Math Problem Statement
If possible, choose k so that the following function is continuous on any interval: f(x) = (4x^3 - 12x^2) / (x - 3) for x ≠ 3 and f(x) = k for x = 3.
Solution
We are tasked with choosing so that the given function is continuous on any interval. The function is defined as:
\begin{cases} \frac{4x^3 - 12x^2}{x - 3}, & \text{if } x \neq 3, \\ k, & \text{if } x = 3. \end{cases}$$ To make this function continuous, we need to ensure that the limit of $$f(x)$$ as $$x$$ approaches 3 from both sides equals $$f(3) = k$$. ### Step 1: Simplify the expression for $$x \neq 3$$ We begin by simplifying $$\frac{4x^3 - 12x^2}{x - 3}$$. Factor the numerator: $$4x^3 - 12x^2 = 4x^2(x - 3).$$ Thus, the expression becomes: $$\frac{4x^2(x - 3)}{x - 3}.$$ For $$x \neq 3$$, we can cancel $$x - 3$$ from the numerator and denominator, leaving: $$f(x) = 4x^2 \text{ for } x \neq 3.$$ ### Step 2: Find the limit as $$x \to 3$$ Now, we compute the limit of $$f(x) = 4x^2$$ as $$x \to 3$$: $$\lim_{x \to 3} 4x^2 = 4(3)^2 = 4 \times 9 = 36.$$ ### Step 3: Set $$k = 36$$ For the function to be continuous at $$x = 3$$, we must have: $$k = \lim_{x \to 3} f(x) = 36.$$ Thus, the value of $$k$$ that ensures continuity is $$k = 36$$. --- Would you like further details, or do you have any questions about the steps? Here are five related questions you can explore: 1. How is continuity of a function defined at a point? 2. What is the difference between continuity and differentiability? 3. How can you check if a piecewise function is continuous at a point? 4. What happens if the function has a discontinuity? How do you classify it? 5. How would you approach continuity for functions with multiple critical points? **Tip:** Always check if a piecewise function is continuous at the boundaries by equating the limit of the function to the function's value at those points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity of Functions
Limits
Factoring Polynomials
Piecewise Functions
Formulas
f(x) = (4x^3 - 12x^2) / (x - 3)
Limit of a function as x → a
f(x) is continuous if lim x→a f(x) = f(a)
Theorems
Definition of Continuity
Factorization Theorem for Polynomials
Suitable Grade Level
Grades 11-12 (Pre-Calculus/Calculus)
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