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Find the quadratic approximation to $f(x)=\sqrt{x+3}$ near $a=1 .$ Graph $f,$ the quadratic approximation, and the linear approximation from Example 1 on a common screen. What do you conclude?

The quadratic approximation is a better approximation of the function even at far away values.

02:36

Grace M.

Calculus 1 / AB

Chapter 3

Derivatives

Section 8

Linear Approximations and Taylor Polynomials

Campbell University

Baylor University

University of Michigan - Ann Arbor

Boston College

Lectures

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.

04:10

Find the linear approximat…

05:19

10:30

Find the best quadratic ap…

02:29

Suppose $f$ is differentia…

02:27

05:51

05:16

in problem. One. We want to give them any approximation of X at X equals X node, and we have X node equals one first step we get. If there's of X, differentiate the function. It was X toe. Get if the tropics we have root X. It's differentiation is half X to the bar of negative off or one divided by two roads. X. The second step is to get the value. If dish off exclude, we have explored equal to one. Then we get if they should one. The substitute here by X equals one. We have one divided by two. Route one, which is half. Third step is to get the aneurysm ation hair off X, which equals it will be if it's not, if one plus f dish off excellent, multiply it by X minus X node, which is woman we can get if one by substituting in the main function. My X equals one. We have Route one plus, if they should. One we've got. If there's one from step to Uh huh, multiplied by X minus one and to simplify, we have one minus half, which is half, then we have half multiplied by X plus one. And this is the linear approximation off F of X X node equals one. Now let's graph the function along with its linear approximation. If we have X axis here and y axis here 1234 and we have year 123 If we have root X, then we can get three points of root of X. The first point is zero on bond zero zero and zero. The first point here. Second point is one on one can point here. The third point, for example, is four on two four to here. Now we can estimate that would of X has this she This is root of X. That's that's a graph l affects hell of X. When we substitute by X equals zero we have they're of X equals half. When we substitute X equals one you have And of X one, they must be the same here at X node equals one and, for example, visibility substitute X Y three. We get a lot of X equals four. Divide by two, which is two. The first point 0.5 here. The second point is here. Third point is three and two Here we can see that the linear approximation is tension. Toe the function F x at the point, exclude equals one.

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