Function Continuity

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Answer Key & Explanations

Function Continuity · Grade 12 · Worksheet 3

1. Mason is an electrical engineer analyzing a signal in a circuit. The signal's voltage (in volts) as a function of time t (in seconds) is modeled by the piecewise function V(t) = { t^2 - 16 for t < 4, a*t + b for 4 ≤ t ≤ 7, 3*t + 1 for t > 7 }. For the signal to be transmitted without distortion, the voltage function must be continuous at both transition points t = 4 seconds and t = 7 seconds. What values must the parameters a and b have to ensure the voltage function is continuous throughout the circuit? Answer: a = 4, b = -16 Solution: For continuity at t = 4, the left-hand limit and the function value from the right must be equal. The left-hand piece is t^2 - 16. At t = 4, this gives 4^2 - 16 = 16 - 16 = 0.
   Full step-by-step solution

   Step 1: For continuity at t = 4, the left-hand limit and the function value from the right must be equal. The left-hand piece is t^2 - 16. At t = 4, this gives 4^2 - 16 = 16 - 16 = 0. The right-hand piece at t = 4 is a*4 + b = 4a + b. Setting them equal gives Equation 1: 4a + b = 0. Step 2: For continuity at t = 7, the left-hand limit from the middle piece and the function value from the right must be equal. The middle piece at t = 7 is a*7 + b = 7a + b. The right-hand piece is 3t + 1. At t = 7, this gives 3*7 + 1 = 21 + 1 = 22. Setting them equal gives Equation 2: 7a + b = 22. Step 3: Solve the system of equations: Equation 1: 4a + b = 0 Equation 2: 7a + b = 22 Step 4: Subtract Equation 1 from Equation 2 to eliminate b: (7a + b) - (4a + b) = 22 - 0 3a = 22 a = 22/3 Step 5: Substitute a = 22/3 into Equation 1: 4*(22/3) + b = 0 88/3 + b = 0 b = -88/3 Step 6: Verify with Equation 2: 7*(22/3) + (-88/3) = 154/3 - 88/3 = 66/3 = 22. This matches. The answer is a = 22/3, b = -88/3.

2. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time using the function C(t) = (5t^2 * e^(-0.3t))/(t^2 + 1), where t is measured in hours. The researchers need to determine if this concentration function is continuous for all t ≥ 0, particularly at t = 0 where the function appears to have an indeterminate form. Analyze the continuity of C(t) at t = 0 and explain your reasoning mathematically. Answer: The function is continuous at t = 0. Solution: Write the function clearly. C(t) = (5 t^2 e^(-0.3 t)) / (t^2 + 1) for t ≥ 0. Check if C(0) is defined by direct substitution.
   Full step-by-step solution

   Let's analyze the continuity of C(t) at t = 0 step by step. Step 1: Write the function clearly. C(t) = (5 t^2 e^(-0.3 t)) / (t^2 + 1) for t ≥ 0. Step 2: Check if C(0) is defined by direct substitution. At t = 0: Numerator: 5 * (0)^2 * e^(-0.3*0) = 5 * 0 * 1 = 0. Denominator: (0)^2 + 1 = 1. So C(0) = 0/1 = 0. The function is defined at t = 0: C(0) = 0. Step 3: Check the limit as t approaches 0. We need to find limit as t -> 0+ of C(t). C(t) = (5 t^2 e^(-0.3 t)) / (t^2 + 1). As t -> 0: - e^(-0.3 t) -> e^0 = 1. - So numerator ~ 5 t^2 * 1 = 5 t^2. - Denominator = t^2 + 1 -> 1. Thus C(t) ~ (5 t^2) / 1 = 5 t^2 as t -> 0. So limit as t -> 0+ of C(t) = 0. Step 4: Compare the limit and the function value. We have: limit as t -> 0+ of C(t) = 0. C(0) = 0. They are equal. Step 5: Conclusion about continuity. Since: 1. C(0) is defined, 2. The limit as t -> 0+ exists, 3. The limit equals the function value at t = 0, the function C(t) is continuous at t = 0. Final answer: The function is continuous at t = 0.

3. Maria is designing a roller coaster track that follows the piecewise function f(x) = { x^2 + 2 for x < 1, ax + b for 1 ≤ x ≤ 3, 4x - 2 for x > 3 }. To ensure a smooth ride, the track must be continuous at both transition points x = 1 and x = 3. What values of the parameters a and b will make the roller coaster track continuous at both points? Answer: a=3, b=0 Solution: For continuity at x = 1, the left-hand limit and function value must equal the right-hand limit and function value.
   Full step-by-step solution

   Step 1: For continuity at x = 1, the left-hand limit and function value must equal the right-hand limit and function value. Step 2: Left-hand limit at x = 1: f(1) from left side = (1)^2 + 2 = 1 + 2 = 3 Step 3: Right-hand limit at x = 1: f(1) from right side = a(1) + b = a + b Step 4: Set them equal: a + b = 3 (Equation 1) Step 5: For continuity at x = 3, the left-hand limit and function value must equal the right-hand limit and function value. Step 6: Left-hand limit at x = 3: f(3) from left side = a(3) + b = 3a + b Step 7: Right-hand limit at x = 3: f(3) from right side = 4(3) - 2 = 12 - 2 = 10 Step 8: Set them equal: 3a + b = 10 (Equation 2) Step 9: Solve the system of equations: Equation 1: a + b = 3, Equation 2: 3a + b = 10 Step 10: Subtract Equation 1 from Equation 2: (3a + b) - (a + b) = 10 - 3 → 2a = 7 → a = 3.5 Step 11: Substitute a = 3.5 into Equation 1: 3.5 + b = 3 → b = 3 - 3.5 = -0.5 Step 12: The values that make the function continuous are a = 3.5 and b = -0.5

4. Consider the function f(x) = (x^2 - 4)/(x - 2) for x ≠ 2. Determine the value that f(2) should be assigned to make the function continuous at x = 2. Answer: 4 Solution: f(x) = (x^2 - 4)/(x - 2) for x ≠ 2. We want to assign f(2) so that f is continuous at x = 2. For continuity at x = 2, we need: lim (x → 2) f(x) = f(2).
   Full step-by-step solution

   Let's go step-by-step. We have the function: f(x) = (x^2 - 4)/(x - 2) for x ≠ 2. --- **Step 1: Understand the problem** We want to assign f(2) so that f is continuous at x = 2. For continuity at x = 2, we need: lim (x → 2) f(x) = f(2). So we must find the limit of f(x) as x approaches 2. --- **Step 2: Simplify f(x) for x ≠ 2** Factor the numerator: x^2 - 4 = (x - 2)(x + 2). So for x ≠ 2: f(x) = [(x - 2)(x + 2)] / (x - 2) = x + 2. --- **Step 3: Take the limit** Since f(x) = x + 2 for all x ≠ 2, lim (x → 2) f(x) = lim (x → 2) (x + 2) = 2 + 2 = 4. --- **Step 4: Assign f(2)** For continuity, f(2) must equal the limit: f(2) = 4. --- **Final answer:** 4

5. A civil engineer is designing a suspension bridge where the cable follows the piecewise function f(x) = { x² + 2 for x < 1, ax + b for 1 ≤ x ≤ 3, 4x - 2 for x > 3 }. To ensure structural integrity, the cable must be continuous at both transition points x = 1 and x = 3. What values must the parameters a and b have to guarantee continuity throughout the bridge cable? Answer: a=3, b=0 Solution: For continuity at x = 1, the left-hand limit (from x < 1) must equal the function value at x = 1.
   Full step-by-step solution

   Step 1: For continuity at x = 1, the left-hand limit (from x < 1) must equal the function value at x = 1. Left-hand limit: lim(x→1⁻) f(x) = lim(x→1⁻) (x² + 2) = 1² + 2 = 3 At x = 1: f(1) = a(1) + b = a + b So a + b = 3 Step 2: For continuity at x = 3, the left-hand limit (from 1 ≤ x ≤ 3) must equal the right-hand limit (from x > 3). Left-hand limit: lim(x→3⁻) f(x) = lim(x→3⁻) (ax + b) = 3a + b Right-hand limit: lim(x→3⁺) f(x) = lim(x→3⁺) (4x - 2) = 4(3) - 2 = 12 - 2 = 10 So 3a + b = 10 Step 3: Solve the system of equations: a + b = 3 3a + b = 10 Subtract the first equation from the second: (3a + b) - (a + b) = 10 - 3 2a = 7 a = 3.5 Substitute a = 3.5 into a + b = 3: 3.5 + b = 3 b = 3 - 3.5 b = -0.5 Step 4: Verify the solution: At x = 1: a + b = 3.5 + (-0.5) = 3 ✓ At x = 3: 3a + b = 3(3.5) + (-0.5) = 10.5 - 0.5 = 10 ✓ The answer is a = 3.5, b = -0.5.

6. An environmental engineer is modeling the temperature of a chemical reaction in a cooling system. The temperature function is given by T(t) = (t^3 - 8)/(t^2 - 4) for t ≠ 2, where t is time in minutes. The engineer needs to determine if the temperature function is continuous at t = 2 minutes, which represents a critical phase transition point. If not continuous, what value should be assigned to T(2) to make the function continuous at that point? Answer: 3 Solution: Numerator: t^3 - 8 = (t - 2)(t^2 + 2t + 4) Denominator: t^2 - 4 = (t - 2)(t + 2) Simplify the function for t ≠ 2 T(t) = [(t - 2)(t^2 + 2t + 4)]/[(t - 2)(t + 2)] = (t^2 + 2t + 4)/(t + 2) Find the limit as t approaches 2 lim(t→2) T(t) = lim(t→2) (t^2 + 2t + 4)/(t + 2) (2^2 + 2(2) + 4)/(2 + 2) = (4…
   Full step-by-step solution

   Step 1: Factor the numerator and denominator Numerator: t^3 - 8 = (t - 2)(t^2 + 2t + 4) Denominator: t^2 - 4 = (t - 2)(t + 2) Step 2: Simplify the function for t ≠ 2 T(t) = [(t - 2)(t^2 + 2t + 4)]/[(t - 2)(t + 2)] = (t^2 + 2t + 4)/(t + 2) Step 3: Find the limit as t approaches 2 lim(t→2) T(t) = lim(t→2) (t^2 + 2t + 4)/(t + 2) Step 4: Evaluate the limit by direct substitution (2^2 + 2(2) + 4)/(2 + 2) = (4 + 4 + 4)/4 = 12/4 = 3 Step 5: Since the limit exists but the function is undefined at t = 2, we have a removable discontinuity Step 6: To make the function continuous at t = 2, we define T(2) = 3 The answer is 3.