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If $f(x)=(1+a x)^{k},$ where $k$ is any real number, show that its linearization near $x_{0}=0$ is given by $T(x)=1+a k x.$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 6

Linearization and Differentials

Derivatives

Campbell 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.

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Show that the linearizatio…

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Find a linearization at a …

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Given $f(x)=x^{2}+x(k-1)+k…

in this problem, were you with the function F X is equal to one plus A. X to both K. Where K is real. And we are to show that this near the point X. S X notice go to zero linear rises As linearize is to TX is equal to one plus AKX. Now to perform a linear ization we need to have prime of A. Where A. Is the point near the function. Now to find a prime of a. First. We have to Yeah. Two differentiate ethics. So have prime of X. Is he called to K comes down Bracket one plus AX Tip of K -1. According to the laws for intuition, more played by A Which is here inside now mhm. We get this as a K Bracket one Plus A. X. To the power of K minus what. Now to find F prime of A. We are now substituting X, not at zero. So it's X F prime of zero. We're just substituting zero where we see A. So we get a K. One to the both K minus one. Since Katie's real. One to the power anything is still one. So we get a K. Next we want to find F. A. Where a zero. So it's F zero. We are substituting zero here. Also we also get one substituting into a linearize expression formula which is F F A. F A plus F. Prime of a. Apply by x minus a. We are going to get um one Because FA is equal to one plus F. Prime of a A K Bracket X zero. And this His record to one plus. Hey, Kay X. Thank you. Which is equal to the given function T. X. Cool. So we have proved that it linearize is to the above given functions.

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