Math Problem Statement
number 58
Determine the x-values where the function in Exercise 58 is differentiable and explain your reasoning.
Solution
Let’s analyze Exercise 58 to determine where the function is differentiable. The piecewise function is defined as:
x^3 + 3 & \text{if } x < 0, \\ x^3 - 3 & \text{if } x \geq 0. \end{cases}$$ ### Steps to Determine Differentiability 1. **Continuity Check**: For the function to be differentiable, it must first be continuous. Let us verify if $$y$$ is continuous at $$x = 0$$. - From the left-hand side ($$x < 0$$): $$\lim_{x \to 0^-} (x^3 + 3) = 0^3 + 3 = 3$$. - From the right-hand side ($$x \geq 0$$): $$\lim_{x \to 0^+} (x^3 - 3) = 0^3 - 3 = -3$$. Since the left and right limits at $$x = 0$$ are not equal, the function is **not continuous** at $$x = 0$$. Therefore, $$y$$ is **not differentiable** at $$x = 0$$. 2. **Behavior Elsewhere**: For all $$x \neq 0$$, each piece of the function ($$x^3 + 3$$ and $$x^3 - 3$$) is a polynomial, and polynomials are differentiable everywhere. Thus, $$y$$ is differentiable for $$x \neq 0$$. ### Conclusion The function is differentiable everywhere except at $$x = 0$$, where it is neither continuous nor differentiable. --- Would you like more details or assistance with other exercises? Here are some related questions to expand your understanding: 1. How do you test for differentiability at a point using derivatives? 2. What is the relationship between continuity and differentiability? 3. How does a piecewise function affect continuity and differentiability? 4. Can a function be differentiable but not continuous? Why or why not? 5. What role does the slope (derivative) from the left and right play in differentiability? **Tip**: Always check for continuity first when determining differentiability, as it is a necessary (but not sufficient) condition.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Differentiability
Limits
Formulas
Limit Definition of Continuity
Limit Definition of Derivative
Theorems
Continuity is a prerequisite for differentiability
Differentiability implies continuity
Suitable Grade Level
Grades 10-12
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