Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Given $f(x)=|2 x-5|$ (a) At what point is the function not differentiable? (b) What is the derivative to the left of this point? (c) What is the derivative to the right of this point?

(a) $(5 / 2,0)$(b) -2(c) 2

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 1

Slope of a Curve

Derivatives

Oregon State University

Baylor University

University of Nottingham

Boston College

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.

02:07

Calculate the derivative o…

00:58

Given $f(x)=\sqrt{x}(\text…

04:45

Determine the derivative a…

01:15

Find the derivative of the…

00:14

Find the derivative of $f(…

01:21

Find the derivative where …

05:19

01:05

Find the value of the deri…

here will be using our knowledge of derivatives to determine that for a function which is an absolute value. Here we have the function F of X is equal to the absolute value of two X minus five. I sketched a quick graph here just to give us a visual representation. As we move forward, let's first find the point at which this function is not differentiable. That would occur as we know at the point of the cusp right here and given the function. This occurs at a point where why is equal to zero. You can see where it's at their so to do this, we can just plug zero into F of X. So zero is equal to the absolute value of two X minus five for purposes Here, though, we don't actually need those absolute value terms because we're just working with the cusp. Let's just bring it to two X minus five, adding five to both sides. We get five is equal to two X, and then we find that X is equal to five divided by two. That would be the point at which this function is non differentiable because you cannot differentiate at the cusp now to determine what the derivative to the left of this point might be. So that's going to be everything within here, right there. Let's determine that by taking our derivative See you. So we have f of X is equal to the absolute value two X minus five. Now, because we're to the left of this, we can see that we have a negative slope. So this is gonna be negative. Two X plus five, actually. Right. You just take you just change your signs to get rid of that absolute value. And you can see that that function makes sense right here at this point there. Why intercept? That would be five. And we have a negative slope, too. Differentiating this, we find that F prime of X is equal to negative. To do that by finding by using the power rule and the sum rule that's going to be our derivative to the left of that point. Now, if we do the same thing, but for to the right at that point we're going to work with this piece of our graph. We're gonna go back to our f of X, which again is equal to the absolute value of two X minus five. But in this case, we're just working with F of X, which is equal to two X minus five. We don't have to change the things with this one will just drop those absolute values. And we find that F prime of X is equal to two again using the power rule and the sum rule to find that, and there we have it.

View More Answers From This Book

Find Another Textbook

05:56

Decide whether or not the function is continuous. If it is not continuous, i…

03:49

Find the equations of the lines and plot the lines from Exercise $48$.

00:51

Compute the indicated limit.(a) $\lim _{x \rightarrow 0^{+}} \frac{1}{x^…

02:14

Determine the equation of the tangent line at the indicated $x$ -value.$…

01:33

Find $y^{\prime}$ if $y=\frac{x^{2}}{x^{2}+1}$

01:12

Find the linear supply function satisfying the following conditions: when th…

01:59

Short segments of the tangent lines are given at various points along a curv…

01:08

You are given a pair of supply and demand equations; identify which is suppl…

02:50

Find the point on the parabola $$y=a x^{2}+b x+c$$ where the tangent line is…

01:53

Use the appropriate rules to determine the derivative.$$f(x)=\sqrt[3]{x}…