Rates of Change and Tangent Lines

Section 2.4

Average Rates of Change• The average rate of change of a quantity

over a period of time is the amount of change divided by the time it takes.

• Example: Find the average rate of change of f(x) = x3 – x over the interval [1, 3]

A line through two points on a curve is a secant to the curve.

The average rate of change is the slope of the secant line.

Use the points (2, 0.368) and (5, 2.056) to compute the average rate of change and the

slope of the secant line.

A line through one point on a curve is a tangent to the curve.

The slope of the tangent line is the rate of change at a particular point.

See p.83

Defining Slopes & Tangents of Curves

• Find the slope of a secant through two points P and Q on a curve.

• Find the limiting value of the secant slope as Q approaches P along the curve.

• 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.

Find the slope of the parabola y = x2 at the point P (2, 4). Write an equation for the tangent to the

parabola at this point.

The slope of the curve y = f(x) at the point P(a, f(a)) is the number

m limh 0

f (a h) f (a)

h

provided the limit exists.

The tangent line to the curve at P is the line through P with this slope.

Let f(x) = 1/x.

Find the slope of the curve at x = a.Where does the slope equal -1/4?

What happens to the tangent to the curve at the point (a, 1/a) for different values of a?

Difference quotient of f at a

f (a h) f (a)

h

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 f(x) = 4 – x2 at x = 1.

Speed Revisited

• Position function: y = f(t)• Average rate of change of position: average

speed along a coordinate axis for a given period of time

• Instantaneous speed: instantaneous rate of change of position with respect to time at time t, or

limh 0

f (t h) f (t)

h

Position function: y = f(t) = 16t2

Find the speed of the falling rock at t = 1 sec

pages 87-89 (2-32 even)========================

pages 87-89 (3, 7, 9, 11, 13, 17,

21, 25, 27, 29)========================

pages 91-93 (2-24 even, 25-30, 32-40 even, 43, 44, 46-50, 52)