Today in Calculus

• Go over homework

• Average Rate of Change (aka slope)

• Tangent Lines

• Homework

Even Answers12a) Yes, f(1)=1

b) Yes,

c, d, e: No

14.Everywhere in[-1,3) except x =0,1,2

16.Reassign f(1)=2 based on the limit

24. 1,2; x=2 removable

26.

lim ( ) 2xf x

2 11,

1

x xFor x y

x

Average Rate 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.

• In general: average rate of change of a function over an interval is the amount of change divided by the length of the interval.

Example: Find the average rate of change of f(x)=x3 – x over the interval [2,4]

Example: Find the average rate of change of f(x)=4x2 – 6x over the interval [1,3]

0

( ) ( )lim

( )h

x af x h f x

So m x ah

f x x a

slope of tangent line at

instantaneous rate of change at

slope of at

ExampleIf y = 2x2, find the slope of the curve and the equation of the line tangent to the curve at x = 3

Example

If , find the slope of the curve and the equation of the line tangent to the curve at x = 2

Find the slope of f(x) at x = 7, x = -3.

At what point is the slope of the tangent line -.25?

At what point is the tangent line horizontal?

1( )

3f x

x

Homework

• Pg 87-88: 1-6, 9-10(a & b only)