Make a Stick Activity Use markers/ colored pencils/ or whatever you like to write your name on a...
Transcript of Make a Stick Activity Use markers/ colored pencils/ or whatever you like to write your name on a...
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Make a Stick Activity
• Use markers/ colored pencils/ or whatever you like to write your name on a stick and decorate it to make it your own.
• When you’re done, bring your stick up to the front table and put it in the box for your class.
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Welcome To Calculus BCExpectations• Effort• Ask Questions – All are valid• RESPECT- Support and help each other• Return books to back shelf• Bring: notebook, pencil, red pen, calculator• Backpacks & purses under desk or in back of room• Cell phones on silent and not on your person
My teacherpage: http://www.husd.org//Domain/1454
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AP Expectations• This class is an AP course. Everyone
takes the AP test in May.
• You will need to spend a significant amount of time outside of class on homework, AP review, and preparation for the AP test.
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Limits and Their Properties1
Copyright © Cengage Learning. All rights reserved.
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BC Day 1
Limits Review
Handout: Limits and their Properties Notes
Copyright © Cengage Learning. All rights reserved.
1.2-1.4A
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Estimate limits numerically, graphically and algebraically.
Learn different ways that a limit can fail to exist.
Special Trig Limits
Define Continuity.
Objectives
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Formal definition of a Limit:
If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit of f(x), as x approaches c, is L.
“The limit of f of x as x approaches c is L.”
limx c
f x L
This limit is written as
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Consider the graph of f(θ) = sin(θ)/θ
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• Let’s fill in the following table
• We can say that the limit of f(θ) approaches 1 as θ approaches 0 from the right
• We write this as
• We can construct a similar table to show what happens as θ approaches 0 from the left
θ 0.5 0.4 0.3 0.2 0.1 0.05
sin(θ)/θ
1sin
lim0
θ 0.5 0.4 0.3 0.2 0.1 0.05
sin(θ)/θ 0.959 0.974 0.985 0.993 0.998 0.9995
θ -0.5 -0.4 -0.3 -0.2 -0.1 -0.05
sin(θ)/θ 0.959 0.974 0.985 0.993 0.998 0.9995
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• So we get
• Now since we have we say that the limit exists and we write
1sin
lim0
1sin
limsin
lim00
1sin
lim0
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3
1
1lim
1x
x
x
An Introduction to Limits
Ex: Find the following limit:
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Start by sketching a graph of the function
For all values other than x = 1, you can use standard
curve-sketching techniques.
However, at x = 1, it is not clear what to expect.
We can find this limit numerically:
An Introduction to Limits
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To get an idea of the behavior of the graph of f near x = 1, you can use two sets of x-values–one set that approaches 1 from the left and one set that approaches 1 from the right, as shown in the table.
An Introduction to Limits
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The graph of f is a parabola that has a gap at the point (1, 3), as shown in the Figure 1.5.
Although x can not equal 1, you can move arbitrarily close to 1, and as a result f(x) moves arbitrarily close to 3.
Using limit notation, you can write
An Introduction to Limits
This is read as “the limit of f(x) as x approaches 1 is 3.”
Figure 1.5
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This discussion leads to an informal definition of a limit:
A limit is the value (meaning y value) a function approaches as x approaches a particular value from the left and from the right.
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Properties of Limits:
Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power.
For a limit to exist, the function must approach the same value from both sides.
One-sided limits approach from either the left or right side only.
x clim f x L
x clim f x L
x clim f x L
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1 2 3 4
1
2
y f x
limx
f x
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Limits That Fail to Exist - 3 Reasons
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Discuss the existence of the limit:
Solution: Using a graphical representation, you can see that x does not approach any number. Therefore, the limit does not exist. 0
limx
20
1limx x
0limx
0limx
DNE
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Properties of Limits
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Properties of Limits
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Properties of Limits
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Compute the following limits
2
1lim
1x x
3
3lim
1x
x
x
sin 2limx
x
x
21
1lim
2x
x
x x
4
lim tanx
x
lim cosx
x x
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• Let’s take a look at the last one
• What happened when we plugged in 1 for x?
• When we get we have what’s called an
indeterminate form
• Let’s see how we can solve it
2
1lim
21
xx
xx
0
0
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• Let’s look at the graph of
Is the function continuous at x = 1?
2
1lim
21
xx
xx
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Strategies for Finding Limits?
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You Try:
Find the limit:
5
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You Try:
3
3
27Evaluate lim
3x
x
x
27
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Find the limit:
Solution:
By direct substitution, you obtain the indeterminate form 0/0.
Example – Rationalizing Technique
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In this case, you can rewrite the fraction by rationalizing the numerator.
cont’dSolution
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Now, using Theorem 1.7, you can evaluate the limit as shown.
cont’dSolution
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A table or a graph can reinforce your conclusion that the
limit is . (See Figure 1.20.)
Figure 1.20
Solutioncont’d
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Solutioncont’d
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You Try:
0
4 2Evaluate lim
x
x
x
1
4
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Example:
0
2
Evaluate lim
1, 0
1, 0
x
f x
x xf x
x x
0
limx
f x
0
limx
f x
1 1
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SPECIAL TRIG LIMITS
You must know these for the AP test!
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Find the limit:
Solution:Direct substitution yields the indeterminate form 0/0.
To solve this problem, you can write tan x as (sin x)/(cos x) and obtain
Example – A Limit Involving a Trigonometric Function
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Solution cont’d
Now, because
you can obtain
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Figure 1.23
Solution cont’d
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AP example
• Find the following:
0
sin 4x
xLim
x
0
sinlim 1x
x
x
4
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Continuity at a Point and on an Open Interval
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Figure 1.25 identifies three values of x at which the graph of f is not continuous. At all other points in the interval (a, b), the graph of f is uninterrupted and continuous.
Figure 1.25
Continuity at a Point and on an Open Interval
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Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil.
A function is continuous at a point if the limit is the same as the value of the function.
This function has discontinuities at x=1 and x=2.
It is continuous at x=0 and x=4, because the one-sided limits match the value of the function1 2 3 4
1
2
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Consider an open interval I that contains a real number c.
If a function f is defined on I (except possibly at c), and f is not continuous at c, then f is said to have a discontinuity at c. Discontinuities fall into two categories: removable and nonremovable.
A discontinuity at c is called removable if f can be made continuous by appropriately defining (or redefining f(c)).
Continuity at a Point and on an Open Interval
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For instance, the functions shown in Figures 1.26(a) and (c) have removable discontinuities at c and the function shown in Figure 1.26(b) has a nonremovable discontinuity at c.
Figure 1.26
Continuity at a Point and on an Open Interval
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Example 1 – Continuity of a Function
Discuss the continuity of each function.
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Example 1(a) – Solution
Figure 1.27(a)
The domain of f is all nonzero real numbers.
From Theorem 1.3, you can conclude that f is continuous at every x-value in its domain.
At x = 0, f has a non removable discontinuity, as shown in Figure 1.27(a).
In other words, there is no way to define f(0) so as to make the functioncontinuous at x = 0.
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The domain of g is all real numbers except x = 1.
From Theorem 1.3, you can conclude that g is continuous at every x-value in its domain.
At x = 1, the function has a removable discontinuity, as shown in Figure 1.27(b).
If g(1) is defined as 2, the “newly defined” function is continuous for all real numbers.
Figure 1.27(b)
cont’dExample 1(b) – Solution
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Figure 1.27(c)
Example 1(c) – Solution
The domain of h is all real numbers. The function h is continuous on and , and, because , h is continuous on the entire real line, as shown in Figure 1.27(c).
cont’d
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The domain of y is all real numbers. From Theorem 1.6, you can conclude that the function is continuous on its entire domain, , as shown in Figure 1.27(d).
Figure 1.27(d)
Example 1(d) – Solutioncont’d
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Removing a discontinuity:
3
2
1
1
xf x
x
has a discontinuity at 1x
Write an extended function that is continuous at 1x
3
21
1lim
1x
x
x
2
1
1 1lim 1 1x
x x xx x
1 1 1
2
3
2
3
2
1, 1
13
, 12
xx
xf x
x
Note: There is another discontinuity at that can not be removed.1x
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Removing a discontinuity:
3
2
1, 1
13
, 12
xx
xf x
x
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Group Work:
Sketch the graph of f. Identify the values of c for which exists. lim
x cf x
lim exists for all values except where 4.x c
f x c
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Homework
Limits and Continuous Functions WS
Get Books! This ppt is on my teacher-page.