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$$f(x)=\left\{\begin{array}{l}x^{2} \text { if } x \geq 0 \\x \text { if } x<0\end{array}\right.$$ (a) Sketch the graph of this function. (b) Determine $f^{\prime}(x)$ if $x<0$ (c) Determine $f^{\prime}(x)$ if $x>0$ (d) What can you say about $f^{\prime}(0) ?$

(a) Graph is answer(b) $2 x$(c) 1(d) Does not exist

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 1

Slope of a Curve

Derivatives

Campbell University

Oregon State University

Harvey Mudd College

Baylor 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.

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.

01:52

$$f(x)=\left\{\begin{align…

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What can you say about the…

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Sketch the graph of each f…

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A function $f$ is given.

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(a) Sketch the graph of th…

04:04

A function $f$ is given. (…

02:33

Sketch the graph of a func…

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01:03

Sketch the graph of $f$.

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03:10

(a) Graph $f(x)=2^{x}-4$

here we have a piecewise function that we want to graph. To start. You can see that where F of X is equal to X squared, it must be in the first quadrant because X needs to be greater than or equal to zero and where F F X is equal to X has to fall in the third quadrant because X must be less than zero. You would see normally that our first piece, X squared, would extend also into the second. But we're limited by those expounds same thing with our ffx equal to X, that would normally be a straight line extending into the first quadrant. But we have to truncate it there at zero because it's not allowed to exist if X is anything but less than zero. So let's start by determining our derivative of f of X. If X is less than zero doing that, we see that we're gonna be using the second piece of our function were f of X equals X, because X being less than zero satisfies its condition. To take the derivative of this, we're just gonna use the power rule, and that's gonna give us that F prime of X is equal to one. That's as if we have X to the first here. Right? And then one gets multiplied into here and we subtract one from our exponent, which leaves us just with one. Now, if we want to do the same thing for our first piece of our function when X is greater than zero, we start with our f of X is equal to X squared, taking distributive. We do the same process as we just did to get f Prime of X is equal to two X And now suppose we wanted to find f prime of zero uh, doing this, it gets a little bit tricky because taking a look at what our bounds are within a piecewise function. Those limitations, it appears as though we should be able to use our first piece where F of X is equal to x squared and how we just found that that's ones derivative is equal to two X because the bounds are that X must just be greater than or equal to zero. So we might think that we could just use that piece. However, given the fact that at zero is the point at which our graph changes directions. That's where these two pieces meat. So we were to make it really obvious We've got the straight line, and then all of a sudden we go like this. So this cusp right here means that we actually cannot have any slope any point on a graph. When you have a cusp, your slope will be non existent. So in this case, F prime of zero does not exist.

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