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For cach 0l the followina: (1sptsl) Funx6.) Y = col,c) > =secI CosxGiven that f() = x sinx; find f"(r) (Spts)Find equation of the line that E tangent to the...

Question

For cach 0l the followina: (1sptsl) Funx6.) Y = col,c) > =secI CosxGiven that f() = x sinx; find f"(r) (Spts)Find equation of the line that E tangent to thegraph of f(x) = 3r2 - x + 1at 1= (Spts)

For cach 0l the followina: (1sptsl) Funx 6.) Y = col, c) > =secI Cosx Given that f() = x sinx; find f"(r) (Spts) Find equation of the line that E tangent to thegraph of f(x) = 3r2 - x + 1at 1= (Spts)



Answers

Find the equation of the line tangent to $y=f(x)$ at $x=1,$ where $f(x)=6 e^{5 x}+e^{-x^{2}}$

Okay, so we're giving our falling X and Y point, and we want to find the equation of the line tangent to our foot long function F of X. So we're going to need to find our slope and already have a point. And once you find those, still, I just find the slope. We can plug that into our point slope form to find the equation intentions to default and function. Okay, so let's start by taking charge of it. Whoever of prime of acts that people to sign in verse derivative of that is one over the square root of one minus are inside, spread in times interpretive over inside, which is 1/4. So we get 1/4 is where roots of one minus X squared over 16. Okay, I know. Let's plug in our exploits of prime at to define conservative or a slope at the point to. So it's one of the four square root of what minus for over 16 before over 16 is one before, so we get 1/4 square roots of 3/4. I can bring out that florist since the square root of force to take it 1/4 or two square bit of three. So this is to square root of three. Okay, so our equation is why minus pi over six is equal to on over to score, but of three times X minus two. Okay, so, multiplying out our term, we get the following. So we get why is equal to one over just as the exit there, minus one over square root of three and then plus pi over six for our equation for attendant like

Okay, so we want to find the equation of our attention line, given her x volume. So for defined, the are changing mine. We're gonna need to use point slope form. So we're gonna need a point on our line as well as I slope when we confined our scope by using this equation here, and we're also giving our exploit. So let's find their course wanting. Why value? So that's going to be Well, um, it's not that f of a well or a value is what's given for X. So this is going to be negative one. So here we have off of negative one. So that's equal to six minus one, which is equal to five. Andi. Um so our point on our line that is next one comma y one. That's equal. Teoh Negative one comma five and it's fine. Doubling smoke. So, um, is equal to the limits as approaches negative one of artworks that six minus X squared minus forbade all over X minus a. That's X minus negatives. That's pushed one. Okay, so let your This was the limits. As X approaches negative. One of one minus X squared over exposed one. So in our numerator we can practice. That's that's the difference of squares. So we get one of minus X times one plus I all over, exposed one. No, he can't go out like term. And I looked used. Except do we have one minus negative one. Who is he here that are? Smoke is equal to one plus one, Which is to Okay, now let's work this into our equation of our life. We're gonna have y minus fine is you with two two times X minus negative one. That's plus one. So it's multiply out that I'm too, and add five on both sides. It's offer way. Okay, but it is going to give us two x was too sudden. Do we get that? Our equation of our line is why you could to act last night.

Okay, so we want to find a slope of our changing line, given our X well, you. Well, it's don't hear that X value is equal to get one that's equal to R A value. So we want to use our equation here. So we have a flex and we need to find every day. But we should that are. Any value is equal to negative one. So let's put that end. We get six overnight, you have one with you do Patil. Negative six. And now let's write out X minus A as well. That's equal to X plus one. So putting this into our crazy in here we get the limits as X approaches. Negative one of us, Lex. That's six over X minus after days, and it's minus native six that its Quest six All Over X plus one. Well, let's combine our numerator into a single fraction, so I'm gonna multiply this by X over X. It's not that we have a six in common so we can fax it out. So we're gonna get the limit as experts is the head of one of six Times X or what that's going to be one plus X over X Times X plus one. Now we can cancel out our like terms and we see that if we used Excel, we get six over negative one. So I slope overtime that mine is equipped to negative six.

Here we have. Why equals f of X at X equals a ever backs equals X science and it equals pi. Over two were asked to find the equation. Police attendant Line four. This function underlined in green. At this point, a equals pi over two. Well, what is the line Look like? A line is y equals MX plus B. We know this formula and what is the director? Derivative shrinks pills he derivative slope park attentions function. So the derivative of a function is the slope of the tangent of that function. And then you couldn't plug in any given point like high over to to get a slope at a particular point. So, yes, we shall start where it makes the most sense. Finding the derivative of the function f of X equals X sine X equals X sign. Next. Yes, we've got some product rule action here X times signing. So remember, it's the first times the derivative of the second plus second time for the derivative of the first. That's always a tongue twister for me with that being you in that being vite. So without mantra, analyze The first is X times the derivative of the second, The derivative of Sine X is co sign. It's plus the second sine x times, the derivative her first, which is one. So you know, that's what that prime of excess. This is the general formula for the derivative, the general formula for the tension. But we wanted at a specific point a equals, however to, But we can't just plug a equals pi over too into eggs. They're not the same variable, but we were told over here that X equals a so that's basically like saying X equals pi over two so we can happily plug it in. F Crime of pi over to my pies. Just get her says the problem goes on pi over two times. Co sign Pi over two plus sign high over too well, Coastline pi over two is zero. This is zero. So zero times anything is zero that's gone. Sign Pi over to think about the unit circle in your mind. Sign Pi over two is one. So remember that the derivative is the slope left basically saying that M equals one. This is our slow, but we're not quite done. We look back at y equals mx plus B, we have em. We don't have beat that. We can't directly find Pete. But what we do have is X. We figured out earlier that X was pi over two. If we plug X in to the original equation F of X equals X, I next. Then we can find why. Because y equals f of X. It's all coming together. So down here, half of X equals X side X Come out when we plug in X equals pi over two. Probably right. Better. What professional color F, uh, high over to just pride Overshoot Time signed by over to sign pot over two is one. So one times pi over two is just piratey. That's a lot of high over twos, so f of X is signed pi over two. We know this is actually why. So we have executed over two Weigel over to, which gives us a great coordinate point of pi over two. Pilot Jim. So now we have everything we need to plug into our y equals MX plus a B base equation. I'll be right to slope. We're slope with one Michael's. I'm explicit. E We know why it's pi over two. You know I am. That's that one we know. Except also pi over two plus speak are variable When we saw for B be nickel zero. So when we rewrite that with everything, we know em being right in the middle, Why equals ex? Ah, lovely diagonal lying. Let's rewrite. That's why to make it better. And that, my friends, is the answer y equals X.


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