Math Problem Statement
Find k so that the following function is continuous: f(x) = {kx if 0 ≤ x < 5; 2x^2 if 5 ≤ x}
Solution
To solve the problem of finding such that the function is continuous, we need to ensure that the two pieces of the function join smoothly at .
The function given is:
\begin{cases} kx, & \text{if } 0 \leq x < 5 \\ 2x^2, & \text{if } 5 \leq x \end{cases}$$ For the function to be continuous at $$x = 5$$, the left-hand limit as $$x \to 5^-$$ must equal the right-hand limit as $$x \to 5^+$$, and both must equal $$f(5)$$. ### Step 1: Left-hand limit ($$x \to 5^-$$) In the interval $$0 \leq x < 5$$, $$f(x) = kx$$. So, $$\lim_{x \to 5^-} f(x) = k(5) = 5k$$ ### Step 2: Right-hand limit ($$x \to 5^+$$) In the interval $$5 \leq x$$, $$f(x) = 2x^2$$. So, $$\lim_{x \to 5^+} f(x) = 2(5)^2 = 50$$ ### Step 3: Setting the limits equal For the function to be continuous, we set the left-hand limit equal to the right-hand limit: $$5k = 50$$ ### Step 4: Solve for $$k$$ Solving for $$k$$: $$k = \frac{50}{5} = 10$$ Thus, $$k = 10$$ ensures that the function is continuous at $$x = 5$$. Would you like further clarification, or do you have any other questions? Here are 5 related questions for further practice: 1. What happens if $$k \neq 10$$ for the function's continuity at $$x = 5$$? 2. How would you verify the continuity of a piecewise function at a general point $$x = a$$? 3. Can you generalize the steps to find $$k$$ for any similar piecewise function? 4. What are the conditions for a function to be continuous at a point $$x = a$$? 5. How does the concept of continuity connect to differentiability? **Tip**: Continuity requires that the value of the function and its limits from both directions are the same at a point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Limits
Formulas
Left-hand limit = Right-hand limit
f(x) = kx for x < 5
f(x) = 2x^2 for x ≥ 5
Theorems
Continuity of a function at a point
Limit Theorem
Suitable Grade Level
Grades 11-12
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