Determine Continuity of a Piecewise Function: f(t) = 48 + t^2

Math Problem Statement

f(t) = {48 + t^2 - (t^3/12) for 0 <= t < 6, for 6 <= t<= 12

g(x)

t(hours) 6 8 10 12

g(t) cuic meters 306 376 428 474

is f continuous for 0<= t <= 12? Justify your answer

Solution

To determine whether the function $f(t)$ is continuous on the interval $0 \leq t \leq 12$ , we need to verify the following:

Piecewise Continuity: We are given $f(t)$ in two different forms for different intervals:
- For $0 \leq t < 6$ : $f(t) = 48 + t^2 - \frac{t^3}{12}$
- For $6 \leq t \leq 12$ : The function expression for this interval is not provided, but we'll check continuity by ensuring the function values match at $t = 6$ .
Continuity at $t = 6$ : Since $f(t)$ is defined differently on the intervals $[0, 6)$ and $[6, 12]$ , we must check if the left-hand limit and right-hand limit of $f(t)$ at $t = 6$ are equal, and if they equal $f(6)$ .

Step 1: Evaluate the left-hand limit as $t \to 6^-$ :

For $t < 6$ , we have $f(t) = 48 + t^2 - \frac{t^3}{12}$ . Let's find the left-hand limit $\lim_{t \to 6^-} f(t)$ :

= 48 + 36 - \frac{216}{12} = 48 + 36 - 18 = 66$$ So, $$\lim_{t \to 6^-} f(t) = 66$$. ### Step 2: Evaluate the right-hand limit as $$t \to 6^+$$: We need to know the expression for $$f(t)$$ for $$t \geq 6$$. Without this expression, we can't fully check the continuity at $$t = 6$$. ### Step 3: Continuity on each interval: - The function $$f(t) = 48 + t^2 - \frac{t^3}{12}$$ is a polynomial, and polynomials are continuous everywhere. Therefore, $$f(t)$$ is continuous on $$[0, 6)$$. - We assume the function for $$[6, 12]$$ is continuous as well, but without knowing the exact form, we can't conclusively determine continuity at $$t = 6$$. ### Conclusion: For $$f(t)$$ to be continuous on $$[0, 12]$$, we need the function values and limits to match at $$t = 6$$. Since the function for $$t \geq 6$$ is not provided, we cannot definitively confirm continuity at $$t = 6$$ without further information. Would you like further clarification or help with another part of the question? Here are some related questions to consider: 1. What is the form of $$f(t)$$ for $$6 \leq t \leq 12$$? 2. How do you determine limits for piecewise functions? 3. What conditions must be met for a function to be continuous at a point? 4. How do polynomials behave in terms of continuity? 5. Can you provide a graph of $$f(t)$$ to visualize continuity? **Tip**: When checking continuity at a point, always compare the left-hand and right-hand limits with the actual value of the function at that point.

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Math Problem Analysis

Mathematical Concepts

Continuity
Piecewise Functions
Polynomials
Limits

Formulas

f(t) = 48 + t^2 - (t^3/12)
Left-hand limit: lim(t → 6⁻) f(t)
Right-hand limit: lim(t → 6⁺) f(t)

Theorems

Continuity Theorem: A function is continuous at a point if the left-hand limit, right-hand limit, and the function value at that point are equal.
Properties of Polynomials: Polynomials are continuous over their entire domain.

Suitable Grade Level

Grades 11-12

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