Math Problem Statement
Given π(π₯)={2π₯^2+ππ₯β6,π₯<5 62βππ₯ ,π₯β₯5 determine the value of βaβ that makes the function continuous
Solution
To make the function continuous at , the left-hand limit as (approaching from the left) must equal the right-hand limit as (approaching from the right). Additionally, the function's value at must match both limits.
Given:
\begin{cases} 2x^2 + ax - 6 & \text{for } x < 5 \\ 62 - ax & \text{for } x \geq 5 \end{cases}$$ ### Step 1: Calculate the left-hand limit as $$x \to 5^-$$ From the first piece of the function, $$f(x) = 2x^2 + ax - 6$$ for $$x < 5$$. The left-hand limit as $$x \to 5^-$$ is: $$\lim_{x \to 5^-} f(x) = 2(5)^2 + a(5) - 6 = 50 + 5a - 6 = 44 + 5a$$ ### Step 2: Calculate the right-hand limit as $$x \to 5^+$$ From the second piece of the function, $$f(x) = 62 - ax$$ for $$x \geq 5$$. The right-hand limit as $$x \to 5^+$$ is: $$\lim_{x \to 5^+} f(x) = 62 - a(5) = 62 - 5a$$ ### Step 3: Set the left-hand and right-hand limits equal to each other For the function to be continuous, the limits must be equal at $$x = 5$$. So: $$44 + 5a = 62 - 5a$$ ### Step 4: Solve for $$a$$ Add $$5a$$ to both sides: $$44 + 10a = 62$$ Subtract 44 from both sides: $$10a = 18$$ Divide by 10: $$a = \frac{18}{10} = 1.8$$ Thus, the value of $$a$$ that makes the function continuous is $$a = 1.8$$. Would you like further details on the solution or have any questions? Here are 5 related questions to explore: 1. How do we ensure continuity at any other point in a piecewise function? 2. What is the geometric interpretation of continuity at a point? 3. How can discontinuity occur in a function, and what types are there? 4. What does it mean for a function to be differentiable at a point, and how does it relate to continuity? 5. Can a function be continuous at all points but not differentiable at some points? **Tip:** Continuity at a point requires that the function's value and both one-sided limits agree at that point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Limits
Continuity
Formulas
Limit of a function as x approaches a point
Piecewise function continuity condition
Theorems
Continuity Theorem
Limit Laws
Suitable Grade Level
Grades 11-12