate the instantaneous rate of increase in temperature at t=35s. Yesterday we used a 5s interval because those were the only values we had in the table. With the formula we can calculate any value so we can use a smaller interval and get more accurate estimate.
Use a 1 second interval. Use a 1-s following interval. Use t=35s and t=35+1=36s The instantaneous rate of change at t=35s is approximately 0.649351 degrees per second. Compare that to 0.67 using a 5 second interval.
You try! Use a 0.1s following interval to estimate the instantaneous rate of change at t=35s. The instantaneous rate of change at t=35s is approximately 0.65996 degrees per second.
Example 2 Given the function y=2x3 a) Find the average rate of change from x=0 to x=1. The average rate of change from x=0 to x=1 is 2.
b) Find the instantaneous rate of change at x=0.5 Given the function y=2x3 b) Find the instantaneous rate of change at x=0.5 Use a 0.1s following interval. The instantaneous rate of change at x=0.5 is 1.82.
Difference Quotient Why is called the difference quotient?
You try Given the function: Estimate the instantaneous rate of change of y with respect to x at x=6.