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Let $f(x)=\sqrt{4+5 x},$ linearize $f$.(a) near $x_{0}=0$.(b) near $x_{0}=1$(c) Using an appropriate linearization, approximate (1.1).

(a) $y=\frac{5}{4} x+2$(b) $y=\frac{5}{6} x+\frac{13}{6}$(c) 3.08 using (b)

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 6

Linearization and Differentials

Derivatives

Missouri State University

Campbell University

Oregon State University

University of Michigan - Ann Arbor

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.

03:23

Find the linearization L(x…

04:00

f(x)=sqrt(1+sin x)Find…

01:01

Calculate the linearizatio…

01:30

find the local linearizati…

01:25

Find the local linearizati…

02:33

01:20

Find the linearization $L(…

02:15

00:31

Find the linearization $ L…

01:24

00:37

$f(x)=\sqrt{x+5}$a. Fi…

01:19

this question. We are given the function of it is you go to the root of four plus five X. And you're chillin arise near X. Not his go to zero and X one is equal to one first. We calculate the derivative of F. Of X. We get F X. Is equal to half pocket 45 eggs, both minus half More. Played by five. When we differentiate inside the brackets, this simplifies to 5/2 Route of four Plus 5 weeks. Yeah is F prime of X. Now To add near X, Nut is go to zero. We calculate F. Of zero. Our goal is to substitute this into the linearize equation. Formula F. A. Plus F. Prime of a. Apply by x minus A. So we need a we need F prime of A. So how A. Here is our eggs not and F half of zero is equal to we substitute zero here we get rid of four which is equal to two. F. Prime of zero. We substitute zero here and We get to the root of four which is two times two which is equal to get us 5/4. As our F. Prime of zero. To continue. We will be substituting these values in our L. Of X. Um which which will give us the linea arised expression here we have L affects is equal to F. of zero. It is too Plus if well c. f prime of 0.5 over four apply by x minus our A. Here is zero and this gives us two plus five. X over four is our dinner arised expression going forward? We are now going to use X. Not his go to one. Mhm. First we calculate F one substituting 1 into the health function. We get um a group of four plus five which is equal to three F. Prime of one. Substituting one into the equation of the derivative of effects substitute one. Here we get four plus five which is nine Hold of nine is 3 to buy 36. So we have 5/6. So now using the L. Affects which is um the equation for the linearize expression we get our Halifax As equal to three plus 5/6 bracket X minus A. O. A. S. one Which gives us three -5/6 plus five over six X. Mhm. This is like 18/6. So subtracting you get 18 over six plus 5/6 X. As our lynnie lynnie arised expression at near external physical too 21. Now we are to to approximate Uh huh. Um when X is go to 1.1 and we have to find which one of the two um is preferable. Either A or B. Now 1.1 is it closer to a or is closer to be since 1.1 is closer to one. So we are gonna use this approximation this expression this equation for approximation which is L 1.1 is equal to 18/6 plus five Over six. Apply by one one, and our solution will be 3.08. Yeah, yeah, that's the answer.

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