Lesson 2-1: Rates of Change and Limits Day #1 AP Calculus Mrs. Mongold.

Lesson 2-1: Rates of Change and Limits Day #1

AP Calculus

Mrs. Mongold

Finding an Average Speed

• To find average speed you divide the distance covered by elapsed time.

• Solid objects dropped from rest will fall y = 16t2 feet in the first t seconds

• Average speed of the rock over any given time interval is the distance traveled, Δy, divided by the length of the interval Δt.

Definition of Free Fall

• Near the surface of the earth, all bodies fall with the same constant acceleration. The distance a body falls after it is released from rest is a constant multiple of the square of the time fallen. At least, that is what happends when a body falls in a vacuum, where there is no air to slow it down. The square-of-time rule also holds for dense, heavy objects like rocks, ball bearings, and steel tools during the first few seconds of fall through air, before the velocity builds up to where air resistance begins to matter. When air resistance is absent or insignificant and the only force acting on a falling body is the force of gravity, we call the way the body falls free fall.

Example

A rock breaks loose from the top of a cliff. What is the average speed during the first 2 seconds of fall?

Solution:

Δy 16(2)2 – 16(0)2 32 ft/sec

Δt 2 – 0

Finding Instantaneous Speed

• Find the speed of the rock in the previous example at the instant t = 2.

• Idea- calculate from t = 2 to slightly later time t = 2+h– Solution

Δy 16(2 + h)2 – 16(2)2

Δt h

Make a chart with values close to 0 because h ≠ 0 or the fraction would be undefined

h Δy/ Δt

1

0.1

0.01

0.001

0.0001

0.00001

Solution

• As h approaches 0 average speed approaches the limiting value 64 ft/sec.

• Algebraically

16(2 + h)2 – 16(2)2

h

Definition of Limit

• Let c and L be real numbers. The function f has a limit L as x approaches c if, given any positive number ɛ, there is a positive number δ such that for all x,

0 < |x – c| < δ => |f(x) – L| < ɛ

We write

* A limit is the “height” of a graph at a particular point, i.e. the y value

Lxfcx

)(lim

Limit From Example 2

64)1664(lim0

hh

Properties of Limits

• Theorem: If L, M, c, and k are real numbers and and then…Lxf

cx

)(lim Mxg

cx

)(lim


1. Sum Rule: The limit of the sum of two functions is the sum of their limits

2. Difference Rule: the limit of the difference of two functions is the difference of their limits

3. Product Rule: the limit of the product of two functions is the product of their limits

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim


4. Constant Multiple Rule: the limit of a constant times a function is the constant times the limit of the function

5. Quotient Rule: the limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero.

6. Power Rule: if r and s are integers, s≠0, then

provided Lr/s is a real number.

Lkxfkcx

))((lim

0,)(

)(lim

MM

L

xg

xfcx

sr

sr

cxLxf

))((lim

Example Using Properties of Limits

• Using the observations and and properties of limits to find the following limits.–

kkcx

lim cx

cx

lim

)34(lim 23

xxcx



kkcx

lim cx

cx

lim

)34(lim 23

xxcx

3lim4limlim 23

cxcxcxxx

Sum & Difference Rules



kkcx

lim cx

cx

lim

)34(lim 23

xxcx

3lim4limlim 23

cxcxcxxx

Sum & Difference Rules

34 23 cc


• Using the observations and and properties of limits to find the following limits.

kkcx

lim cx

cx

lim

5

1lim

2

24

x

xxcx



kkcx

lim cx

cx

lim

5

1lim

2

24

x

xxcx

5limlim

1limlimlim2

24

cxcx

cxcxcx

x

xx



kkcx

lim cx

cx

lim

5

1lim

2

24

x

xxcx

5limlim

1limlimlim2

24

cxcx

cxcxcx

x

xx

5

12

24

c

cc

Theorem Polynomial and Rational Functions

1. If f(x) = anxn + an-1xn-1 + ∙∙∙+ a0 is any polynomial function and c is any real number, then

01

1)()(lim acacacfxf nn

nn

cx

Theorem Polynomial and Rational Functions

1. If f(x) = anxn + an-1xn-1 + ∙∙∙+ a0 is any polynomial function and c is any real number, then

2. If f(x) and g(x) are polynomials and c is any real number , then

provided g(c)≠0

01

1)()(lim acacacfxf nn

nn

cx

)(

)(

)(

)(lim

cg

cf

xg

xfcx

Examples

• A) )]2([lim 2

3xx

x

Examples

• A)

32(2 – 3)

)]2([lim 2

3xx

x

Examples

• A)

32(2 – 3)

-9

)]2([lim 2

3xx

x

Examples

• A)

2

42lim

2

2

x

xxx

Examples

• A)

2

42lim

2

2

x

xxx

22

4)2(222

Examples

• A)

3

2

42lim

2

2

x

xxx

22

4)2(222

**NOTE**

• As with polynomials, limits of many familiar functions can be found by substitution at points where they are defined. These properties apply to trig functions, exponential and logarithmic functions, and composites of the functions.

Example

• Determine solve graphically and confirm analytically

• Graphically…graph f(x)• Analytically

x

xx

tanlim

0

xx

x

xx

x

x

xxxxx cos

1lim

sinlim

cos

1sinlim

tanlim

0000

10cos

11

sinlim

0

and

x

xx

111

Exploring a Nonexistent Limit

• Use a graph to show that does not exist. 2

1lim

3

2

x

xx

Exploring a Nonexistent Limit

• Use a graph to show that does not exist.

• Solution: Notice that the denominator is 0 when x is replaced by 2, so we cannot use substitution to determine the limit. The graph suggests that as x2 from either side, the absolute values of the function values get very large. Therefore the limit does not exist.

2

1lim

3

2

x

xx

Homework

• Pg 62-63/ 7-30 ALL skip #12

Lesson 2-1: Rates of Change and Limits Day #1 AP Calculus Mrs. Mongold.

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Transcript of Lesson 2-1: Rates of Change and Limits Day #1 AP Calculus Mrs. Mongold.