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A9 Biv Sketch thegraph of f(x) =log (~x)+5. 274010 pqrtatsd? 9d Illw 7oco Rb0 JUvMJA oamoyf 9 #1 r #endt anp rlonet, 6 Jn~ (D (0,1 /9 nollusolS AJal e ( '[ IoD...

Question

A9 Biv Sketch thegraph of f(x) =log (~x)+5. 274010 pqrtatsd? 9d Illw 7oco Rb0 JUvMJA oamoyf 9 #1 r #endt anp rlonet, 6 Jn~ (D (0,1 /9 nollusolS AJal e ( '[ IoDetermine the dorain ard Grge = fk) - Use jritervel @ottionFt4R0T ; ,

a9 Biv Sketch thegraph of f(x) =log (~x)+5. 274010 pqrtatsd? 9d Illw 7oco Rb0 JUvMJA oamoyf 9 #1 r #endt anp rlonet, 6 Jn~ (D (0,1 /9 nollusol S AJal e ( '[ Io Determine the dorain ard Grge = fk) - Use jritervel @ottion Ft 4R0T ; ,



Answers

Given that $f(x)=F^{\prime}(x),$ use $\mathrm{Ta}-$ ble 7.1 to evaluate the expression correct to three decimals places. Table 71. $$\begin{array}{c|c|c|c|c|c|c|c}\hline x & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\\hline F(x) & 2 & 3 & 16 & 21 & 24 & 26 & 31 \\\hline\end{array}$$ $$\int_{\pi / 6}^{\pi / 2} f(\sin x) \cos x d x$$

Okay, We've given a function off two variables X and y. Where is the function of you to sign X Y? So we have The function is for the first value. You refer to what we need to plug it for X on second value, Refer to what we put into our Why so in this, the first instance will have two times pi over six. And what are the sign of that now, do you prop 60 private three. So we're gonna be left with Sign Pai Who? That three. I feel out your calculator sign. Part three should come out at route 3/2. So next up, we're gonna have minus three lots off pi over 12 minutes. Could be a sign Se 12 Divide by minus three is gonna leave us with minus for, So we'll be left with pie before I remember that. Minus there off a minus sign. But I'm sure you can get just so it provides a calculator. You should get a value of minus route to To on then in this one. We've got pie time. 1/4. Where is gonna exact simmers? We're gonna sign pie before, but this time could be positive for. And if you put that into calculator, you should just get the positive version off the one before, as expected. So we've got route to over to and then finally, we've got minus pi of it, too, on minus seven. So you're gonna end up with sign. It's really a postive got minus over, minus times a minus. Say he's going to end up with seven pi over two on the sign off. Seven pi over two. It's just going to be equal to minus one. On these are are different values.

This problem. Number two of the Stuart Calculus eighth edition section in two point eight. Use the given graft to estimate the value of each derivative, then sketched the graph of F crime. So this is a graph of F given as this certain shape. And this is a graph for this problem, and we're hoping to sketch a graph of F prime or the derivative of dysfunction. And we're going to do that by finding the derivative for each point from zero to seven. X equals zero tech sickle seven and then using each of the individual values on DH this plot here, tio, plot what our function will look like. Okay, so we begin with X equals zero. What is the slope? What is the derivative at the origin we look at our function on? We see that it has a pretty steep, positive soap. It goes approximately up little more than three. This is with attention. Lang would look like, uh so it's not to re over one the slope. It probably is more closer to four over one or five over one from as to how steep the slope of this tangent, Linus. So as an estimate. We're going to say that the rise over run is for for the slope at the origin. Carping at X equals one. What? It's a derivative. What is the tension? I ll look like tension in line at X equals one. It looks like this all right, touches the function at X equals one only at one place and the line is horizontal, meaning that it has a slope of zero at X equals one. Add X equals two. We take a look at where the function is at X equals two. It's right here. Let's say that we went down one and over one thats not quite a line that is tangent to the function. And so we go down to approximately and over one that seems to be a little too steep. So when they get one over one, it's not steep enough. Negative two or one a little too steep. How about halfway halfway between negative one Negative too. That seems more appropriate as attention line, uh, for this function. So half way we can make it a one or negative to three over to, and that will be our estimate for the Slope X equals two at X equals three. What is the derivative? What is this stuff of the attention line X equals three. This is the point here, crossing that Texas. And we see that if we draw attention, line on that it is closely associate ID approximately ah, equal to a slip of negative one. So that is our best estimate. We're going to go with a slip of negative one for At X equals three and X equals four. We look at where that point is. Two. Three, for this is the point X equals form. We're trying to estimate what potential and my look like we'd go down one and we'LL go over one thats not quite steep enough or it's a little too steep. So we go over to and this seems to be a bit more appropriate if we were to draw this tension line here. That seems to be a pretty appropriate tension. Lame at X equals four. And this slope here is down one over to a rise of negative one over two. That is a slope of the negative one house annexe Eagles fry. What is the derivative of F one, two, three, four, five This is the point for the function. There's a slope world. Well, tryingto as to me by going up one in going over one, two. Uh, it seems to be a good estimate. How about three? I think that is a little better. We know that the slope add five as it gets closer to the next value. I'm The slopes are going from steeper tonight. A Steve. So this seems to be a better estimate. Um, the sloping arise of negative on and a run of three or closer to that than negative one over to. So we're going to see that they're slow. Estimate at X equals five is negative. One over three at X equals two six. This is where the function is at X equals six. We see that it is it horizontal pendant horizontal line on the slope of zero. So six there's a minimum and the slope of the tangent line zero. So the dirt of zero and finally at X equals seven, one, two, three, four, five, six, seven. This is where the point is. All right, if we go down and then over one too steep for the soap over two Still seep three, four five at five. We imagine it a more approximate tangent. So this is what ah, run of five and a rise of naked one would look like. So this being a slow a rise of one. So this is a rise of one and a run of five. That is a slope of one over five or one fifth. And this seems to be more amore. Appropriate estimate for the tension. This liberal potential in an X equals seven. So we're going to go with one over five. Great. So finally, we're going to plant this function in this F prime function, and we're going to supply each of these points for X equals zero through X equals seven. Starting with X equals zero. The point. The value of the derivative is for the next point at X equals one. The value of the derivative is zero for X equals. So we're gonna draw a couple things here in a draw, make it one, continue the function a little lower, and then draw negative too. Okay, being this in mind, what is the next point at X equals two. The stop is thinking of three over to our negative one half that's approximately right here at X equals three. The value of the dirt of his negative one right about there at X Eagles before the value is approximately negative. One half right right there had X equals to find the value of the dirt is a personal maid of one third. So just a little higher closer to the X axis. An ex eagles to six, the value the dirt of zero. So now across the X axis. And Alex, he was the seventh valued. The jury was one fifty. So this is what the shape of the function will look like. Now we found thie. Sure, if it is at each of the points. And now we plotted them separately and we join them with the single smooth curve to estimate and sketch the parent behavior of this function of crime. So this is a sketch of the graph of Prime that we found it right from the individual irritants. Eight different points of the graph of F given here and that completes this problem

Surface. Question says to find the specific function value So you're in the function of X Y is equal to sign of X Y since over party, and we want to find the value of death of two in pi over six. So this is just going to be the same thing as if this were just like f of to put to you and Rex. So now we're gonna put into and for X and his tie over 64 Why so plugging that into our function? I get sign of two times pi over six. Which sickle? The sign of pie over three, which is just the same thing as the square root of three for two for part B breast. Fine as a negative three and power 12. It's again plugging in negative three for X and power 12 for why I get negative. Three six time size. I get negative three times prior 12 which is negative pirate, before which that is just going to be equal to negative square root of two. But him they were the same thing for all these. So first see, we have f bellow it up high and when force so plugging that into my function, I get pi times 1/4 which is prior before which we just found to be square root of two over too. Over the last one We are evaluating the function at X equals negative pie over to and why does make it a seven. So when I multiply my ex and my wife here, I get a positive seven pi over too and sign of seven pi over to is going to be equal to negative one.

Given the function um F of X equals one minus cosine X. Um And the points we want to find the slopes of the secret lines through A. D, A. C. And A B. So A. D. Um What we're going to have is FFP i -F of pi over two. Mhm. Yeah divided by um I might inspire him. So the slope of this second line would be about .636. And then we're going to look at A C. Which is going to be pi plus repeat over two plus 0.5. Mhm. Mhm. This will be private two plus 0.5 And this is about .958. Now we're going to get even closer so .05.05. And we see that with the slope of the secret lines we see that we're approaching a slope of one.


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