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If $ f(x) = 3x^2 - x^3 $, find $ f'(1) $ and use it to find an equation of the tangent line to the curve $ y = 3x^2 - x^3 $ at the point $ (1, 2) $.
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01:35
Carson Merrill
01:50
Madi Sousa
02:28
Ashley Boni
Calculus 1 / AB
Chapter 2
Limits and Derivatives
Section 7
Derivatives and Rates of Change
Limits
Derivatives
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November 4, 2020
Harvey Mudd College
Baylor University
University of Nottingham
Boston College
Lectures
04:40
In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.
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.
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Alright so here we have a function epithets equals three X squared minus X cubed. And we want to find the tangent line. Alright, so the general form of the tangent line can be re written as Y equals F of all. We have to know where we're doing it at our tangent line is going to be at um X equals one. Okay, so we're going to find F of one plus F. Prime of one times X minus one. Alright, so for F. Of well we can plug into our expression and I'll do it up here on the right. F. F one is basically three times one squared minus one. Cute just plugging into the function. So three minus one is two. Alright, so F of one is two. Now we need to find F prime of one but let's go ahead and just do F prime of X first. We're going to do power roll on our function F. So we will get three times two are six X to the first power minus three X squared by power role. So if we want if prime at one we'll plug in one for X. So it's six times one -3 times one squared. So that's 6 -3 or three. Excellent. Okay, so now we're almost there. Why then is f of one which is two Plus three times X -1. So that is our tangent line. If we decide to put it in more standard form Then let's see that's three X -1. So we can write it as y equals three x minus one as our tangent line. Alright? There it is, and hope you have a wonderful time and a great rest of your day.
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