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Let $f(x)=3 x^{4}-7 x+10 .$ Approximate $f(1.01)$ by linearizing $f$.(a) near $x_{0}=1$(b) near $x_{0}=0$ and computing $T(1.01)$(c) Explain why one of the above choices for $x_{0}$ yields a better approximation then the other.

(a) 6.05(b) 2.93(c) 6.05 is better because the 1 is closer to 1.01 than is 0

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 6

Linearization and Differentials

Derivatives

Oregon State University

Harvey Mudd College

Idaho State University

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.

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in this question we are going to linearize F X. Is equal to three x. to the about 4 -7 x plus 10. To find. Yeah the approximation of F. Um 1.01. Yeah we know that. To linearize we use the formula of the linearize equation. Lx is go to F. Okay plus F. Prime of a applied by X minus a. Now we are given the points X. Not equal to one. An ex not easy. Go to zero. To use is the function approaches these points. Um First we have to find F. Prime of X Which is equal to 12 x. to the power of 3 -7 at X. Not is it what you want? F prime of one is equal to 12 minus seven Which is equal to five and F of one is equal to 3 -7 past 10 which is equal to six. Now substituting into the formula for the linearize equation. We get our our equation is X. Mhm. Is equal to If one which is six plus flies Bracket X -1. And this gives our lx yeah linearize equation is six minus five plus five. Six. Geico to one plus five X. Is our legalized equation Now to find the approximation of four FX. We have to substitute. Yeah 1.01. Yeah into Alex can we get our Alex as he called to six 0.5. Yeah is our approximation. Yeah it's X. Is equal to zero. We have f prime of zero. We are substituting into there into their um prime affects we get um cough its growth yeah huge minus seven. And this gives us minus seven for f of X. Where X is go to zero. Mhm. We get you're substituting zero into the function of uh fx fx. We get um 10. Now substituting into the formula Alex. We get our Alex has equaled two of 0 to 10 plus F. Prime of a. Which is minus seven bracket X minus zero. And this gives us how Alex is equal to 10 -7. X. Now substituting the substituting the 1.31 X Becomes 1.01. Then we get our approximation is 10 -7 seven, multiplied by 1.01. Yeah This approximation then becomes two 93. Mhm. Mhm. Next we asked why is um why is one um one approximation here, why is one um point better for the approximation than the other? So the better one here is X note is go to one why? Because it is closer to 1.1 than zero is close to 1.01. So x notice got one is the better uh point to use for this approximation

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