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Determine the equations of the tangent lines from the point (a) (1,10) (b) (4,17) to the curve $y=2 x-x^{2}$

(a) $y=6 x+4, y=-6 x+16$(b) $y=-16 x+81, y=4 x+1$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 2

Derivatives Rules 1

Derivatives

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04:40

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.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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here we have a calculus problem which will be using derivatives to solve for the functions of the tangent lines, which start from points 1 10 and 4 17, which are tangent to the curve y equals two x minus X squared to begin. Let's acknowledge that we're working with a function of X, which is equal to two X minus X squared. It's derivative is equal to two minus two x, which is also the slope of our tangent lines, which is equal to two minus two A for working with some general value of A and our point would also end up being as it follows that a is a function of with a fib. Forex is in our wise, giving us a general point to work with of a two a minus a squared at that by plugging a into our ffx equation into our function. Now we're going to start by using the point slope form, which is why minus y one is equal to m times X minus X one plugging in our points we just developed. We're going to have y minus two a minus. A squared is equal to M, which is two minus two a times x minus x one x one In this case is a simplifying. We end up with y minus to a plus. A squared is equal to two x minus two a minus two a x plus to a squared, further simplifying and getting why by itself we find that why is equal to two X minus two a X plus a squirt. Highlight this equation so that we can't lose it when we're going to be using. Now it's time to input are X and y values that we were given that are tangent lines cross through. So starting with our point 1 10. So at this point, we'll plug that into our equation. Where we have 10 is equal to two times one minus to a times one plus a squared. Our goal here is to solve for a so simplifying we have 10 is equal to two minus two a plus a squared subtract 10 from both sides so we can set the whole thing equal to zero. To get zero is equal to negative eight minus to a plus a squared. We were to break this apart. We would get a minus four times a plus two equal to zero. If we set each of these values in each of these parentheses, sets equal to zero themselves will end up finding that A is equal to four and A is equal to negative two. Using these values, we can then plug them back into our wide equation so we can do take ry equation. We'll say Why is equal to two X minus two? A. Let's use four to start times four times x plus four squared. Doing that, we get that Y is equal to negative six X Plus 16 that gives us one of our functions of the tangent line doing the same thing with our value of negative two we have y is equal to two X minus two times negative two X plus negative two squared. We find that Y is equal to six X plus four. Here is our second function of the tangent line that starts or crosses through the 0.1 10 doing the same now for the point 4 17, we're going to plug these values into ry equations. Giving US 17 is equal to two times four, minus two a four plus a squared following the same steps we did in our last problem. We have 17 is equal to eight minus a A plus a squared setting this equal to zero. It equals negative nine minus eight a plus a squared breaking this apart. We have a minus nine times a plus. One is equal to zero. If we were set each of those equal to zero on their own, we would find that a is equal to nine and A is equal to negative one. Using these values again, we're going to plug them back into our Y equation to solve for the function of the tangent line we have. Why is equal to two X minus two times nine to use our value of nine for a X plus nine squared. Giving us why is equal to negative 16 x plus 81. Mhm. There's our first tangent line that starts at 4 17 or crosses through it. Doing the same thing with for A is equal to negative one. We've got Y is equal to two X minus two times negative one X plus negative one squared. Giving us why is equal to four x plus one. Mhm

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