Section 2.4 Questions
Question
Is anyone else having trouble with these problems on Homework 4? I'm confused as to how find a number based on the graph that satisfies the condition. Any help is much appreciated! Maddie Mitchell
Answer
Problem 2.4.1 (WebAssign HW 4, Problem 11)
Here is my version of the problem (if the figure doesn't show up, load misc/Problem2-4-1.png into your browser).
Consider the interval on the x axis between the two numbers 10/7 and 10/3. These are the x values that result in function values 1/x that are in the "epsilon window" around 0.5. That is, if x is in the interval (10/7, 10/3), then 1/x will be in the interval (0.3, 0.7). Now, since 10/7 is closer to 2 than 10/3 is, we can ignore 10/3 and focus on 10/7.
We want to make sure x is closer to 2 than 10/7. That is, we want the distance between x and 2 to be less than the distance between 10/7 and 2. Here is how you say this mathematically: we want x to satisfy
|x - 2| < |10/7 - 2| = |-4/7| = 4/7.
So, for my version of the problem, I would let delta be 4/7.
Does this make sense? If not, please post again and point to the part you find confusing.
Problem 2.4.2. (WebAssign HW 4, Problem 12)
This is very similar to Problem 11. Also, Problem 12 is the only one of the three questions you asked about that does not have a "Watch it" button, so I would really like you to try to solve it on your own. But feel free to post again if you still have questions.
Problem 2.4.13. (WebAssign HW 4, Problem 14)
Here is my version of the problem (if the figure doesn't show up, load misc/Problem2-4-13.png into your browser).
We want to find the largest number delta such that, if the distance between x and 2 is less than delta, then the distance between 4x and 8 will be less than 0.5 = 1/2.
Notice that if we factor 4 out of |4x-8|, we have 4|x-2|. So the condition we want to satisfy is equivalent to 4|x-2| < 1/2. Dividing both sides of this inequality by 4, we have |x-2| < 1/8. So an appropriate delta is 1/8.
Does this make sense? If not, please post again and point to the part you find confusing.