AP Calculus BC

Chapter 2

2.1 Rates of Change & Limits

Average Speed = distanceelapsed time Instantaneous Speed is at a

specific time - derivative

Rules of Limits:1. If you can plug in the value,

plug it in.2. Answer to a limit is a y-value.3. Holes can be limits.

lim ( )x c f x L

Sandwich Theorem

0sinlim

xxx

1Example: A rock is dropped off a cliff. The equation: 216y t

Models the distance the rock falls. FIND:1. The average speed during the 1st 3 seconds.2. The Instantaneous speed at t=2 sec.

2.1 cont’d.#1 – slope, #2 – Definition of Derivative

0( ) ( )lim '( )

hf x h f x f x

h

Properties of Limits:1. Sum/Difference2. Product3. Constant Mult.4. Quotient5. Power

lim ( ) ( )

lim ( ) ( )

lim ( )

( )lim( )

lim( ( ))

x c

x c

x c

x c

r rs s

x c

f x g x L M

f x g x L M

kf x kL

f x LMg x

g x M

lim ( )

lim ( )x c

x c

f x L

g x M

GIVEN:

1-sided limits & 2-sided limits

lim ( )

lim ( )

lim ( )

x c

x c

x c

f x

f x

f x

Rt. HandLeft HandOverall

Do some examples, including Step-Functions

2.2 Limits involving InfinityHorizontal Asymptote occurs if: lim ( )

xf x b

H.A. –> y = b

Compare Powers:( )lim( )xN xD x

If N(x)=D(x)-> y = coeff.

If N(x)<D(x) -> y = 0

If N(x)>D(x) -> y = slant (use leading terms)

Infinity as an answer:

lim ( )

lim ( )

x a

x a

f x

f x

Then, x = a is a V.A.

End-Behavior Models:Right & Left End Models

lim

lim

x

x

or

2

( ) 3

( ) 3 7

xf x e x

g x x x

2.3 ContinuityBeing able to trace a graph without lifting your pencil off the paper.

Draw a graph, answer questions. 2-sided limits, 1-sided limits.

Continuity at a point:Interior point:Rt.End point:Left End point:

lim ( ) ( )

lim ( ) ( )

lim ( ) ( )

x c

x c

x c

f x f c

f x f c

f x f c

Types of Discontinuities

Removable Jump

InfiniteOscillating

A continuous (cts.) function is cts. at every point in its domain.

An example of an extended function.Composition of functions.

Intermediate Value Thm. for cts. Functions.

2.4 Rates of Change & Tangent Lines

Average Rate of Change:(think : SLOPE)

( ) ( )f b f ab a

Definition of the Derivative: 0( ) ( )lim '( )

hf x h f x f x

h

The first derivative will give you the slope of the tangent line at any x-value.

Normal Line is perpendicular to the Tangent Line

Examples:Find the T.L. and N.L. at x = 1.

2

2

( ) 4

( ) 4 5 10

f x x

f x x x

Long way