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Chapter 2: Problem 57

Use the alternative form of the derivative to find the derivative at \(x=c\) (if it exists). \(h(x)=|x+5|, \quad c=-5\)

Short Answer

Expert verified
The derivative of \(h(x) = |x + 5|\) at \(x = c = -5\) doesn't exist due to the non-equality of the left-hand and right-hand limits.

Step by step solution

01

Express the function without absolute value

The absolute function can be expressed without the absolute value as \(h(x) = (x+5)\) for \(x \geq -5\) and \(h(x) = -(x+5)\) for \(x < -5\).
02

Calculate the derivative using the alternative form

For \(x \geq -5\), the derivative \(h'(x) = 1\), obtained by standard methods. For \(x < -5\), the derivative \(h'(x) = -1\).
03

Consider the behavior at \(x = c\)

At \(x = c = -5\), the behavior of the function changes. Therefore, calculate the left-hand and right-hand limits at \(x = -5\) to check the existence of the derivative. \[ \lim_{x \to -5^+} h'(x) = 1 \] \[ \lim_{x \to -5^-} h'(x) = -1 \] Since the two limits aren't equal, the derivative at \(x = -5\) doesn't exist.

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Most popular questions from this chapter

In Exercises \(81-88\), (a) find an equation of the tangent line to the graph of \(f\) at the indicated point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. \(\frac{\text { Function }}{y=\cos 3 x} \quad \frac{\text { Point }}{\left(\frac{\pi}{4},-\frac{\sqrt{2}}{2}\right)}\)

In Exercises \(81-88\), (a) find an equation of the tangent line to the graph of \(f\) at the indicated point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. \(\frac{\text { Function }}{f(x)=\frac{1}{3} x \sqrt{x^{2}+5}} \quad \frac{\text { Point }}{(2,2)}\)

In Exercises 107-110, (a) use a graphing utility to find the derivative of the function at the given point, (b) find an equation of the tangent line to the graph of the function at the given point, and (c) use the utility to graph the function and its tangent line in the same viewing window. \(s(t)=\frac{(4-2 t) \sqrt{1+t}}{3},\left(0, \frac{4}{3}\right)\)

Let \(L\) be any tangent line to the curve \(\sqrt{x}+\sqrt{y}=\sqrt{c}\). Show that the sum of the \(x\) - and \(y\) -intercepts of \(L\) is \(c\).

Find equations of all tangent lines to the graph of \(f(x)=\arccos x\) that have slope -2
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