Grapher Failure Graph the function f(x)=|x| x in the window [-5,5] by [-3,3] and explain why f^'

Grapher Failure Graph the function f(x)=|x| x in the window...

Key Concepts

Numerical Differentiation and Calculator Limitations

Graphing calculators often estimate derivatives using numerical methods, such as difference quotients over small intervals. At points where the function has a corner or involves a nondifferentiable component, the numerical approach may produce misleading results. In this case, even though the analytic limit may exist, the calculator’s NDER function might be misled by the behavior of the absolute value function used in the expression, leading to an incorrect or inconsistent value for f?'(0).

Limit Definition of the Derivative

The derivative of a function at a point is defined as the limit of the difference quotient as the interval approaches zero. Even if one component of a function (like the absolute value) is not differentiable at a given point, the overall function can still be differentiable if the limit from both sides exists and is equal. For f(x) = |x|x, computing the limit separately for x positive and negative shows that both one–sided limits yield the same result at 0, despite the derivative of |x| not existing in isolation.

Piecewise Function Representation

Functions involving absolute values are often best understood when rewritten as piecewise functions. In the case of f(x) = |x|x, the expression can be split into different formulas depending on whether x is positive or negative. This method provides clarity on the behavior of the function on either side of the point of interest (x = 0) and helps in the analysis of continuity and differentiability.

Absolute Value Function

The absolute value function, by definition, outputs the non?negative value of its input and has a “corner” at x = 0. This corner is a point where the left? and right–hand slopes differ (for |x| itself, they are –1 and 1, respectively), so while the function is continuous, it is not differentiable at that point. When absolute value expressions are part of a larger function, care must be taken to analyze whether cancellations or other operations may remove the non–smooth behavior.

Transcript

00:01 We are going to plot the function absolute value of x over x in the interval negative 5 -5 and on the window negative 5 5 times negative 3 3 and we will explain why the derivative of f at 0 does not exist and if we try to find the numerical derivative at 0 of this function on a calculator we are going to discuss about obtaining any numerical value of that.

00:40 So in this case we we know that the function f given us the absolute value of x over x is not defined as zero because at zero we will have sorry you will have zero over zero which is not defined so for positive x we know this function is equal to 1 because let's see that for positive x, the absolute value of x is x x, we then divide x by itself equals 1 if x is positive and if x is negative, absolute value of x is negative x over x and that is negative 1.

01:42 So this function is equivalent to the function 1 for the positive and negative 1 for the negatives.

01:50 So the graph is more or less this graph here at the left where the blue line is a function and we know that this is infinite to the right to the left.

02:03 And at the point zero there is no definition of the function.

02:10 So this function is clearly not continuous at zero.

02:22 Simply because it's not there is no limit at that point and with that already we know that it's not continuous but it's not it also not defined at zero so in any way this function is discontinuous at zero and this implies that if derivative at zero cannot exist because if a function has a derivative at a point it get to be continuous at that point.

03:01 And in fact there is no reasonable way of defining a tangent line at zero.

03:08 Not possible to do that in this case...

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