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Use linearization to approximate the given quantity. In each case determine whether the result is too large or too small.(a) $(4.001)^{4};$(b) $(1.003)^{5}.$

(a) $256.256 \mathrm{small}$(b) 1.015 small

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 6

Linearization and Differentials

Derivatives

Oregon State University

Baylor University

University of Michigan - Ann Arbor

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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.

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04:17

in this question. We are going to estimate the the number 4.1 to the power of four. Using linear approximation, linear ization approximation or approximation by linear ization. Now we know that this 4.001 is very close to the hole number four. So we are going to use our point Our ex not is four and our function will be people to X to the pole. For now by the formula of the linea arised equal Lynnie, lynnie arised uh expression equation will be go to F A. In this case if it's not plus I have prime. It's not. Yeah, multiply by x minus X. Not no. So obviously we need to find F prime of X. Note. So first we have to find the F prime of X. It is found by differentiating effects And that will be four x to the power of three. And therefore if prime of X not now substituting X. Note which is four. Um we are going to find four by 4 to the whole three. He gives us 2 56. So this is our f prime of excellent. Then this is our f primary fixed. Yes. Now to find ethics, We know that if it is it's about four and F X not We substitute our four into FX then that would give us um 256. So both fx and fx we're F prime X and F prime And FX not to 56 and plugging into our formula to find the linearize expression equation we have L X is equal to 2 56 Plus 2 56 bracket. Yeah, X minus It's not. Which is four. Now substituting in our yeah 4.001. Mhm. Which is our X. That we had substituted with the next year. Now going forward Get to 56 Bracket without teaching our 4.001 minus four. And uh this is going to give us 256 Time, And solving This will give us to 56.256. And if we just press this into a calculator to find them, The the solution the exact solution to 4.001 to the power before. We will find a number that is greater than they saw. This is actually smaller then what would get from me chocolate. Next we are going to mhm. We're going to um solve To approximate 1.003 To the power of five. Using a linear ization. Now we know that 1.003 is very close to the number one. So we're gonna use we are going to use X not is equal to one and how F X will be equal to X. to the ball five. Now we know that we need F prime of X. So we differentiate extra about five we get the bar comes down five X. To the power four. And how formula given before is F X not plus if prime of X. not bracket x minus x. Not no um solving for X. Not in our fx we're gonna substitute F. X. Not and this will give us substituting in one. You just give us one have prime of X. Not Start shooting in one. We're going to get in five. No substituting everything into your formula. Get L. X. Is he called to fx note which is one plus F. Prime of X. Not he's five Bracket X -1. Now substituting in 1.003. To estimate the final solution, We'll get five plus 1.003 minus one. And this will give us um our solution is one 015 and This solution is actually smaller than what you get by plugging in 1.003245 into a calculator. This is your final solution. You're always

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