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Problem 62 Hard Difficulty

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

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Video Transcript

this is problem number sixty two of the story Calculus eighth edition, section two point eight Part a sketched A graph of the function G of X is equal to X plus the absolute value of X. So one way we can interpret this, maybe we can rewrite it because we have the absolutely rex and the Ah and the function. We can rewrite it as an absolute or a piece wise function for, ah, two different functions where X is greater than or equal to zero and one where X is less than zero. And we were called that specifically the absolute value of X. Yeah, can be X is given as exit when excreted into and is given his negative X when X is less than zero. Okay, so this is true executed and zero means that volume X We're going to take us just ex so x plus X is too X and one X is less than zero. In this region, we get explosive negative X meaning that the function G is equal to zero, which is in the region ex listen zero so we can test a graph from knowing this information. We see that the function are, we plot the function at zero up until the reaches. Ah, the origin where it's equal zero For both cases, X equals zero. We get zero, so it's continuous and then afterwards it goes, uh, into a who linear a function with the slope of two two X that pass through the origin. And this is our graph of G of X purpose for what values of X is G differential. All G is defensible in this region. Here we see that this horizontal line is a constant value. And if for a constant, a horse on the line, the tensions are all horizontal on their slopes are zero. So it's a printable everywhere here, less network for X is less than zero and Rex is greater than zero. We have this linear slope which is to X, and all the tangent lines have soup of two for this part of the function. So differential everywhere they're the only poor words not defensible is here at the origin because there is a corner or a kink in the function that's not smooth. We go from a zero slope to it to a slip of two immediately. Abruptly meaning that at X equals zero It is not different people. So party for what values is X of exes. She differential all values except X is not zero example party. Find a formula for G. Well, we just discussed the exactly slopes of G so formal for G prime cheap primaries are the derivatives A ll, ah are all the slopes of the tension lines. So it's a derivative of function of Jean and we confirmed that the slope should be to all the tension lines are exactly the same. Is this function to X? All those slopes of the stench lines is too because ah, is it linear relationship. So this is true for X is greater than zero The derivative functions equal to two And then over here the slopes of all the tension lines are zero because it's a horizontal line. So jia cheap prime of X zero for excess less than zero and then we were called X cannot be zero. It's not defensible addicts equal zero. So we don't include that in our domain for the G prime of X function. And this is exactly the formula for G proper vex another way. If we would like to rewrite and our formula for vex is in the form one plus and again taken advantage of absolute value. Vics are absolutely please, we write it. Is that politics over X And we see here that as long as excess positive this quantities of a positive one. So we get one plus one. And as long as excess negative, this quantity is always equal to native ex native X rex at eight one one plus one zero. So this is the same as this piece wise function, and both of these are acceptable answers.