Warm UpDetermine

a) ∞b) 0c) ½ d) 3/10 e) 1

2.4 – Rates of Change and Tangent Lines

GOAL

I will be able to calculate the slope of a line tangent to a curve through the

definition of the slope of a curve

Average rate of change Average rate of change on an interval [a,b] is (slope)

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

• Need to find f(1) and f(3)• f(1)=0 and f(3)=24

• Answer= 12

You try• Find the average rate of change of on [1,6]

• Answer: 1/5

Remember Geometry?• Secant line is a line which passes through at least 2

points on a curve

• Tangent line is a line which passes through exactly one point on a curve.

Average rate of change=slope of the secant line

Instantaneous rate of change = slope of tangent line

Example• Find the slope of the parabola at the point (2,4). Write an equation

for the line tangent to this point.

• Insert pic from page 89

Solution• We will first find the slope. In order to find the slope, we need 2

points, so we will use the point given, and some other nearby point on the curve

• P(2,4) and Q(

• Slope=

Continued

• =

• Slope of the tangent line is the

• =

Continued•What 2 things do you need for an equation of a line?•Now, we have a slope, and a point (given at beginning)

so our answer is

In general: The slope of the curve f(x) at the point (a, f(a)) is

Provided the limit exists.

You try• Find the slope of the curve f(x) = x2 + x at x = 5.

•When you are asked to find the slope at a point a, use the interval [a, a+h]. So in this case, use [5, 5+h]

• Find f(5) and f(5+h). Then plug those into the slope equation.

Solution

Example If the function f given by f (x) = x2+ x has an average rate of change of 7 on the interval [0, k], then k = ?

(a) -8(b) 2(c) 6(d) 30(e) k cannot be determined from the information given.

Definition• The normal line to a curve at a point is the line perpendicular to the

tangent line at that point.

• So: to find the equation of the normal line, follow the process of finding slope, then use the opposite reciprocal.

Time check!

Example• Find the slope of f(x) = x2 + x at the point x = a.

Continued• When is the slope equal to 9?

• 9= 2a+1

• At a=4

Continued• What is the equation of a line tangent to the curve at this point?

• I know the slope, and the x value. I need the y value, so plug x into f(x)• so point is (4,20)

• Equation: y-20=9(x-4)

Continued• When is the tangent horizontal?

• 0=2a+1

Homework 2.4