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StepThe slope of the tangent line at any point on the graph of f(x) = 972 4x is given by the derivative of f at x To find the derivative of the given function, we w...

Question

StepThe slope of the tangent line at any point on the graph of f(x) = 972 4x is given by the derivative of f at x To find the derivative of the given function, we will use the four-step process_Step 1: Compute fx + h) and expand the expression_fx + h)(x + h)2 + 4(x + h)xh + h2) + 4(x + h)xh + 9h2 + 4x + 4hFinally, simplify (if possible) bY combining any like terms_{ + h)B1nS42 6eueannot come back)

Step The slope of the tangent line at any point on the graph of f(x) = 972 4x is given by the derivative of f at x To find the derivative of the given function, we will use the four-step process_ Step 1: Compute fx + h) and expand the expression_ fx + h) (x + h)2 + 4(x + h) xh + h2) + 4(x + h) xh + 9h2 + 4x + 4h Finally, simplify (if possible) bY combining any like terms_ { + h) B1n S42 6eueannot come back)



Answers

Use the four-step process to find the slope of the tangent line to the graph of the function at any point. $$f(x)=8-4 x$$

Yeah. In a given question, we need to find the slope of dysfunction at that point minus two comma 16. A slope of the function is nothing but the derivative of the function at the given point. So first we need to find the derivative, which is where the X L g fx. And that cases divide the X f X to the power four, which is equal to four times X. To the power four minus one, which is equal to four X cubed. Then the slope at the given point will be liberty X of G f x 37 point excess colder minus two. Whatever you call to for the four multiplied by -2 to the power three. It's called the four. Multiply and minus eight. It's called minus started to you see this will help to function.

In this video, we're going to be working with secret lines and tangent lines and for those of you that need a refresher. A second line is a straight line that joins two points on a function. While the tangent line is a straight line that touches the function at one point. So in this question we are given the graph X squared plus X. And we are asked to first find the slope of the second line joining two, f. 2 And four F 4. So, you know by looking at our graph, these values are going to be 26 and 4 20. So the first thing we're going to do is bring out our slope formula which is m is equal to change in Y over change in X. And we're going to plug in our values. So the change and why it is simply 20-6 And the change in X is 4 -2. Mhm. This simplifies to 14/2 or simply seven, Meaning that our final answer for part eight is 7. Mhm. Mhm. Now in part B. They want us to find the slope of the second line joining two F two and two plus H f F two plus H. So what we're first going to do is evaluate what FF two plus H is. So F of two plus H is going to be mhm two plus H squared Plus two Plus H. Further evaluate this, we get H squared plus six H. Sorry five H plus six. Mhm. Now we're going to plug this back in to our original slope value so each squared Plus five h plus six minus six Over two Plus H -2. Now all of these cancel out and we see that the H and the non majors going to cancel out the ages in the numerator, there's a final answer for the soap we get em is equal to H plus five. So the final answer for heartbeat is H. post five. Mhm. What they ask the sport in part C. Is the slope for the tangent line at two F two. And we know that the slope of the tangent line at any point on a function is the limit. As a jew approaches zero of the post h minus F. They all over. H. So this part of the limit should ring a bell to you because it is exactly what we calculated in party. Because if you see what we did in part B was f of two plus H minus F two. And we know that our denominator before we knocked it out simplified to H. So we know that if we want to find the slope of the tangent line at two F two. But we have to do is find the limit of this part H. Plus five. And we know that if we want to evaluate that, What we have to do is plug in zero for H. Yeah and when we do that We get five as our final answer, Meaning that our slope is five. Mhm. Now what they ask us to do in part B is to use all the information that we found in part C. To write the equation of this tangent line at two F 2. So we know that the equation for a line is why is it good to mx plus B? We already know that our end value is going to be five. So why is the go to five x plus b. And because of the definition of a tangent line where we know that 26 is going to be the only point that intersects this line. What we can do is we can plug in to six to get our B value, so we get six is equal to five times two plus B. And as a final answer we get that beat is equal to negative for, So our final answer is why is 0-5 X -4? And this wraps up our question Yeah.

Okay, so rest defying the equation of a tangent line. Well, let's note that in order to find our tangent line, we're going to need some point what's given. And we also need to find our slope so we can use the following equation to find our spoke. So we'll start with that. And once we have the to weaken use, points don't form to plug in our slope and our point to find our equation overtimes in life. So we want to find F prime at sea, which she is equals R X points. So that's needed to. So that's equal to the limits as X approaches negative, too, of negative four X squared, minus negative for times negative, two squared. So it's time to four, so it's a negative 16. But we have times that I gotta. So this is plus 16 over X Plus two and now it's not that in our numerator we can talk. They're all a negative four, and we're left with X squared minus war, and we can use the difference of two squares Factor dust into the fallen. Okay, now let's cancel like service and use Jack sub so we have negative four times major Tu minus two. So this gives us a They get a four times I get a four, which is equipped to, um 16 for our slope. Okay, so now that we found ourselves, let's plug this into our point. So points, folks. Which is why I minus white one, is equal to M times X minus X one to find our tank in line. So we have why minus negative 16. So that's close. 16 is equal to em. What, you 16 times X minus negative. Two. So that's plus two. So this is equal to 16 x plus 16 times two. Which is you 32? No, all we have to do. It's a track 16 from both sides, and then we get our equation of or tangent line. And that's going to be equal to why is equal to 16 x on dirty to what it's 16 is equal to positive. 16

Okay, so we want to find the equation over a tangent line. So we're going to need some point and our spoke and to find I slope, we can use this equation here. So let's start by finding art slope. So we have time at sea where C is our ex point. So that's one. So that's gonna be equal to the limit as X approaches one of our fallen function. Negative two X squared plus X minus three. And now we want to find F evaluated at one. So that's gonna be a native to plus a one minus three. So that gives us a negative four. So we have minus and that you don't force. That's plus for so we actually have a positive one here. So this is all over X minus one, and I Let's factor out our new Rita. So we convey activists into a negative two x, um, times something, and then an x. I'm something. We want this to be a positive Valium. So let's take negative one for both of these. So we get negative two x plus two x minus accident plus one. So that's good. And this is over. X minus one. And now we can cancel like times and I'll let's use ducks up. So we get negative two minus one, and that's equal to negative three. So the series are slow. So using point Slope form, let's plug in or slope and art points to find our equation over attention line. So we have why. Plus four is equal to negative three times X minus one. So that's equal to negative three X plus three. Now let's subtract for on both sides and we get that or equation of our tensions line that's going to be equal to y is equal to negative three x minus one


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