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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) $(3.99)^{4};$(b) $(32.04)^{3 / 5}.$

(a) 253.44 small(b) 8.006 large

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 6

Linearization and Differentials

Derivatives

Missouri State University

Campbell University

Harvey Mudd 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.

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02:38

In this question, we are going to estimate 3.99 To the power of four. Using approximation by linear authorization. Now we know that 3.99 is very close to four. So we're going to use our X not as equal to four and our effects will be equal to X. To the power for so that we can estimate this. Um This number. Now we know that by the linear arised uh miniaturization equation we will need F. Of X not plus F. Prime of X. Note apply by X minus X. Not so we will need F. Prime of X. And that is found by differentiating our ethics By the Parlow before comes down X to the power of three. And if prime of X not substituting in our X not which is four, we get four by 4 to 12 3, That's 2 56 and F. X not submitting in four X. Not here we get four to the power four. It's also to 56. And plugging this into our Alex formula. We get Alex As equal to 256 Plus 2 56. Well played by X -4. Now we know that you want to approximate 3.99. So we plug in 3.99 into our Alex and then we get to 56 plus 2 56 bracket 3.99 subtract four. And uh this will give us 2 56 plus 2 56 bracket minus 0.1 And sold him for this will get 5: 56.256. Um And this number, if you actually plug in 3.99 to power four into your calculator, you find that this number is actually um I made an error right there. It's 256 4: 53.44. If you plug that, If you calculate that it's 253 0.44. Now you find that this number is actually um smaller than what you would find by plugging in this into into a calculator. Next we are going to to find the mhm solution or they were going to estimate 32.04 To the power of 3/5. Now we know that 32.04 is very close to 32 And we're going to use 32 is Rx not the formula for Alex is still the same physical to F. Of X. Note plus F. Prime of X note. Mhm. More play by X minus X. Not now, we're going to use our fx Similarly as X to the power of 3/5. Now finding F prime of X. The poll comes down three is equal to 3/5. X. To the power of the new power is The Old Power -1. Get -2/5. And this can be simplified to look like these three over five. The fifth root of X squared like this and this is our primary facts. Now we will need to find fx not by substituting in our it's not 32 to the power of 35. And The 5th root of 32. That's two to the power three. That's eight. So FX Notice eight F. Prime of X. Not we are subjecting in um Ex not into this. So we get three over five. The few food of 32 To the power of two. Now we can see that we have a new power here. We're going to get three over five. The fifth root of 32. That's two to the power of two, that's four. And we're going to get 3/20 as our f. Prime of X. Next we're going to plug everything into our Alex formula. Which will give us L. X. Is equal to FX. note is eight plus. If prime affects not go to 3/20 bracket picks -1. Not our eggs. Not 32. Now plugging in our 32 0.4, what do you want to estimate? eight plus 3/20 drug hit now. 32. My necessary 32.04 -32. This will give us some 0.04. And if we complete this equation we are going to get eight eight point zero. Mhm 006. And this figure is actually uh larger than what you would get by plugging in 32.04 to the power of 3/5 into a calculator. And this is our solution. Mhm. Yeah.

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