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(a) Find the slope of the tangent to the curve $ y = 1/\sqrt{x} $ at the point where $ x = a $.

(b) Find equations of the tangent lines at the points $ (1, 1) $ and $ (4, \frac{1}{2}) $.

(c) Graph the curve and both tangents on a common screen.

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a) $-\frac{1}{2} a^{-\frac{3}{2}}$b) $x+2 y=3$$x+16 y=12$c) see solution

09:44

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 7

Derivatives and Rates of Change

Limits

Derivatives

Brooke H.

December 11, 2020

Awesome!

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

Boston College

Lectures

04:40

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.

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we have a question and it's we need to find the slope of turned into the curve. Who is equal to one Bundle. Tax or accelerated power -1 x two. At a point where X equal to a. So strong contingent means we need to find first Dy by dx because this is the slope attention this will become minus two X. Raised to the power minus one by two minus one minus one by two X. Raised to the power minus three by two. So D. Y by dx at that's equal to a. Will be equal to -1 x two areas to the Power -3 x two. Some miners one by two areas to the power three x 2. This could Britain as -1 by two. A under route. So this is a slope attendant at X equal to a. We have to find equation of tension lies at the point of 1:01 and for commerce one by two If point is 1:01. This is excellent. There's a Y. One. So the web I. D. X. That is low attendance at 1:01 will be equal to -1 Byte to simply because he is one here. My question would be why minus y one equal to? Do you have idea at this point? An X minus Excellent. So why -11 will be -1 x two. X -X one. Okay so why would we call to -1 by to work Plus one x 2 Plus one. So I will be equal to minus one by two weeks Plus three. Back to this is because I'm tangent at one. Common. Similarly equation of tangent at four comma one x two for that we have to find the slope of the tangent ah four comma one by two. Well that explains the value of X as fall over here so that it will be -1 x two -1 x two and x rays to the par minus two by two which is four raised to the power minus three by two. So -1 x two. For this to the power provide to and four years to risk wire. So this will be eight -1 x two and 2. A two slope is -1 x 16. The question will be why -1 equal to -1 x 16 X minus excellent. So why minus why one will be equal to minus one by 16. PL one x 4. So I will be equal to minus one by 16. X plus one by four Plus one x 2. Uh So I will be equal to minus one by 16 X plus 3. +54 resistant. They are offended at this point. No. Yeah. Uh graph the curl and both attendance on the common screen. So graphing the curl will be okay. No, no, we'll be first trying to grapple Why? Well too. And behind her back we can see that when X approaches to zero, Why will approach to in the finite? And when X approaches to zero. when y approaches zero, X will approach to in finite. So at access and y axes both are there are some totes it should go like this more precisely and it will definitely be in only first squadron because X cannot be negative over here because it is under the square root. So we have to draw the attendant both attendance. First engine days. Mine's one x 2 x plus three x 2. Okay, Standard # one is uh well I got two minus one by two plus three by two, Y tu minus and by two words Let's see by two. So when X approaches X equal to zero, Y 43 by two, this is 1.5 and it is tangential at one one comma one 0.1 comma one it is the intention at 10.1 common for most passes to one Former 1. So uh you have to redraw the curve. Uh huh Right. This it is one comma and superb must pass through this point apart from her one like this. Okay. Oh this will be the first danger. Which is why people too. But this is the first engine second is Why go to -1 x 16 x plus three x 4. So when x equal to zero while three x 4 When it's equal to zero. So I went to be back for and point is point of tendencies Full comma one x 2. So for summer went by too. So this is our second indent, which is so I called to minus one by 16 X plus one by two. Uh, um Let's be 5 4. So these are the answers. Thank you. Mhm.

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