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QUESTIONWhich of the following is the equation of the tangent line to the graph = ofv=2x2+3x- [at x = -27 OAV= -Sx+1 O8.V = ~Bx - 21 Ocv=-Sx-2 D,v= ~Sx-9 Ev= 42 5*+...

Question

QUESTIONWhich of the following is the equation of the tangent line to the graph = ofv=2x2+3x- [at x = -27 OAV= -Sx+1 O8.V = ~Bx - 21 Ocv=-Sx-2 D,v= ~Sx-9 Ev= 42 5*+

QUESTION Which of the following is the equation of the tangent line to the graph = ofv=2x2+3x- [at x = -27 OAV= -Sx+1 O8.V = ~Bx - 21 Ocv=-Sx-2 D,v= ~Sx-9 Ev= 42 5*+



Answers

For the following exercises, find the equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.
$$
\frac{x}{y}+5 x-7=-\frac{3}{4} y,(1,2)
$$

Okay, This question asks us where the slope of the tangent line is equal to four on this curve. So to do this, let's find why Prime. So that's just the derivative of our function with respect to acts and again going turn by turn with the power rule derivative of X squared. His two ex, the derivative of three X, is three and derivative of negative 70 So now that we know that, why prime equals two X Plus three, we want to find where why promise? Four? So four equals two X plus three and solving for X is 1/2 because you subtract three and divide by two. So we know the slope is four at the point 1/2. So where is that on the graph? So after 1/2 equals 1/2 squared plus degree times 1/2 minus seven and that works out to negative 21 over four. So why prime equals four at point 1/2 negative, 21 over four. And that's our answer

So argument function here is f of X is equal to four minus X squared. So to find the equation of the tangent line to the graph, we first. While we need to find the slope of the curve at any given point, that's finding the derivative. So this is our function. Well, then are derivative would be f prime of X. Okay, well, that's just gonna be equal to We can take the derivative here term by term, on DSI the 12, the derivative of 40 the derivative of negative X squared is negative two x to the one that's just negative. Two x So the derivative here is just negative two x okay. And then four party. We are looking to find the equation of the tangent line to the graph at the point. Negative one, comma three. So we find the slope way have a point on the line, which would give it a point on the line, and we have the slope of the line. We can then use point slope form to write the equation of the line. So we have a point right to find slope. Well, the slope of the Kansas Line is just evaluating the derivative at that point at the X value. So we find F prime off negative one that's equal to negative two times negative one. Look, you could have positive too. So the slope of the tangent line when actually negative one. So at the point, negative 13 is too. So then we use point slope form, which is why minus y one is equal to a slope. M times x minus x one for any given point, x one y one online and the slope so are given point. Here is, um, negative 13 So therefore we have y minus three is equal to the slope two times x minus the X one. So times X minus and negative one which is X plus one. We can then solve this for why and get that y is equal to two X plus five. All right, and then four part beat. We're looking to find the equation of the tangent line to the graph now at the point zero comma four. So we find the derivative when x zero that's evaluating f prime of zero that's equal to negative two times zero, which is zero so into our point slope form we have why, minus y one. So why minus four? Hopes my minus four, um, is equal. Two of the slope zero times. Well, X minus X one times X minus zero. Well, this become zero minus whatever. This is just zero. Um, and so I get why minus four is equal to zero. Therefore, our equation of the tender line here is just why is equal before, right? And then forward, part seat. You want the equation of the tangent line now to the graph at the point five comma Negative. 21. Okay, So again, we find the slope of the tangent line at this point, which is evaluating the derivative now at the X value. So that five f prime of five that's gonna be equal to, um, negative two times five, which is negative. 10. Okay, so then our point slope form is well, why, minus by one. So why? Why? Why? Minus a native 21 is why plus 21 is equal to theme slope, which is negative 10 times x minus X one, sometimes x minus five. We can sell for why he gets that. Why is equal to We have a negative 10 X and then we start over 21 AB abstract Another five and we get the t. Um, Why is equal to negative 10 X um, plus 29. Yeah, right.

Now this question wants us to find the equation of the tangent line of this curve at the point X equals two. So first let's find what why value This equation has X. Goes to And that's two to the third, which is 8 -7, which is 11 to the fifth. So it's one. So this point is to one. Now let's find the derivative which is a slope of the tangent line to the power rule. Now we need to take the derivative of the inside, which is three X squared. And that gives us 15 X squared times X cubed minus seven to the fourth. And we want to find the derivative When x equals two. So that's 15 times four times two to the third, eight minus 711 to the fourth. So 15 times four which is 60. So our line is going to have a slope of 60, so y equals 60 x plus some value B. And we want to know what this intercept is. So we know that at X equals two. Why is one? So let's plug in one and two to find this be, so B is one minus 1 20 which is negative 1 19 to our final line is why ICO 60 x minus 1 19.

So the theme of a tangent line problem is that you need a point and a slope. Eso as you look at this problem although they give you, is that the X coordinate is to they don't they don't even give you the y coordinate. Uh and then the second piece is the slope, which is finding the derivative and then plugging in X equals two into the problem. So, um, at this point, we have all of our information where we can go ahead and solve three x of the fourth minus two X squared plus seven and the whole premise for the at least finding the Y value is plugging into and through all these values. Mhm. Um, I thought about doing this in my head, but I'll probably make an Al Jerome mistake's. So I'm gonna go ahead and use a calculator to to the fourth. Unless, yeah, it's pretty much to to the third plus seven, and you get a, like coordinate of 47. So now what you need to do is find the derivative of this. So do you y the X. Let's bring the four in front. Three times four is 12 x to the third minus two times two is four. And now you need a plug in X. He goes to here and again. I'd be afraid that I'm doing this incorrectly. So I'm using a calculator and plugging into and for both these excess, I get a slope of 88. Now, I typically do this change, align problem in point slope form where you write down the slope and then X minus the X coordinate and then plus the y coordinate. I leave my answer like that. Otherwise you could distribute this in and combine like terms to get an answer of 88 x 88 times negative two is negative. 1 76 plus 47 is negative. 1 29 is a second option for you.


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