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Sketch the parabolas $ y = x^2 and $ y = x^2 - 2x + 2. $ Do you think there is a line that is tangent to both curves? If so. find its equation. If not, why not?

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04:02

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 1

Derivatives of Polynomials and Exponential Functions

Derivatives

Differentiation

Campbell University

Oregon State University

Idaho State University

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.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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he is Claire's. So when you're in here, so we're gonna sketch are two parabolas. I'm gonna sketch our first. Why is equal to X square with black and the other one in red? There are two problems. Look us follows and ver y is equal to x square. You get the derivative to be two x confer. Why is equal to X square nine is two X plus two we get to X minus two. So we're gonna use the equation. Why is equal to X plus be? And that's tension to why is equal two X square at X sub one comma ex subway in square and for why is equal to X square minus two X plus two We get X sub too on the except to square minus two except two plus two. So we get the equation. A is equal to a to accept one which is equal to to accept to minus two Your ex Someone square is equal to a X sub one It must be we get except to square minus two except two plus two is equal to a except two plus B And from our equation we get except one is equal to a over to on except to is equal to a over to plus one. And when we put it into our equations, we gotta be value of negative. 1/4 a square and B is equal to needed of 1/4 a square minus a plus one. We're a second equation. So when we set these equal to each other older right here, negative 1/4 a square is equal to negative 1/4 a square minus a plus one. We get a is equal to one B is equal to negative 1/4. So the equation we get is why is equal to X minus 1/4?

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