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Problem 32 Hard Difficulty

(a) Sketch the graph of $ f(x) = \sqrt{6 - x} $ by starting with the graph of $ y = \sqrt{x} $ and using the transformations of Section 1.3.

(b) Use the graph from part (a) to sketch the graph of $ f' $.

(c) Use the definition of a derivative to find $ f'(x) $. What are the domains of $ f $ and $ f' $?

(d) Use a graphing device to graph $ f' $ and compare with your sketch in part (b).

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Video Transcript

Okay, first we want a graph of the square root of X. So the square root of X goes through 00 When X is one, Y is one when X is four. Why is too So it looks like this? And so it says now use that grab to get the graph of six minus square root of six minus X. All right. I have to work on a little bit. Have to first make it minus X plus six. Then have to factor the minus sign up. All right, so the X -6 woman would six to the right and then the minute sound will flip it vertically across the Y axis. Okay, 123456. So it looked like that. So it looked like that. That's just the square root of X -6. And that's the square root of -6 -6. Canal graph prime of X. Okay, I'm gonna draw me another picture of it. 123456. Goes through Over one up going on over four. Shoot. Oops. Act yeah. Okay, so to graph the derivative, all you have to do is look at the slope of the tangent lines and make some guesses. So we'll make this swim 12,345,678,910. Okay, so here we have a tangent line and its slope looks like maybe negative one half. And here I have a teenager line And its slope is like maybe -1. And here I have a tangent line and slope with something between those two. And then over here the tangent line is getting even smaller and then here is getting way bigger. So here this one slope looks like down 1/1 half. So slope of negative two, negative five, -9. Until I get right to this point and the slope is vertical. Okay, So it doesn't have any kind of the slope is undefined. Okay, so That was at 612345, 6. Okay, so Here have a slope of negative one half. Well and then I had a slope of negative 3/4 and then had a slip of negative one And then I had to sleep with -2. Awesome. And then right in here, girls like this. Mhm. All right. So I say that's the derivative and this one is pointing down because it's going to negative infinity right there. Alright, next says find the derivative using the limit. Okay, sure. Prime of X. For limit As a church goes to zero square to six minus X plus H minus the square is six minus eight X. Oliver, H. Now we plug H equals zero in here. You get 0/0, which tells us we have some algebra to do. But there's just a trick and the trick is to multiply the top and bottom by the conjugate the top. Contact. It means the same exact stuff except for different sign. Yeah. And don't forget to multiply by it on the bottom two. Okay. The reason you multiplied by the Collins you get is when you multiply a minus B. A plus B. You get a squared plus B squared. No you don't get a squared you get minus and you get b squared no square roots. Which which is what we're trying to get. So you get 6 -1 plus H. Okay not squared because I have the square root and I squared it Minour parentheses 6 -1 over age. Don't really multiply on the bottom. Just leave it like it does. Okay so on the top we have 6 -6. So those are gone negative X minus minus X. So those are gone and you're left with minus H over eight times six minus x minus H Plus six -X. And now you can cancel the ages which will leave us a -1 on top. Now take the limit plug in H equals zero and you end up with minus one over six minus x plus the square six minus x. Which is -1/2 square roots of six -X. Mm. All right, so that's the derivative. And then the last thing was find the domain K. Domain means which numbers did you use to draw it? That's the easiest way. So um for square root of six minus x I used all the negative X values up to negative six. So domain f minus infinity to minus six including minus six. And then when I drew the derivative I used all the same numbers except for -6. So Okay. And then it's to check your calculator. Make sure you drew them, right? So do that? Make sure I drew them right? If I didn't. Bad, bad me. Okay. I'm sure they're good. Okay. I hope that helps.