Math Problem Statement
Solution
The question asks for which values of and the function is continuous everywhere. The piecewise function is defined as follows:
\begin{cases} a \sin(\pi x) + b & \text{if } x \leq 2 \\ x^2 + a & \text{if } 2 < x < 5 \\ b \cos(2\pi x) + a & \text{if } x \geq 5 \end{cases}$$ To ensure that $$f(x)$$ is continuous everywhere, we need to check the continuity at the points where the definition of the function changes, specifically at $$x = 2$$ and $$x = 5$$. ### Continuity at $$x = 2$$: For $$f(x)$$ to be continuous at $$x = 2$$, the left-hand limit (as $$x$$ approaches 2 from the left) must equal the right-hand limit (as $$x$$ approaches 2 from the right), and both must equal $$f(2)$$. 1. **Left-hand limit as $$x \to 2^-$$:** $$\lim_{x \to 2^-} f(x) = a \sin(2\pi) + b = a(0) + b = b$$ 2. **Right-hand limit as $$x \to 2^+$$:** $$\lim_{x \to 2^+} f(x) = 2^2 + a = 4 + a$$ 3. **For continuity at $$x = 2$$, we need:** $$b = 4 + a$$ ### Continuity at $$x = 5$$: Similarly, for $$f(x)$$ to be continuous at $$x = 5$$, the left-hand limit (as $$x$$ approaches 5 from the left) must equal the right-hand limit (as $$x$$ approaches 5 from the right), and both must equal $$f(5)$$. 1. **Left-hand limit as $$x \to 5^-$$:** $$\lim_{x \to 5^-} f(x) = 5^2 + a = 25 + a$$ 2. **Right-hand limit as $$x \to 5^+$$:** $$\lim_{x \to 5^+} f(x) = b \cos(10\pi) + a = b(1) + a = b + a$$ 3. **For continuity at $$x = 5$$, we need:** $$25 + a = b + a$$ Simplifying, we get: $$b = 25$$ ### Solving the System of Equations: We now have the following system of equations: 1. $$b = 4 + a$$ 2. $$b = 25$$ Substituting $$b = 25$$ into the first equation: $$25 = 4 + a$$ Solving for $$a$$: $$a = 21$$ Thus, the values of $$a$$ and $$b$$ that make $$f(x)$$ continuous everywhere are: $$a = 21, \quad b = 25$$ Would you like more details on this solution, or have any further questions? Here are five related questions: 1. How do you determine if a function is continuous at a point? 2. What is the importance of limits in ensuring continuity? 3. How does the piecewise nature of the function affect its continuity? 4. Can a function be continuous everywhere but still non-differentiable? 5. What role do trigonometric identities play in solving piecewise functions? **Tip:** Always ensure both the left-hand and right-hand limits match the function's value at the point when checking for continuity at a specific point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Trigonometric Functions
Limits
Formulas
lim_{x→c} f(x) = f(c) for continuity at x=c
sin(πx)
cos(2πx)
Theorems
Limit definition of continuity
Sine and cosine periodicity
Suitable Grade Level
Grades 11-12, First-year Calculus
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