Problem 51
Question
Describe the \(x\) -values at which the function is differentiable. Explain your reasoning. $$ y=|x+3| $$
Step-by-Step Solution
Verified Answer
The function \(y = |x + 3|\) is differentiable for all real numbers except \(x = -3\).
1Step 1: Identify the Corner Point
The function \(y = |x + 3|\) is not differentiable at \(x = -3\) since here the function changes direction abruptly, forming a corner. This is where the input of the absolute value function becomes zero.
2Step 2: Determine Range of Differentiability
Since the function \(y = |x + 3|\) is only non-differentiable at \(x = -3\), the function is differentiable for all other real numbers, \(-\infty < x < -3\) and \(-3 < x < \infty\).
Other exercises in this chapter
Problem 51
Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\frac{1}{x-2} $$
View solution Problem 51
Use a graphing utility to graph \(f\) and \(f^{\prime}\) on the interval \([-2,2]\). $$ f(x)=x(x+1) $$
View solution Problem 51
$$ \lim _{x \rightarrow 2} f(x), \text { where } f(x)=\left\\{\begin{array}{ll} 4-x, & x \neq 2 \\ 0 & x=2 \end{array}\right. $$
View solution Problem 52
Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ s(t)=\frac{1}{t^{2}+3 t-1} $$
View solution