Let f(x)={ 2-x if x ≤ 1 x^2-2 x+2 if x>1 . Is f differentiable at 17 Sketch the

Let f(x)={ 2-x if x ≤ 1 x^2-2 x+2 if x>1 ....

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions over distinct intervals of the domain. This requires examining each segment separately, especially where the defining expressions change. Understanding piecewise functions is crucial as they often model situations with different behaviors in different ranges.

Continuity and Differentiability at Boundary Points

When dealing with piecewise functions, checking continuity and differentiability at the points where the definition changes (the boundaries) is essential. Continuity requires that the left-hand and right-hand limits of the function agree at the boundary, while differentiability additionally requires that the left-hand and right-hand derivatives match. Although the question asks about differentiability at an interior point of one branch (x = 17) where the function is smooth, these concepts are generally key when examining potential transitions.

Polynomial Differentiation

The branches of the piecewise function often include polynomial expressions, which are differentiable over their entire domains. Knowing how to differentiate polynomials is fundamental; for example, the derivative of a quadratic function is a linear function. This basic rule helps in determining the derivative function (f') and analyzing the slope at various points.

Graphical Representation of Functions and Their Derivatives

Sketching graphs of functions and their derivatives requires understanding how the original function's behavior (such as increasing, decreasing, turning points, and smooth transitions) is related to its derivative. The derivative graph indicates the slope of the function at each point. For smooth segments (like those given by polynomials), the graphs provide insights into rates of change, while any abrupt changes or discontinuities in derivatives can signal non-differentiable points.

Transcript

00:01 In this question, we have the function of f, and it's a parvoise function.

00:05 So given that, we need to find the f prime function and determine if the function is differentiable at 1.

00:15 So first, let's take the derivatives.

00:18 F prime of x is going to be, what is the derivative of 2 minus x? well, it's going to be 0 minus 1 because the derivative of 2 is a 0, and derivative of negative x is negative 1.

00:34 And that is for x less than one, not less than or equal to, because you have a parovized function and you have a pairwise function and you have a break in between the two functions.

00:48 So it's not going to be a smooth transition.

00:50 That means there's not going to be any slope.

00:54 And what's the derivative of x squared minus 2x plus 2? so the derivative of x squared is going to be 2x, negative 2x is going to be negative 2.

01:04 And for 2, is going to be zero.

01:11 So, as you can see, f is not differential at x equals to 1.

01:26 And let's also graph it and see it and be sure of it.

01:30 So let's first graph f.

01:33 So f is 2 minus x for x less than or equal to 1.

01:39 So i'm going to mark it over here.

01:44 So at 1, right, our value is 1, and the function is 2 minus x.

01:52 It's going to look like this.

01:55 And the continuation is going to be x squared minus 2x plus 1 plus 2.

02:03 It's going to be a quadratic equation, and it's going to be of the following form...