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

Determine any function which is discontinuous at $x=1$ and $x=5$ but which has a derivative at every other point.

$f(x)=\frac{\left(x^{2}-1\right)\left(x^{2}-25\right)}{(x-1)(x-5)}$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 3

Limits and Continuity

Derivatives

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

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.

01:03

Determine the numbers, if …

02:22

Determine the values of $x…

01:17

Determine for what numbers…

01:05

Find the $x$ -values (if a…

06:48

Decide whether or not the …

Determine all values of $x…

01:53

this problem we have to find we have to determine such a function, which is discontinuous, which is discontinuous, which is discontinuous at X is equal to when and by and X is equal to five but has a derivative at every other point. So let us take this function. F of X is equal to X super minus one into execute two in minus 25 divided by X minus one into X minus five. So let us check this function as when you substitute X is equal to one and X is equal to five in the function. In the function we get, denominator is equal to zero for access equal to one or for accessible to fight. The denominator of the given function becomes zero. It means that the UN function f of X is not defined, not defined at these pipes at this point. So we can say that so we can say that the function is discontinuous at X is equal to one and X is equal to five. But this discontinuity is removable discontinuity, so we can remove this discontinuity by simplifying like human function. So let us simplify the U. N. Function. Therefore, facts is equal to X. U K minus. One can be written as X plus one into X minus one and exhale minus 25. Can be written as X minus five into X plus five, divided by X minus one into X minus five so we can cancel from numerator and denominator. So we have F of X is equal to X plus one into X plus five and which is equal to exa kit plus six X plus five. Now we will check the different she ability of this function. So let us find the derivative for the given function. So this will be equal to there were two of of execution is two X plus six so we can see that f beige of x is define. Do you find for all real numbers for all real numbers? So we can say that the given function is the human function has the derivative at all points other than X is equal to one and five because this function is discontinuous at this point. So if a function is discontinuous, that cannot be differentiated, but it can work this possible. A different differentiable function is always continuous, so this is this is the required function which is continuous, which is discontinuous at X is equal to one and five and having their veto at all other pipes.

View More Answers From This Book

Find Another Textbook

01:09

Use the zeros, vertical and horizontal asymptotes along with the sign of the…

02:41

sketch the graph of the given ellipse, labeling all intercepts.$$\frac{x…

01:52

Find the slope of the line passing through each pair of points.$$(1,4) \text…

Find the equation of the parabola with (a) focus at $(h, k+p)$ and directrix…

04:11

Determine (a) $f(x)$ ) and the domain of the composite function, (b) $g(f(x)…

Find $f^{\prime}(x)$.$$f(x)=3 / x$$

01:16

Find the indicated limit.$$\lim _{x \rightarrow 2}(7.1-3.4 x)$$

You are given a pair of equations, one representing a supply curve and the o…

01:14

Find the equation of the parabola with focus on the line $y=x$ at $(h, h)$ a…

01:36

The revenue (in thousands of dollars) resulting from the demand for a given …