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The slope of the tangent line at any point $(x, y)$ on a curve is given by $2 \sqrt{x}$, if the point (4,10) is on the curve, find the equation of the curve.

$$y=\frac{4}{3} x^{3 / 2}-\frac{2}{3}$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 2

Applications of Antidifferentiation

Integrals

Baylor University

University of Michigan - Ann Arbor

Boston College

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

04:41

Find the slope of the tang…

04:37

Find the equation of a cur…

02:08

Tangent line Find the equa…

02:49

Find an equation of the ta…

and this problem has to find the slope of the tangent line to the kerb y equals one over the square root of X at the point where X is equal four. So to do this, I find it kind of easier sometimes to rewrite. This is a CZ more of a function notation f of X equals one over square root of ax at X equals four. Because now, in order to find the slope we know that the slope is given is a limit. And I just like to have the f so that I can write f of X plus age minus f of X over age, which is a nice expression. Now we just need to plug in our function so f of X plus h has won over the square root of X plus H minus one over the spirit of X all over age. But in this problem, we're given what exes were given. X equals four. So we can just plug that in for ex who right here and here. So one over the square root of two plus age, as you mean four plus age, not to plus age minus one over the square to four all over age. So this is a limit as H coast zero of one over the square root of four plus age minus one half minus one half all over age. So now, in order to solve this limo, we're going to need to combine the fractions in the numerator so that we can get it down to like a single expression. This is a limited H ghost, zero of to over two times the square root of four plus h. So we're just going to put everything in the numerator with a common denominator square root of four plus age over two times the square root of four plus h all over H. So once we combine this, this is limited HBO's zero now of two, minus the square root of four plus age over two times square into four plus age all over age. Now let's combine the denominators. Make this a little bit simpler is limited. H code zero of two minus the square root of four plus H divided by two times a tch Times Square into four plus age. So all we've done here, if we we've combined the two denominators by multiplying them together. So to route two times greater four plus h times H is to age times greater four plus age. So these two have been combined into this greatest emitting. Something still looks a little Marley. We're not entirely done yet. Whenever, Let's just just go ahead and go to a new page and rewrite our limit. We need a little bit more space. This was two minus the square root of four plus age over to age. Time to square two four plus h Have to go back and check. Yep, this looks good. So now whenever we have a square root in a limit like this, we want to multiply by the conjugating. So is going to put a plus instead of a minus two plus the square to four plus age over two plus the square root of four plus h. When we multiply this out, the numerator is goingto be a ble to cancel really nicely the numerator we have ah four minus four plus h all over two times a tch Times square to four plus age times two plus the square to four plus age denominator is still a little little gnarly, but it's hanging there. We're getting this nice cancellation in the numerator of four minus four. And so we're just left with a minus h and the numerator. Let's write that out. Minus h over to age squared of four plus age times two plus square to four plus age. Now we have this age that we can cancel. We haven't h in the numerator and h in the denominator. Go ahead and cancel those out. And so we're left with the limit as H goes to zero of negative one, divided by two times the square in a four plus age times two plus the square root of four plus age. Even though this still looks kind of bad, we're at the point where all we have to do is plug in h equals zero. So this is a negative one over two times the square root of four times two plus the square to four. So negative one over two times two times four, which is negative. One over sixteen. So the slope of this tangent line to this graph is negative. One over sixteen

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