Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Use linearization to approximate the given quantity. In each case determine whether the result is too large or too small.(a) $\sqrt[3]{26.9};$(b) $\sqrt[3]{-27.2}.$

(a) 2.9963 large(b) -3.00741 large

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 6

Linearization and Differentials

Derivatives

Missouri State University

Baylor 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.

04:04

Use linearization to appro…

06:34

06:00

05:20

Let $f(x)=\sqrt{4+5 x},$ l…

04:34

Use the linearization $(1+…

01:57

01:25

Find the local linearizati…

in this question we are going to estimate the value of um Thank you brute of 26.9 Using a linear ization. Now we know that 26.9 is very close to It's not easy equal to 27 30. Close to 27. So we're gonna use 27 is our it's not next. We are going to formulate our effects as X cube root of X. Mhm Which is you go to X tip of one of the three. Now we know that by the equation to linearize we use the formula F. X not plus F. Prime of X not Mhm. Multiplied by X minus A. And therefore we need to calculate F prime of X. It is found by the following cheating fx. Now by differentiating the power comes down 1/3 More played by X. The New Power Becomes One of the 3 -1. You get -2 or three. This can be rearranged to look like 1/3 U. Boot Of X. to the power of two. Mhm. Mhm. Now because we need to substitute X. Not if it's much is equal to the cube root of 27 which is three F prime of X. Note Is equal to 1/3. The cube brooch off X. It is 27 okay To the ball three devolved to in this case And this would give us one over The cable of 27 is 33 squared. That's 9 9 x three. We get 27 again. So we now have our f. Prime fix. Not we have our effects note we have now we can substitute into our Alex And this is the co two. Effects note we get three plus F. prime of x. note is one of 27 More played by X- or x note which is 27. Okay, this is X note as well. 27. No because we want to Approximately 26.9 we substitute eat into Alex. Then here we have three plus 1/27 bracket 26 0.9 minus 27 solving for this, we will find our solution To be two nine nine nine 63 and this is actually larger than the value that we find by plugging in 26.9 and Finding the Cube root of 26.9 from a couple it. So this is our first solution next we're going to to solve for we're going to estimate the number minus the q boot of -27 point two. Now if we are going to we know that it's almost the same as the last one. You know that 12 -27.2 is also close to the number 27. So that is going to be our export and our ethics, He's still going to be extra power of 1/3. Um expertise actually minus since Mike It's -27.2. So it's close to -27. We're going to use our fx as you go to access to the power 11 3rd and F. Prime of X. Obviously it's gonna be 1/3 X 2 -2/3 which is a sequel to the previous average meant that we have cube root of X squared. Now we are substituting it's not as they were his into the equation. To find uh to solve for Alex which is previously given affects note plus. If prime of X not my play by X minus X. Not no for f. Prime of X. Note We are plugging in -27 Into this formula, we get 1/3 Q boot of -27 then we squared and since we are squaring that goes our our miners so we are going to have one over three. The Cube Root of -27 -3 Squared. That gives us nine this gives us 27 over here. So continuing we are going to oh we need F. X. Note Our FX note is going to be the cube root of -27 Which is equal to -3. Now plugging in into our Alex we're going to find alecks Equal to our efforts not -3 Plus, our prime affects note is one of 27. More play by X minus our X note which is minus 20 seven in judo, plugging in our Figure is -27.2 we get minus three plus one over 27 Bracket -27.2 -17, the same as 27 27.2 bracket. And solving this, we actually find our in of -27 is equal to minus three .00741. And this figure is actually larger than what we would find By plugging -27.2 the Cube root into a calculator, or it will be larger than the exact value. So this is our solution Yeah.

View More Answers From This Book

Find Another Textbook

Numerade Educator

05:05

The area of a square is increasing at the rate of 1 square inch per minute. …

01:55

Sketch the graph of the function defined by the given equation.$$f(x)=3\…

01:05

Determine whether or not the function determined by the given equation is on…

02:31

While we have stated the chain rule, for the most part we examined the speci…

03:00

How much should be deposited into an account today if it is to accumulate to…

02:01

Sketch the graph of the function defined by the given equation.$$f(x)=2.…

02:00

Sketch the graph of the function defined by the given equation.$$f(x)=3^…

01:10

Determine if the given graph represents a one-to-one function.

02:40

A large cube of ice is melting uniformly at the rate of 6 cubic inches per s…

Show that if two functions $f$ and $g$ have the same derivative on the same …