The graph above is of the function: 9 Value of € = 8 (enter as a decimal). 7 6 5: 4 (2+5) 5+c (2)-5 1454 (2-5) 2-5 f(z)=z²+1, = x +4, 2 I≤2 I> 2 5-1.50 2+5 The red dot indicates that the point (2, 5) lies on the graph. In order for the function f to be continuous at z = 2, you have to show that given any € > 0 you can find a > 0 such that the interval (2-8,2+8) is mapped entirely within the interval (f(2), f(2) + €) =(5-€, 5+ €). In this example, you have to show f is not continuous at z = 2. So you have to find an € >0 such that there is no such 8. Using the diagram above, this boils down to finding an € >0 so that the interval (2-8,2+8) is not mapped inside the interval (5-€,5 + €) (in red), for any 8 > 0. = 1.50 Experiment by changing the value of using its slider to a value where no matter what positive value of you choose by using its slider, the interval (2-8,2+8) is not mapped by f inside (5-€, 5+ €). This gives a value of you can use in a formal proof.
The graph above is of the function: 9 Value of € = 8 (enter as a decimal). 7 6 5: 4 (2+5) 5+c (2)-5 1454 (2-5) 2-5 f(z)=z²+1, = x +4, 2 I≤2 I> 2 5-1.50 2+5 The red dot indicates that the point (2, 5) lies on the graph. In order for the function f to be continuous at z = 2, you have to show that given any € > 0 you can find a > 0 such that the interval (2-8,2+8) is mapped entirely within the interval (f(2), f(2) + €) =(5-€, 5+ €). In this example, you have to show f is not continuous at z = 2. So you have to find an € >0 such that there is no such 8. Using the diagram above, this boils down to finding an € >0 so that the interval (2-8,2+8) is not mapped inside the interval (5-€,5 + €) (in red), for any 8 > 0. = 1.50 Experiment by changing the value of using its slider to a value where no matter what positive value of you choose by using its slider, the interval (2-8,2+8) is not mapped by f inside (5-€, 5+ €). This gives a value of you can use in a formal proof.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Transcribed Image Text:The graph above is of the function:
9
Value of € =
8
(enter as a decimal).
7
5:
4
(2+5)
5+c
(2)-5
1454
(2-5)
2-5
2
f(z)=z²+1, z≤2
= x +4,
I> 2
5-1.50
2+5
The red dot indicates that the point (2, 5) lies on the graph.
In order for the function f to be continuous at z = 2, you have to show that given any € > 0 you can find a > 0 such that the interval (2-8,2+8)
is mapped entirely within the interval (f(2), f(2) + €) =(5-€, 5+ €).
In this example, you have to show f is not continuous at z = 2. So you have to find an € >0 such that there is no such 8.
Using the diagram above, this boils down to finding an € >0 so that the interval (2-8,2+8) is not mapped inside the interval (5-€,5 + €) (in red),
for any 8 > 0.
= 1.50
Experiment by changing the value of using its slider to a value where no matter what positive value of you choose by using its slider, the interval
(2-8,2+8) is not mapped by f inside (5-€, 5+ €). This gives a value of you can use in a formal proof.
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