(a) Sketch the graph of the function f(x) = x | x | . (b) For what values of x is f differentiab

(a) Sketch the graph of the function f(x) = x | x

Key Concepts

Absolute Value Function

This concept involves understanding the function |x|, which returns the non?negative magnitude of x. It often results in piecewise definitions because it behaves differently for positive and negative x, which is crucial when simplifying functions that include an absolute value.

Piecewise Function Representation

When a function includes an absolute value, it is often useful to express it in piecewise form. In this problem, rewriting f(x) = x|x| as f(x) = x^2 for x ? 0 and f(x) = -x^2 for x < 0 is essential for analysis, particularly for graphing the function and examining its differentiability.

Graph Sketching Techniques

Graph sketching involves understanding the shapes, intercepts, and continuity of the different pieces of a function. Here, recognizing that the function represents two parabolic curves (one opening upward and one downward) on different intervals helps in accurately plotting the graph.

Differentiability of Piecewise Functions

This key concept is about determining whether a function has a derivative at a point, especially at the boundary where the formula changes. It involves checking if the left-hand and right-hand derivatives at the transition point (in this case, x = 0) are equal, ensuring the function is smooth and differentiable at that point.

Derivative Computation

Calculating the derivative of a function, especially when it is defined piecewise, involves differentiating each piece separately and making sure that the derivative at the juncture (if any) is consistent from both sides. This approach is critical to find the formula for f' by applying basic differentiation rules to each piece.

Transcript

00:01 This is problem number 61 of the stewart calculus 8th edition, section 2 .8.

00:05 Part a, sketch the graph of the function f of x is equal to x times the absolute value of x.

00:12 Well, one approach we could do is to rewrite our function as a piecewise, which is a common approach when dealing with an absolute value function or a function that's related to an absolute value function.

00:27 We know that the absolute value function is x for x is greater than 0 and negative x when x is less than 0.

00:45 Therefore, if this is what we take and we separate this into x is greater than or equal to 0, and x is less than 0, then the function can be rewritten as x times x or x squared when x is greater than or equal to 0, and x times negative x or negative x squared when x is less than 0.

01:07 Now, this should help with graphing our function here, because what we have is we have an upward -facing parabola x squared for x is greater than equal to zero.

01:21 So the general shape we'll have.

01:25 And instead of having the other half of the parabola in the positive direction, when x is less than zero, we have a negative downward -facing parabola.

01:34 So we'll have a parabola reflected across the x -x.

01:39 X is just like that.

01:48 Ok, so this is our graph for the function f of x, which is x times the absolute value of x.

02:00 Part b.

02:00 Well, for what values of x is f differentiable? so that was part a, part b.

02:07 We look at the graph and we see that it's a pretty smooth function.

02:10 We don't see any corners, kinks, or cusps.

02:13 We also don't see any opportunity for a vertical tangent line, meaning that there are non, there don't seem to be any non -differentiabilities associated with those.

02:26 The last one to check is continuity while it's always continuous.

02:31 So we suspect that this is fully differentiable.

02:34 But for one check, just to make sure we suspect maybe at the origin, there may be a non -differentiability.

02:42 So what we need to do is we need to evaluate f prime of a.

02:50 Or f from 0, we're going to check two different ways.

02:53 We're going to check the limit as x approaches 0 from the right, meaning that we're using this function, x squared, minus the function evaluated at x squared or at 0.

03:12 So it's 0 squared over x minus 0.

03:16 This is equal to x, and as x approaches 0, this approaches 0.

03:23 So that limit is 0, and then we also have to do the limit as x approaches 0.

03:28 The left, meaning that we choose the other function, negative x word.

03:34 We have negative x squared minus the other function, evaluated at 0, divided by x minus 0.

03:43 And we end up, this limit is equal to negative x evaluated at 0, which ends up being 0...

Please give Ace some feedback