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Let $ \displaystyle g(x) = \int^x_0 f(t) \,dt $, where $ f $ is the function whose graph is shown.

(a) Evaluate $ g(0) $ and $ g(6) $(b) Estimate $ g(x) $ for $ x $ = 1, 2, 3, 4, and 5.(c) On what interval is $ g $ increasing?(d) Where does $ g $ have a maximum value?(e) Sketch a rough graph of $ g $.(f) Use the graph in part (e) to sketch the graph of $ g'(x) $. Compare with the graph of $ f $.

a. Estimate integralb. Estimate integralc. $9^{\prime} \geq 0 \Rightarrow f \geqslant 0 \quad[0,3]$d. $g^{\prime}=0 \Rightarrow f=0 \quad x=3$e. graphf. graph

02:18

Frank L.

Calculus 1 / AB

Chapter 5

Integrals

Section 3

The Fundamental Theorem of Calculus

Integration

University of Michigan - Ann Arbor

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we are given a function F. For given its graph. And we're told that G is to function defined to be the integral from zero to X of F F T D. T. In part a were asked to evaluate G zero and GF six. He was great. Well, by definition, this guy's a gold G of zero is the integral from 0 to 0 of F F T D. T. Since the limits of the integral are the same. It follows that this is just zero. Some. Likewise, G of six. By definition Is the integral from 0 to 6 of F F T D. T. However, notice that the graph of F. Who remembers Stephen Yeah. Do the big pedophile out. Famous pedophiles. Are you accusing romano has rotational symmetry about? Nobody's going to believe you. Nobody's going to believe this happened. Sure, you can fucking cry all you want. Nobody's gonna believe it was. Hey look, The .30. So the part from X equals 0 to 3 is equal in area but opposite and sign to the part from X equals 3 to 6. Well and therefore this is equal to G. F. Six is in fact zero as well. Yeah, that's it. Hello? Mm. Then in part B were asked to estimate G of X for X from 1- five. Like sort listen to me, you fucking idiot. So to do this, you want to count the squares between the graph of F and the X. Axis. And so these are gonna be pretty rough estimates. Therefore the squares above the X axis. We count as positive, the squares below the X axis. We count as negative. And then and so for example you might calculate G of one. Well this is the area under f from 0 to 1 which is about 2.8. Calendar was likewise counting squares. Again, G of two is approximately G of one, which was 2.8 Plus. The area from x equals 1- two under the graph of F, which is about 1.9. So this is approximately 4.7. Yes. My this impression of I don't want to do look it up. Likewise, GF three is approximately the area From x equals 0 to 2 under the curve. 4.7 Plus the area from x equals 2 to 3 under the curve, which is about .6. So this is about I've had so many fucking goddamn chores. 5.3. Yes. And we can do the same processes for G four and GF five. Yeah. GF four is about 53 but now we subtract about six Which is about 4.7. Yeah. And likewise, g of five Is about GF four which was 4.7 About 1.9 which is About 2.8. Yeah. Then we Yeah, the HBO shows Then in part C were asked them what interval is the function G increasing? Oh yeah warehouse Well, from x equals 0- three. The area between the curve and the X axis is all positive. Yes. And then negative afterwards. Yeah. So as X goes From 0 to 3, it follows that a cumulative area increases. That's too easy. Yeah. Yeah. On the other hand, past x equals three. Total cumulative area decreases most of was on that for me, you had a boy I'm saying. Yeah. Yeah. And therefore It follows that the function G is increasing on the interval 03 that have to sit on uh what twinkle little star birthday to you. Yeah. No. Then in part D were asked where the function G has a maximum value. Yeah well as we pointed out in Part C G. Is increasing On the interval 0-3 and is then decreasing On the Interval 3 6. Therefore it follows that G. Has A maximum value at x equals three. So mm you know I cried then in part E. Whereas to sketch a rough graph of G I'll be upset you don't want to because I like myself. So because I actually said said well we have our table of values essentially from parts A. And B. So we can simply plot these points along a smooth curve. Mhm. But that was from her. That's yeah I guess it's I guess I would be possible. He's saying you should be happy. I guess I'd be happy. I don't know what I do. Glad the ability to stop it's okay it's okay to take pleasure. So look dude man get a wax. I I would be lying if I said. Mhm. Yeah. So for example I have a point at 00 Appointed about 1 2.8 private islands at about two 4.7. Mhm. Three five point three. It is trying to fuck kids. The median price. And it's symmetrical about the line x equals three. Let's get dinner. And and so the graph looks something like this. Yeah but I thought we got good for the live show maybe. But let's God yeah. Then in part F. Were yet asked to use the graph in part E. To sketch the graph of the derivative G. Prime of X. And then to compare this graph with the graph of F. And I'm going to start. Yeah. But for the average August 12 out of what is the boston is well from part D. We have the graph of G. We already saw the G has a maximum at x equals three. Take it? And the slope of the tangent line X equals three is zero. In other words, G prime of three equal zero. Now on the left hand side of X equals three. G is increasing and therefore G prime is going to be positive on the interval 03 it's decreasing to the right of X equals three. Is that G prime is negative on 3 to 6 words in Seattle notice also that the rate at which G is increasing Gets smaller and smaller as ex moves from 0 to 3. Come on. So it follows that G prime of zero is a maximum he's been doing it. I was like he thinks it's fun. Like the funny thing is sex crying because just a suggestion and G prime is decreasing On the interval 03 like straight. Also. Likewise on the right hand side, the rate at which G is decreasing, gets larger and larger at least an absolute value As ex moves from 3 to 6. So So it follows that G prime of six is a minimum of G and that G prime is increasing Tuesday. He said monday. Yes, I'm sorry not increasing but still decreasing warriors. So On the interval 3-6, what's up on? You know what dude? Yes. With all this information we can sketch a graph of G prime. Yes. All right. Yeah. So here is a graph of G. Prime. We start out with our maximum at X equals zero. Yeah. Men we decrease until We reached G prime of three which is zero. Now we have a minimum which is symmetrical below the X axis. And yeah, you put. So if we draw a smooth curve in these points, it looks something like this. Just didn't think also. But I shouldn't dress because in reading this subreddit I see are claiming that bitching about some. Yeah. Were asked, compare this with the graph of F. Well, whatever it's things happen, I would rather have show we see that at the very least the graph of G. Prime just point of fucking argument resembles the graph of F. The subreddit was 100 problem.

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