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(a) Find the slope of the tangent line to the curve $ y = x - x^3 $ at the point $ (1, 0) $ (i) using Definition 1 (ii) using Equation 2
(b) Find an equation of the tangent line in part (a).
(c) Graph the curve and the tangent line in successively smaller viewing rectangles centered at $ (1, 0) $ until the curve and the line appear to coincide.
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a) (i) -2(ii) -2b) $y=-2 x+2$c) graph unavailable
Calculus 1 / AB
Limits and Derivatives
Derivatives and Rates of Change
September 21, 2020
What is a numerator and denominator?
Well the top number in a fraction. Shows how many parts we have. (The bottom number is the Denominator and shows how many equal parts the item is divided into.)
Can someone explain what the tangent line is?
Hey Samantha In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve.
Anyone else confused by the linear function, can someone explain?
I know this one! In mathematics, the term linear function refers to two distinct but related notions: In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one.
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In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
(a) Find the slope of the …
a. Find an equation of the…
Alright so here we have the function Y equals x minus X cubed. And we're going to find the slope of the tangent line at 10 using two different methods, both of them are limit definitions of the derivative. Um But we're gonna go ahead and just show how to do both. Okay, so let's first do the one on the left and this is where we basically do rise over run. We do the lameness. X goes to one of our differences between F of x minus fo one over x minus one. So for us then let's go ahead and plug in our F of X X minus X cubed minus f of one. When we plug in one we get minus one minus one or zero. Okay so our goal, if we plug in one right now is is 0/0 so we can't solve that yet. So we're gonna go ahead and see if we can factor, we can definitely factor on the X. Out of the top. And that leaves behind um one minus X squared which we go, oh well that's the difference of square. So I can factor that and that will give me x times one plus X and one minus X. So that's the top, fully factor the bottom. I noticed that I have something very similar on top to this. So if I pull out a minus That I can rewrite the denominator as 1 -1 and then times of -1 and this is what we're looking for because to solve this and not get a 0/0 case, we need to be able to cancel something. Such that when we plug in one it's no longer a 0/0 case. Now when I plug in one I'll get one times one plus one Over a -1. So basically we get -2. So our slope of the tangent line at the .10. Using the This definition of the derivative is um solved -2. So now the goal is if we go ahead and do the blue one that we will um let's changes to blue. Oh that we will get the same answer right? That would be ideal. So let's go ahead and substitute in our function but one plus H. For every act. So we get one plus H where X goes and then we have X cubed. So we have one plus H cubed minus F. 01. No one is still zero. So it's like that in. Okay so here we're going to need to do a on the top, we're going to have one plus age and then we've got to attract off one plus H cubed. And hopefully you remember pascal's triangle because this gives us the coefficients. We need we need this rope. So therefore our big foil is one plus three H plus three H squared plus H. Cute. Um All over age and we're looking to see hopefully some things can cancel out. Let's see the ones do cancel out because we have one and then minus distributed and um what else do we have that could cancel out. I guess that's it. But what we do have is an age comment to everything. So I'm gonna divide through every term now by age And when I do that I will get one And I can distribute the negative as we go, one three minus three h minus eight squared all over one because I divide through by age. So um If now I plug in zero for age, I will end up with zeros for these terms and I will end up with -2 worked. Okay so both methods lead to a slope of the tangent line of -2. Alright, so now we're going to actually create the tangent lines. Let's do that below here. Okay so our tangent line in general is in this case we're doing it at one. So f of one Plus f. prime of one Times X -1. So we get f of one is zero, so zero plus minus two for our slope. And then X -1. So it looks like we get -2 x plus two. So our tangent line then I'll just rewrite it is minus two X plus two. That is our tangent line. Whoops. I will fix that. That is our tangent line to ffx At x equals one. All right, excellent. And if you have a calculator you can have fun zooming in um on this. If you make it pretty big, you get something that looks like this with a tangent line. Um Something like in read, if you zoom in, if you use your calculator to zoom in a couple of times, right at X equals one, you could really get it to where the black line. Uh because you're so zoomed in, look straight and your red line as well, so they really start becoming the same line. So anyway, have fun with your calculator zooming in. Um I found that when I zoomed in two of x. distribution of .975 And 1.0025, that was just too zooms in on 90 84 for X. And why I ended up being between -106-5 And .06-5. So for that zooming I was able to really get a nice where they look like the same line. So anyway, lots of fun. Hope you're doing well. Have a great rest of your day.
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