Determining Rates of Change from an Equation

1
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.

2
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).

3
Temperature Example revisited
4
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 yf(x).

5
Average rate of change

• What was the average rate of change of the
  temperature with respect to time from t0s to20s
  ?

How did we do this yesterday?
The average rate of increase in temperature is
2.5 degrees per second.
6
Estimating instantaneous rate of change

• Estimate the instantaneous rate of increase in
  temperature at t35s.
• 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.

7
Use a 1 second interval.

• Use a 1-s following interval.
• Use t35s and t35136s

The instantaneous rate of change at t35s is
approximately 0.649351 degrees per
second. Compare that to 0.67 using a 5 second
interval.
8
You try!

• Use a 0.1s following interval to estimate the
  instantaneous rate of change at t35s.

The instantaneous rate of change at t35s is
approximately 0.65996 degrees per second.
9
Example 2

• Given the function y2x3
• a) Find the average rate of change from x0 to
  x1.

The average rate of change from x0 to x1 is 2.
10

• Given the function y2x3
• b) Find the instantaneous rate of change at x0.5

Use a 0.1s following interval.
The instantaneous rate of change at x0.5 is
1.82.
11
Difference Quotient
Why is called the difference quotient?
12
You try

• Given the function

Estimate the instantaneous rate of change of y
with respect to x at x6.