2.4 Rates of Change and Tangent Lines

Calculus

Finding average rate of change

• Find the average rate of change of over the interval [1, 3].

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Slope of a secant line

• Use points P(23, 150) and Q(45, 340) to compute the average rate of change and the slope of the secant line PQ. • 8.6 flies/day•We can always think about average rate of change as the slope of a secant line.

Instantaneous rate of change

• What about the growth of the population on day 23? We move point Q closer to point P to get a better estimate.

• Notice the secant line appears to be approaching the tangent line.

• So we could use the slope of the tangent line as the instantaneous rate of change at

Steps for finding the slope of the tangent

1. Start with what we can calculate- the slope of the secant through a point P and a point nearby (Q) on the curve.

2. Find the limiting value of the secant slope (if it exists) as Q approaches P along the curve.

3. Define the slope of the curve at P to be this number and define the tangent to the curve at P to be the line through P with this slope.

Definition: Slope of a curve at a point

• The expression is the difference quotient of f at a.

Example: Finding slope and tangent line

• Find the slope of the parabola at the point P(2, 4). Write an equation for the tangent to the parabola at this point.

Example:

• Find the slope of the curve at .

• Where does the slope equal -1/4?

Lines normal to a curve

• The normal line to a curve at a point is the line perpendicular to the tangent at that point.•Write an equation for the normal to the curve at

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Free fall…again

• Find the speed of the falling rock (discussed earlier in this chapter) at sec. • Remember:

• 32 ft/sec