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Linearize $f$ near $x_{0}$.$f(x)=3 x^{2}+12 x-3$(a) $x_{0}=0$(b) $x_{0}=-2$

(a) $y=12 x-3$(b) $y=-15$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 6

Linearization and Differentials

Derivatives

Oregon State University

University of Michigan - Ann Arbor

Idaho State University

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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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in this problem we are given the function F of X. Is equal to three, exclude Plus Cough X -3. And we are to linearize near the points. X not is good to zero And it's not difficult to -2. 1st. We have to the formula for the linear arised function is L. Of X. Is equal to F of a plus F. Prime of a. Multiply by x minus a. No 1st. We calculate f prime of X. Can you get a form of X equal to six X plus 12. For the first point we have X not being equal to zero. We calculate F of zero substituting 0. In the end of its function We get our zero is equal to -3 and our f prime of zero as equal to substituting in the f prime of X. Function Is equal to 12. No substituting these values into our inner as function equation. We get our Type of a -3 plus 12 player by here we have x minus seal Which gives us 12 eggs minus three. Has our linearize function equation. Mhm. For the 2nd 1 We have our initial point is equal to -2. Um Calculating FF -2. F of -2 is equal to three -2 squared plus 12 -2 -3 which gives us This is 12 -24 minus three. Yeah minus three which gives us- Cough -3 -15. So we have half of -2 being equal to minus 15. Next we calculate our f prime of -2 Subsequuting -2 into six of -12. We catch this is minus go plus Hope, which is equal to zero, substituting me to R L. Of X equation. We get f prime f of a is equal to -15 plus F. Prime of A is going to zero, so everything becomes zero. We get our L. Of X S. 15 and it's our final solution.

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