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Problem 45 Easy Difficulty

Graph the function $ f(x) = x + \sqrt{| x |} $. Zoom in repeatedly, first toward the point $ (-1, 0) $ and then toward the origin. What is different about the behavior of $ f $ in the vicinity of these two points? What do you conclude about the differentiability of $ f $?

Answer

As we zoom in toward (-1,0) , the curve appears more and more like a straight line, so $f(x)=x+\sqrt{|x|}$ is differentiable at $x=-1 .$ But no matter how much we zoom in toward the origin, the curve doesn't straighten out-we can't eliminate the sharp point (a cusp). So $f$ is not differentiable at $x=0$.

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Video Transcript

This is a problem. Number 45 of the Steward Calculus eighth edition section 2.8 graph The function F of X equals X plus the square root of the absolute value. X zoom in repeatedly first toward the point native 10 and then towards the origin. What is different about the behavior of F in the vicinity of these two points? What do you conclude about the different ability of F? So we take a look at the function here, plotted using a graphic calculator or any plotter X plus the square root of the value of X. And here we're looking at the two points in question negative 10 and the origin 00 The behavior of F is different at each of these two points, although it's continuous at every point here, we see the function is definitely continuous, as in any number within this domain, are is part of the domain of the function, so negative one will give an answer of zero as well. X equals zero, so this is a continuous function. However, the behavior is different at negative 10 because it is a smooth function. The function does not abruptly change doesn't have a corner or kink, and it is differentiable or and it is, uh, continuous. And it does not have a vertical tangent. So none of the different ability restrictions apply. Therefore, the function is differentiable and negative 10 however, that there, however, at the origin the behavior of the function is different. It has this curve towards the origin and then away from the origin. This is This can be described as either having a vertical tangent at X equals zero or what we call a custom. But essentially what we have is we have a negative derivatives negative slopes of tangent lines to the left of Mexico zero and then positive tangent lines positively sloped tangent lines to the right of X equals zero. Therefore, it is not differential. Politics equals zero for the reason that this function curves and it's cusp to here at X equals to zero. So it is differentiable in negative 10 But the function F is not differentiable at X equals zero or at the origin