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Problem 61 Hard Difficulty

(a) Sketch the graph of the function $ f(x) = x | x | $.
(b) For what values of $ x $ is $ f $ differentiable?
(c) Find a formula for $ f' $.

Answer

(a) $f(x)=x|x|=\left\{\begin{array}{ll}x^{2} & \text { if } x \geq 0 \\ -x^{2} & \text { if } x<0\end{array}\right.$
(b) So $f$ is differentiable at $0 .$ Thus, $f$ is differentiable for all $x$
(c) $2|x|$

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Video Transcript

this problem. Number sixty one of this tour calculus eighth edition Section two point eight Party Sketched A graph of the function F of X is equal to X attempted camps. Loot Value of X Well, one approach we could do is to rewrite our function as the piece wise, which is common approach when dealing with an absolute value function or function that's related to maps of the wave function. We know that the absolute value function is eggs for X is greater than zero and negative x when X is listen. Zero. Therefore, if this if this is what we take and we separate this into X is created in our equal to zero and excess lessons room than function can be re written as X times X or X squared. When X is greater than zero an extent, think of X or negative X squared when X is less than zero. Now this should help with graphene. Our function here because what we have is we have aboard facing problem X squared for X is greater than equal to their own, so that's the general shape will have. And instead of having the other half of the problem in the positive direction. When X lessons There we have it. Negative, Downward facing problem still have a problem reflected across I'm the X axis. Just think. Okay, so this is our graph for the function of X, which is extended absolutely of X, Part B. Well, for what values vexes have different chewable. That was part of a part B. We look at the graph and we see that it's a pretty smooth function. We don't see any corners, kings or cusp. So I also don't see any opportunity for a vertical tangent line, meaning that there are none. They don't seem to be any non different abilities associated with those. The last one to check his continuity. Well, it's always continuous s. So we suspect that this is police. They're French mobile, home before one check, just to make sure we suspect may be at the origin there, maybe non different ability. So what we need to do is we need to evaluate every crime of aim or from zero are in check two different ways. We're going to check the limit as experts zero from the right, meaning that we're using this function X squared minus the function evaluated X squared are at zero. So what's your squared? Over X minus zero. This is equal to X And as experts zero, this approaches zero. So that led me to zero. And then we also have to do the limit. Its experts is zero from the left meaning that we choose thie Ah, other function Negative X word. I have negative X squared minus the other function in Hollywood. It's zero divided by X minus zero and we end up His limit is equal Tio negative X evaluated at zero, which ends up in here. So our limit from left and from the right agree with each other. These are the limits associated with the definition of the derivative. Essentially, the derivatives are, uh, the same her lift on the right, which means that it is different trouble at the origin. So we checked every single possible non different ability. But we can conclude that empathy. French hable for X belonging to all real numbers. Two exes differential everywhere. Finally, part C is what is the formula for halftime? Well, what we're going to do is we essentially do the same procedure here, and we want to find this at paramedics. But we have to do for two cases. One for this region X is greater than zero and one. For this reason. Exits Lesson zero. Throw into H purchases. The room of this function evaluated H minus the function itself kind of age. This gives us the limit. Is each purchase is zero X squared plus two ex age H squared minus X squared or ridge X squared is cancelled Ah, one age councils in each turn and as each part zero this term ghost zero Meaning that our druid or our prime of X for this first region is too x. Ah, if we repeat the procedure for the other function, we have the limit as H coaches zero negative x where it worked We're evaluating negative X squared of experts each a minus negative X squared. So we'll have the limit as each approaches zero Thank you X squared minus two x h minus h squared plus X werder divided by age Then hear negative X squared in positive Export Council One age counsels with each of the other terms in the numerator And since h Burt zero this H term purchase zero meaning that ordered of his neighbor two x in the region for exes. Lesson zero. So we conclude the derivative function is can be written as a piece of dysfunction. It is two acts when X is greater than zero. It is negative. Two extra nexus. Listen, zero. And we've already confirmed that the derivative is zero they're at X equals zero. And so we conclude that that bit of information here, here, here, including X equals zero and the domain of the dirt of function on. So this is exactly a formula for half primer and another way that we can actually write this since we have to x in the positive region and the tricks in the negative region, another way would write. This is two times the absolute value of X. So this is the formula for the derivative F crime.