Calculus - Class 1 (introduction To Limits).docx
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Mike’s House of Math Calculus: It Pushes Me to My Limit
Road Map #1: Graphical Limits Calculus is the study of functions and how they change. Calculus I (“Calculus AB”) is divided into three parts, two of which we’ll introduce during this preview course. The first is called the _______________, abbreviated __________. A. Limit Notation__________
lim f(x) x 2 Read: “The limit as x approaches 2 from the left of f(x).” _______________ limit:
lim f(x) x 2 Read: “The limit as x approaches 2 from the right of f(x).” _______________ limit:
lim f(x) x 2 Read: “The limit as x approaches 2 of f(x).” “total limit”:
Note: Here, x is approaching 2, though x could approach any number, even infinity. Problem 1: Write in limit notation: “The limit as x approaches 4 of x2 is equal to 16.”
Today, we’ll look at several graphs to help us explore this concept. In each example, we’ll assume the function graphed is f(x). A. “The Nice, Neat Function” __________
1. f(3) 2.
lim f(x) x 3
3.
lim f(x) x 3
4.
lim f(x) x 3
Ask yourself, “What happens as we get closer and closer to x = 3?”
Note that in order for the “total limit” to exist, the ______________ limit must equal the ______________ limit. Let’s look at another (a piecewise function). B. “The Jump”__________
1. f(2) 2.
lim f(x) x 2
3.
lim f(x) x 2
4.
lim f(x) x 2
Isn’t this exciting? You don’t have to memorize these but rather know their principles. Though calculus is algebra-driven, we also need to understand the concepts. C. “The Gap”__________
1. f(2)
2.
lim f(x) x 2
3.
lim f(x) x 2
4.
lim f(x) x 2
The above example illustrates an important point: A function does not necessarily have to be _______________ at a point in order for the limit to exist at that point. This next one looks almost identical to the previous one except THE FLOATING POINT. D. “The Floating Point”__________
1. f(2) 2.
lim f(x) x 2
3.
lim f(x) x 2
4.
lim f(x) x 2
Another thing to notice—even if the function is defined at a point, it does not necessarily ________________________________________. PIT STOP
A limit is the value of a function as it approaches a certain point.
There are three types of limits: left-hand, right-hand, and “total.”
In order for the “total limit” to exist, the left-hand and right-hand limits must be equal.
A function does not necessarily have to exist at a point for the limit to exist.
Even if the function does exist at a point, it may be different than the limit.
YAY CALCULUS E. “Vertical Asymptotes – Case I”__________
1. f(2) 2.
lim f(x) x 2
3.
lim f(x) x 2
4.
lim f(x) x 2
Please note that a limit going to _______________ or _________________________ is different than the limit _________________________. F. “Vertical Asymptotes – Case II”__________
1. f(2) 2.
lim f(x) x 2
3.
lim f(x) x 2
4.
lim f(x) x 2
Problem 2: In order for the “total limit” to exist, what must be true about the left-hand and right-hand limits?
Name: ________________________________________
calculus: power practice #1 For questions 1 – 20, use the graph of f(x) below to calculate what is provided.
1. f(2)
2.
lim f(x) x 2
3.
lim f(x) x 2
4.
lim f(x) x 2
5. f(4)
6.
lim f(x) x 4
7.
lim f(x) x 4
8.
lim f(x) x 4
9. f(5)
10.
lim f(x) x 5
11.
lim f(x) x 5
12.
lim f(x) x 5
13. f(8)
14.
lim f(x) x 8
15.
lim f(x) x 8
16.
lim f(x) x 8
17. f(10)
18.
lim x 10
f(x)
19.
lim x 10
f(x)
20.
lim x 10
f(x)
PERFECT TEN Calculus – Class 1 1. What is the slope of a line perpendicular to 2x + 3y = 6 ?
2. The graph of y = x2017 has which of the following symmetries? Circle ALL that apply. x-axis
y-axis
origin
3. State the difference quotient.
4. What is the domain of y = 4 - x ? State using interval notation.
5. What is the average rate of change of the cubic function on the interval [1, 4] ?
6. Give an angle coterminal to
7. Evaluate: 4 sec 120°.
8. What is the period of y = 2 cos 4x + 6 ?
9. Give the Pythagorean identity involving the cosecant and cotangent functions.
10. For what values of x on the unit circle is sin x = –
3 ? 2
19π radians. 6
Road Map #1: Graphical Limits Calculus is the study of functions and how they change. Calculus I (“Calculus AB”) is divided into three parts, two of which we’ll introduce during this preview course. The first is called the _______________, abbreviated __________. A. Limit Notation__________
lim f(x) x 2 Read: “The limit as x approaches 2 from the left of f(x).” _______________ limit:
lim f(x) x 2 Read: “The limit as x approaches 2 from the right of f(x).” _______________ limit:
lim f(x) x 2 Read: “The limit as x approaches 2 of f(x).” “total limit”:
Note: Here, x is approaching 2, though x could approach any number, even infinity. Problem 1: Write in limit notation: “The limit as x approaches 4 of x2 is equal to 16.”
Today, we’ll look at several graphs to help us explore this concept. In each example, we’ll assume the function graphed is f(x). A. “The Nice, Neat Function” __________
1. f(3) 2.
lim f(x) x 3
3.
lim f(x) x 3
4.
lim f(x) x 3
Ask yourself, “What happens as we get closer and closer to x = 3?”
Note that in order for the “total limit” to exist, the ______________ limit must equal the ______________ limit. Let’s look at another (a piecewise function). B. “The Jump”__________
1. f(2) 2.
lim f(x) x 2
3.
lim f(x) x 2
4.
lim f(x) x 2
Isn’t this exciting? You don’t have to memorize these but rather know their principles. Though calculus is algebra-driven, we also need to understand the concepts. C. “The Gap”__________
1. f(2)
2.
lim f(x) x 2
3.
lim f(x) x 2
4.
lim f(x) x 2
The above example illustrates an important point: A function does not necessarily have to be _______________ at a point in order for the limit to exist at that point. This next one looks almost identical to the previous one except THE FLOATING POINT. D. “The Floating Point”__________
1. f(2) 2.
lim f(x) x 2
3.
lim f(x) x 2
4.
lim f(x) x 2
Another thing to notice—even if the function is defined at a point, it does not necessarily ________________________________________. PIT STOP
A limit is the value of a function as it approaches a certain point.
There are three types of limits: left-hand, right-hand, and “total.”
In order for the “total limit” to exist, the left-hand and right-hand limits must be equal.
A function does not necessarily have to exist at a point for the limit to exist.
Even if the function does exist at a point, it may be different than the limit.
YAY CALCULUS E. “Vertical Asymptotes – Case I”__________
1. f(2) 2.
lim f(x) x 2
3.
lim f(x) x 2
4.
lim f(x) x 2
Please note that a limit going to _______________ or _________________________ is different than the limit _________________________. F. “Vertical Asymptotes – Case II”__________
1. f(2) 2.
lim f(x) x 2
3.
lim f(x) x 2
4.
lim f(x) x 2
Problem 2: In order for the “total limit” to exist, what must be true about the left-hand and right-hand limits?
Name: ________________________________________
calculus: power practice #1 For questions 1 – 20, use the graph of f(x) below to calculate what is provided.
1. f(2)
2.
lim f(x) x 2
3.
lim f(x) x 2
4.
lim f(x) x 2
5. f(4)
6.
lim f(x) x 4
7.
lim f(x) x 4
8.
lim f(x) x 4
9. f(5)
10.
lim f(x) x 5
11.
lim f(x) x 5
12.
lim f(x) x 5
13. f(8)
14.
lim f(x) x 8
15.
lim f(x) x 8
16.
lim f(x) x 8
17. f(10)
18.
lim x 10
f(x)
19.
lim x 10
f(x)
20.
lim x 10
f(x)
PERFECT TEN Calculus – Class 1 1. What is the slope of a line perpendicular to 2x + 3y = 6 ?
2. The graph of y = x2017 has which of the following symmetries? Circle ALL that apply. x-axis
y-axis
origin
3. State the difference quotient.
4. What is the domain of y = 4 - x ? State using interval notation.
5. What is the average rate of change of the cubic function on the interval [1, 4] ?
6. Give an angle coterminal to
7. Evaluate: 4 sec 120°.
8. What is the period of y = 2 cos 4x + 6 ?
9. Give the Pythagorean identity involving the cosecant and cotangent functions.
10. For what values of x on the unit circle is sin x = –
3 ? 2
19π radians. 6
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