[Technology Exercise]

Graph y = 1/(2√x) in a ...

Question

[Technology Exercise]

Graph y = 1/(2√x) in a window that has 0 ≤ x ≤ 2. Then, on the same screen, graph

y = (√(x + h) − √x)/h

for h = 1, 0.5, 0.1. Then try h = −1, −0.5, −0.1. Explain what is going on.

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. The graph of Y equals 1 divided by the square root of 2 X is shown. Alongside the graph of Y equals the square root of 2 multiplied by X plus H minus the square root of 2 X all divided by H for H is equal to curly brackets of 1.0.5.0.2 is also shown on the same coordinate plane. Describe what happens to the graph of Y equals the square root of 2 multiplied by X plus H minus the square root of 2 X all divided by H as X approaches 0. Awesome. So it appears for this particular problem, we're asked to describe what happens to our graph of Y as it as this value of X approaches 0. So that's what we're ultimately trying to solve first. We're trying to figure out how to describe what is happening to our graph of Y as X approaches. 0. So now that we know what we're ultimately trying to solve for, let's quickly look at our graph that is provided to us. So let us know we have a legend for our graph of Y, which once again, Y is equal to the square root of 2 multiplied by X plus H minus the square root of 2 X all divided by H. And as you can see, we have our graph where our Y axis is the vertical axis, and our horizontal axis is our x-axis. And as you can see, we got Three different curves labeled in our legend. So we have our curve that's represented in red is when H equals 1, and our curve in orange is for when H equals 0.5, and our curve in blue is when H is equal to 0.2, and our green curve, which is on our graph represents Y is equal to 1 divided by the square root of 2 X. And as you can see, they all are spaced between, they both have a little bit of a gap between them when they're stretching out towards our Y axis. However, when they start to curve downwards to run parallel with our X-axis, they start to get closer and closer and closer to one another. So now that we have a good visualization of what's happening in this particular prong, let us recall and note that from our provided graph, let us note that H gets smaller and will approach 0. And the graph of Y equals the square root of 2 multiplied by X plus H minus the square root of 2 X all divided by H will get closer to Y equals 1 divided by the square root of 2 X. So let us note that our function Y equals the square root of 2 multiplied by X plus H minus the square root of 2 X all divided by H will represent the difference quotient of the. Function of F of X is equal to the square root of 2 X, and this is a form of the derivative definition, which suggests that as H approaches 0, the function of Y equals the square root of 2 multiplied by X plus H minus the square root of 2 X all divided by H. Will approximate the derivative of F of X is equal to the square root of 2 X. So this can be proven by taking the limit of the square root of 2 multiplied by X plus H minus the square root of 2 X all divided by H as H approaches 0. So when we take the limit as H approaches 0 of the square root of 2 multiplied by X plus H. So we're taking once again the limit as H approaches 0 of the square root of 2 multiplied by X plus H minus the square root of 2 X, all divided by H. And this will be equal to the limit as H approaches 0 of the square root of 2 multiplied by X plus H. Minus the square root of 2 X divided by H multiplied by the square root of 2 multiplied by X plus H. Plus the square root of 2 X, all divided by the square root of 2 multiplied by X plus H plus the square root of 2 X. Which is going to be all equal to the limit as H approaches 0 of. So the limit as H approaches 0 of, when you multiply it out 2 multiplied by X plus H minus 2 X. Awesome, all divided by H multiplied by parentheses, the square root of 2 multiplied by X plus H. Plus the square root of 2 X. Which is going to be equal to the limit as H approaches 0 of 2H, and we're gonna take 2 H divided by H multiplied by parenthesis square root of 2 multiplied by X plus H plus the square root of 2 X. Fantastic. Which will be all equal to the limit, as H approaches 0 of 2 divided by the square root of 2 multiplied by X plus H. Plus the square root of 2 X, which is going to be equal to when we use the direct substitution method plugging in a value of 0 in for H, we will get 2 divided by 2 multiplied by the square root of 2 X, which is going to be equal to 1 divided by the square root of 2 X. Fantastic. So that will mean, without a doubt, that our final answer will have to be to quickly recap. So our final answer. Once again, quickly recapping. Is that as H gets smaller and approaches 0, the graph of Y equals the square root of 2 multiplied by X plus H minus the square root of 2 X all divided by H will get closer to Y equals 1 divided by the square root of 2 X. And that's it. That will have to be our final answer. Hooray, we did it. So without a doubt the quickly say our final answer once again. Is that as H gets smaller and approaches 0, the graph of Y equals the square root of 2 multiplied by X plus H minus the square root of 2 X all divided by H gets closer to Y equals 1 divided by the square root of 2 X. Fantastic. And that's it. That is how you solve for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

[Technology Exercise]

Graph y = 1/(2√x) in a window that has 0 ≤ x ≤ 2. Then, on the same screen, graph
y = (√(x + h) − √x)/h
for h = 1, 0.5, 0.1. Then try h = −1, −0.5, −0.1. Explain what is going on.

Key Concepts

Graphing Functions

Limits and Continuity

Derivative Approximation

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