Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem
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Transcript of Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem
![Page 1: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/1.jpg)
![Page 2: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/2.jpg)
The Squeeze Theorem
![Page 3: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/3.jpg)
The Squeeze Theorem
Let’s say f and g are two functions such that one is always belowthe other. Let’s say that g is always above:
![Page 4: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/4.jpg)
The Squeeze Theorem
Let’s say f and g are two functions such that one is always belowthe other. Let’s say that g is always above:
f (x) ≤ g(x)
![Page 5: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/5.jpg)
The Squeeze Theorem
Let’s say f and g are two functions such that one is always belowthe other. Let’s say that g is always above:
f (x) ≤ g(x)
Also, let’s say that their limits are equal in some point:
![Page 6: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/6.jpg)
The Squeeze Theorem
Let’s say f and g are two functions such that one is always belowthe other. Let’s say that g is always above:
f (x) ≤ g(x)
Also, let’s say that their limits are equal in some point:
limx→a
f (x) = limx→a
g(x) = L
![Page 7: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/7.jpg)
The Squeeze Theorem
![Page 8: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/8.jpg)
The Squeeze Theorem
Now, let’s suppose we squeeze a third function between them:
![Page 9: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/9.jpg)
The Squeeze Theorem
Now, let’s suppose we squeeze a third function between them:
![Page 10: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/10.jpg)
The Squeeze Theorem
Now, let’s suppose we squeeze a third function between them:
Let’s call this third function h:
![Page 11: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/11.jpg)
The Squeeze Theorem
Now, let’s suppose we squeeze a third function between them:
Let’s call this third function h:
f (x) ≤ h(x) ≤ g(x)
![Page 12: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/12.jpg)
The Squeeze Theorem
Now, let’s suppose we squeeze a third function between them:
![Page 13: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/13.jpg)
The Squeeze Theorem
Now, let’s suppose we squeeze a third function between them:
The Squeze Theorem says that:
![Page 14: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/14.jpg)
The Squeeze Theorem
Now, let’s suppose we squeeze a third function between them:
The Squeze Theorem says that:
limx→a
h(x) = L
![Page 15: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/15.jpg)
The Fundamental Trigonometric Limit
![Page 16: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/16.jpg)
The Fundamental Trigonometric Limit
We’re going to prove the following limit using the SqueezeTheorem
![Page 17: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/17.jpg)
The Fundamental Trigonometric Limit
We’re going to prove the following limit using the SqueezeTheorem
limx→0
sin x
x= 1
![Page 18: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/18.jpg)
The Fundamental Trigonometric Limit
![Page 19: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/19.jpg)
The Fundamental Trigonometric Limit
![Page 20: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/20.jpg)
The Fundamental Trigonometric Limit
![Page 21: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/21.jpg)
The Fundamental Trigonometric Limit
∆MOA < Sector MOA < ∆COA
![Page 22: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/22.jpg)
The Fundamental Trigonometric Limit
sin x =
![Page 23: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/23.jpg)
The Fundamental Trigonometric Limit
sin x =BM
1
![Page 24: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/24.jpg)
The Fundamental Trigonometric Limit
sin x =BM
1= BM
![Page 25: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/25.jpg)
The Fundamental Trigonometric Limit
sin x =BM
1= BM
∆MOA =
![Page 26: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/26.jpg)
The Fundamental Trigonometric Limit
sin x =BM
1= BM
∆MOA =1.BM
2
![Page 27: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/27.jpg)
The Fundamental Trigonometric Limit
sin x =BM
1= BM
∆MOA =1.BM
2=
sin x
2
![Page 28: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/28.jpg)
The Fundamental Trigonometric Limit
∆MOA < Sector MOA < ∆COA
![Page 29: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/29.jpg)
The Fundamental Trigonometric Limit
����:
sin x2
∆MOA < Sector MOA < ∆COA
![Page 30: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/30.jpg)
The Fundamental Trigonometric Limit
sin x
2< Sector MOA < ∆COA
![Page 31: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/31.jpg)
The Fundamental Trigonometric Limit
![Page 32: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/32.jpg)
The Fundamental Trigonometric Limit
tan x =
![Page 33: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/33.jpg)
The Fundamental Trigonometric Limit
tan x =AC
1
![Page 34: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/34.jpg)
The Fundamental Trigonometric Limit
tan x =AC
1= AC
![Page 35: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/35.jpg)
The Fundamental Trigonometric Limit
tan x =AC
1= AC
∆COA =
![Page 36: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/36.jpg)
The Fundamental Trigonometric Limit
tan x =AC
1= AC
∆COA =1.AC
2=
![Page 37: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/37.jpg)
The Fundamental Trigonometric Limit
tan x =AC
1= AC
∆COA =1.AC
2=
tan x
2
![Page 38: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/38.jpg)
The Fundamental Trigonometric Limit
tan x =AC
1= AC
sin x
2< Sector MOA < ∆COA
![Page 39: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/39.jpg)
The Fundamental Trigonometric Limit
tan x =AC
1= AC
sin x
2< Sector MOA <���
�:tan x2
∆COA
![Page 40: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/40.jpg)
The Fundamental Trigonometric Limit
tan x =AC
1= AC
sin x
2< Sector MOA <
tan x
2
![Page 41: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/41.jpg)
The Fundamental Trigonometric Limit
![Page 42: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/42.jpg)
The Fundamental Trigonometric Limit
The area of a circular sector is:
![Page 43: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/43.jpg)
The Fundamental Trigonometric Limit
The area of a circular sector is:
A =θr2
2
![Page 44: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/44.jpg)
The Fundamental Trigonometric Limit
The area of a circular sector is:
A =θr2
2
A simple proof is the following:
![Page 45: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/45.jpg)
The Fundamental Trigonometric Limit
The area of a circular sector is:
A =θr2
2
A simple proof is the following:
A(2π) = πr2
![Page 46: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/46.jpg)
The Fundamental Trigonometric Limit
The area of a circular sector is:
A =θr2
2
A simple proof is the following:
A(2π) = πr2
If we multiply both sides of this equation by θ2π :
![Page 47: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/47.jpg)
The Fundamental Trigonometric Limit
The area of a circular sector is:
A =θr2
2
A simple proof is the following:
A(2π) = πr2
If we multiply both sides of this equation by θ2π :
θ
2πA(2π)
![Page 48: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/48.jpg)
The Fundamental Trigonometric Limit
The area of a circular sector is:
A =θr2
2
A simple proof is the following:
A(2π) = πr2
If we multiply both sides of this equation by θ2π :
θ
2πA(2π) = A(
θ2π
2π)
![Page 49: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/49.jpg)
The Fundamental Trigonometric Limit
The area of a circular sector is:
A =θr2
2
A simple proof is the following:
A(2π) = πr2
If we multiply both sides of this equation by θ2π :
θ
2πA(2π) = A(
θ��2π
��2π)
![Page 50: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/50.jpg)
The Fundamental Trigonometric Limit
The area of a circular sector is:
A =θr2
2
A simple proof is the following:
A(2π) = πr2
If we multiply both sides of this equation by θ2π :
θ
2πA(2π) = A(
θ��2π
��2π) = A(θ) =
![Page 51: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/51.jpg)
The Fundamental Trigonometric Limit
The area of a circular sector is:
A =θr2
2
A simple proof is the following:
A(2π) = πr2
If we multiply both sides of this equation by θ2π :
θ
2πA(2π) = A(
θ2π
2π) = A(θ) =
θπr2
2π
![Page 52: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/52.jpg)
The Fundamental Trigonometric Limit
The area of a circular sector is:
A =θr2
2
A simple proof is the following:
A(2π) = πr2
If we multiply both sides of this equation by θ2π :
θ
2πA(2π) = A(
θ2π
2π) = A(θ) =
θ�πr2
2�π
![Page 53: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/53.jpg)
The Fundamental Trigonometric Limit
The area of a circular sector is:
A =θr2
2
A simple proof is the following:
A(2π) = πr2
If we multiply both sides of this equation by θ2π :
θ
2πA(2π) = A(
θ2π
2π) = A(θ) =
θ�πr2
2�π=θr2
2
![Page 54: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/54.jpg)
The Fundamental Trigonometric Limit
![Page 55: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/55.jpg)
The Fundamental Trigonometric Limit
![Page 56: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/56.jpg)
The Fundamental Trigonometric Limit
Now, we have a circle of radius 1 and angle x , so, our area is:
![Page 57: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/57.jpg)
The Fundamental Trigonometric Limit
Now, we have a circle of radius 1 and angle x , so, our area is:
Sector MOA =
![Page 58: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/58.jpg)
The Fundamental Trigonometric Limit
Now, we have a circle of radius 1 and angle x , so, our area is:
Sector MOA =x .12
2=
![Page 59: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/59.jpg)
The Fundamental Trigonometric Limit
Now, we have a circle of radius 1 and angle x , so, our area is:
Sector MOA =x .12
2=
x
2
![Page 60: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/60.jpg)
The Fundamental Trigonometric Limit
Now, we have a circle of radius 1 and angle x , so, our area is:
sin x
2< Sector MOA <
tan x
2
![Page 61: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/61.jpg)
The Fundamental Trigonometric Limit
Now, we have a circle of radius 1 and angle x , so, our area is:
sin x
2<���
����:
x2
Sector MOA <tan x
2
![Page 62: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/62.jpg)
The Fundamental Trigonometric Limit
Now, we have a circle of radius 1 and angle x , so, our area is:
sin x
2<
x
2<
tan x
2
![Page 63: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/63.jpg)
The Fundamental Trigonometric Limit
Now, we have a circle of radius 1 and angle x , so, our area is:
sin x
2<
x
2<
tan x
2
sin x < x < tan x
![Page 64: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/64.jpg)
The Fundamental Trigonometric Limit
![Page 65: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/65.jpg)
The Fundamental Trigonometric Limit
sin x < x < tan x
![Page 66: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/66.jpg)
The Fundamental Trigonometric Limit
sin x < x < tan x
1 <x
sin x<
1
cos x
![Page 67: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/67.jpg)
The Fundamental Trigonometric Limit
sin x < x < tan x
1 <x
sin x<
1
cos x
1 >sin x
x> cos x
![Page 68: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/68.jpg)
The Fundamental Trigonometric Limit
sin x < x < tan x
1 <x
sin x<
1
cos x
1 >sin x
x>���:
1cos x
![Page 69: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/69.jpg)
The Fundamental Trigonometric Limit
sin x < x < tan x
1 <x
sin x<
1
cos x
1 >sin x
x>���:
1cos x
Conclusion:
![Page 70: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/70.jpg)
The Fundamental Trigonometric Limit
sin x < x < tan x
1 <x
sin x<
1
cos x
1 >sin x
x>���:
1cos x
Conclusion:
limx→0
sin x
x= 1
![Page 71: Day 5 of the Intuitive Online Calculus Course: The Squeeze Theorem](https://reader034.fdocuments.us/reader034/viewer/2022051514/548975a2b47959050d8b59c8/html5/thumbnails/71.jpg)