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Discuss (i) the continuity and; (ii) the differentiability of the given function. (iii) If a discontinuity is removable, redefine the function so as to make it continuous there. (iv) Is it now differentiable?$$\text { (a) } L(x)=\left\{\begin{array}{cl}x-1 & \text { if } x<3 \\$$\text { (c) } N(x)=\left\{\begin{array}{cl}x-1 & \text { if } x \leq 3 \\2 & \text { if } x>3\end{array} \quad \text { (d) } P(x)=\left\{\begin{array}{cl}x-1 & \text { if } x<3 \\2 & \text { if } x \geq 3\end{array}\right.\right.$$2 & \text { if } x>3\end{array} \quad \text { (b) } M(x)=\left\{\begin{array}{cl}x-1 & \text { if } x<3 \\1 & \text { if } x=3 \\2 & \text { if } x>3\end{array}\right.\right.$$

(a) and(b) (i rem. disc at $x=3$ (ii) not diff at $x=3$ (iii) $L(3)=2$ (iv) not diff. at $x=3$(c) and(d) cont. everywhere not diff. At $x=3$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 3

Limits and Continuity

Derivatives

Campbell University

Harvey Mudd College

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.

01:08

Consider the function $f(x…

01:11

Let $f(x)=x^{3}-x^{2}+x+1$…

So in this question we have F of X is occurred to is occur two. This is X cubed. This is X less than zero. This is the excess care. This is extroverted than occur to zero less than one. This is two X minus one. This is X greater than incurred to one, less than two. This is X squared minus two X plus three. This is extra to learning per 2 2. This is F dash of X. Is equal to F dash of X will be equal to vote. Will be core to rise of excess care x less than zero. This is two. X. X is greater than incurred to zero less than one. This is true. This is true. One expert in the nickel tour in less than two this is two X minus two. X is greater than the 32. F of X is continuous and and differentiable and differentiable for X belongs to our. So which option is correct And and he is correct and say is correct.

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