Function End Behavior

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Answer Key & Explanations

Function End Behavior · Grade 12 · Worksheet 3

1. lim(x→∞) (3x³ - 4x² + 2)/(2x³ + 5x - 1) = ? Answer: 3/2 Solution: Identify the highest degree terms in numerator and denominator Numerator: 3x³ Denominator: 2x³ For rational functions where degrees are equal, the limit equals the ratio of leading coefficients Ratio = 3/2 Numerator degree: 3 Denominator degree: 3 Since degrees are equal, limit = 3/2 The answer…
   Full step-by-step solution

   Step 1: Identify the highest degree terms in numerator and denominator Numerator: 3x³ Denominator: 2x³ Step 2: For rational functions where degrees are equal, the limit equals the ratio of leading coefficients Ratio = 3/2 Step 3: Verify degrees are equal Numerator degree: 3 Denominator degree: 3 Step 4: Since degrees are equal, limit = 3/2 The answer is 3/2.

2. Matiu is a marine biologist studying the population dynamics of a certain fish species in a large ocean ecosystem. He models the population size (in thousands) as a function of time t (in years) using the rational function P(t) = (5t^4 - 2t^3 + 7t + 11) / (10t^4 + 3t^2 - 8). As t approaches positive infinity, what population size (in thousands) does Matiu's model predict the fish population will approach? Answer: 0.5 Solution: Identify the highest power of t in the numerator and denominator. In the numerator (5t^4 - 2t^3 + 7t + 11), the highest power is t^4 with coefficient 5.
   Full step-by-step solution

   Step 1: Identify the highest power of t in the numerator and denominator. In the numerator (5t^4 - 2t^3 + 7t + 11), the highest power is t^4 with coefficient 5. In the denominator (10t^4 + 3t^2 - 8), the highest power is also t^4 with coefficient 10. Step 2: For a rational function where the degrees of the numerator and denominator are equal, the horizontal asymptote (end behavior as t approaches infinity) is the ratio of the leading coefficients. Step 3: Compute the ratio: 5/10 = 1/2 = 0.5. Step 4: Therefore, as t approaches infinity, P(t) approaches 0.5 thousand fish. The answer is 0.5.

3. Noah is a climate scientist studying the long-term concentration of a pollutant in a lake. The concentration C(t) in parts per million (ppm) is modeled by the rational function C(t) = (9t^2 - 7t + 12) / (3t^2 + 2t - 5), where t represents time in years. As t approaches positive infinity, what steady-state concentration does the pollutant approach according to this model? Express your answer as a simplified fraction or integer. Answer: 3 Solution: Identify the degrees of the numerator and denominator. The numerator is 9t^2 - 7t + 12, which is degree 2 with leading coefficient 9. The denominator is 3t^2 + 2t - 5, which is degree 2 with leading coefficient 3.
   Full step-by-step solution

   Step 1: Identify the degrees of the numerator and denominator. The numerator is 9t^2 - 7t + 12, which is degree 2 with leading coefficient 9. The denominator is 3t^2 + 2t - 5, which is degree 2 with leading coefficient 3. Step 2: Since the degrees of the numerator and denominator are equal, the horizontal asymptote (the value the function approaches as t → ∞) is the ratio of the leading coefficients. Step 3: Calculate the ratio: 9 / 3 = 3. Step 4: Therefore, as t approaches infinity, C(t) approaches 3 ppm. Final answer: 3

4. A biomedical researcher is modeling the concentration of a new medication in a patient's bloodstream using the function C(t) = (3t^3 - 2t^2 + 5)/(t^3 - 4), where t represents hours after administration. As time approaches infinity, what value does the medication concentration approach, and what does this tell the researcher about the long-term behavior of the drug in the system? Answer: 3 mg/L Solution: To find the long-term behavior of the concentration function C(t) = (3t^3 - 2t^2 + 5)/(t^3 - 4) as time approaches infinity, we need to determine the limit of C(t) as t → ∞.
   Full step-by-step solution

   To find the long-term behavior of the concentration function C(t) = (3t^3 - 2t^2 + 5)/(t^3 - 4) as time approaches infinity, we need to determine the limit of C(t) as t → ∞. Step 1: Identify the highest power of t in the denominator. The denominator is t^3 - 4. The highest power of t here is t^3. Step 2: Divide both the numerator and the denominator by this highest power. We divide every term in the numerator and denominator by t^3. Numerator: (3t^3 - 2t^2 + 5) / t^3 = 3t^3/t^3 - 2t^2/t^3 + 5/t^3 = 3 - 2/t + 5/t^3 Denominator: (t^3 - 4) / t^3 = t^3/t^3 - 4/t^3 = 1 - 4/t^3 So the function becomes: C(t) = [3 - 2/t + 5/t^3] / [1 - 4/t^3] Step 3: Take the limit as t approaches infinity. As t → ∞, terms with t in the denominator approach 0: - 2/t → 0 - 5/t^3 → 0 - 4/t^3 → 0 Therefore: lim(t→∞) C(t) = [3 - 0 + 0] / [1 - 0] = 3/1 = 3 Step 4: Interpret the result. The concentration approaches 3 mg/L as time goes to infinity. This tells the researcher that in the long term, the drug concentration stabilizes at 3 mg/L in the patient's bloodstream, rather than continuing to increase or decrease to zero. This represents a steady-state concentration for the medication.

5. Mason is analyzing the end behavior of the function f(x) = -4x^5 + 9x^3 - 7x + 12. Describe the behavior of f(x) as x approaches positive infinity and as x approaches negative infinity using limit notation. Answer: As x → ∞, f(x) → -∞; as x → -∞, f(x) → ∞ Solution: Identify the leading term of f(x) = -4x^5 + 9x^3 - 7x + 12. The term with the highest exponent is -4x^5. The degree is 5 (odd) and the leading coefficient is -4 (negative).
   Full step-by-step solution

   Step 1: Identify the leading term of f(x) = -4x^5 + 9x^3 - 7x + 12. The term with the highest exponent is -4x^5. The degree is 5 (odd) and the leading coefficient is -4 (negative). Step 2: For an odd-degree polynomial with a negative leading coefficient: - As x → ∞, the leading term -4x^5 dominates. Since x^5 is positive for large positive x, -4 times a large positive is a large negative. So f(x) → -∞. - As x → -∞, x^5 is negative (odd power), so -4 times a negative is positive. Thus f(x) → ∞. Step 3: Write in limit notation: lim_{x→∞} f(x) = -∞ lim_{x→-∞} f(x) = ∞ The answer is: As x → ∞, f(x) → -∞; as x → -∞, f(x) → ∞.

6. An environmental scientist is modeling the population growth of an endangered species using the function P(t) = (4t^4 - 3t^3 + 2t - 7)/(2t^4 + 5t^2 - 1), where P represents the population in thousands and t represents time in years. As time extends indefinitely into the future, what population level will the species approach according to this model? Answer: 2 Solution: Identify the highest degree terms in numerator and denominator Numerator: 4t^4 Denominator: 2t^4 As t approaches infinity, the lower degree terms (-3t^3, +2t, -7 in numerator and +5t^2, -1 in denominator) become negligible compared to the highest degree terms The function approaches the ratio of…
   Full step-by-step solution

   Step 1: Identify the highest degree terms in numerator and denominator Numerator: 4t^4 Denominator: 2t^4 Step 2: As t approaches infinity, the lower degree terms (-3t^3, +2t, -7 in numerator and +5t^2, -1 in denominator) become negligible compared to the highest degree terms Step 3: The function approaches the ratio of the coefficients of the highest degree terms P(t) ≈ (4t^4)/(2t^4) = 4/2 = 2 Step 4: Therefore, as t approaches infinity, P(t) approaches 2 The answer is 2.

7. A right circular cone has a height of 12 cm and a base radius of 5 cm. A horizontal plane cuts through the cone parallel to its base, creating a smaller cone at the top and a frustum below. If the smaller cone has a volume exactly one-eighth the volume of the original cone, what is the height of the smaller cone? Answer: 6 cm Solution: Volume = (1/3) * π * r^2 * h Height H = 12 cm Base radius R = 5 cm Volume_original = (1/3) * π * (5)^2 * 12 = (1/3) * π * 25 * 12 = (1/3) * π * 300 = 100π cm³ So Volume_original = 100π Volume_small = (1/8) * Volume_original So Volume_small = (1/8) * 100π = 100π/8 = 12.5π cm³ Relating dimensions…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Volume of the original cone** The formula for the volume of a cone is: Volume = (1/3) * π * r^2 * h For the original cone: Height H = 12 cm Base radius R = 5 cm Volume_original = (1/3) * π * (5)^2 * 12 = (1/3) * π * 25 * 12 = (1/3) * π * 300 = 100π cm³ So Volume_original = 100π --- **Step 2: Volume of the smaller cone** The problem says: Volume_small = (1/8) * Volume_original So Volume_small = (1/8) * 100π = 100π/8 = 12.5π cm³ --- **Step 3: Relating dimensions of small cone to original cone** When a cone is cut parallel to the base, the small top cone is similar to the original cone. Let h = height of the smaller cone (what we want to find). Let r = base radius of the smaller cone. By similarity: r / R = h / H So r / 5 = h / 12 Thus r = (5/12) * h --- **Step 4: Volume of small cone in terms of h** Volume_small = (1/3) * π * r^2 * h = (1/3) * π * [ (5/12) * h ]^2 * h = (1/3) * π * (25/144) * h^2 * h = (1/3) * π * (25/144) * h^3 --- **Step 5: Set equal to known volume** We know Volume_small = 12.5π So: (1/3) * π * (25/144) * h^3 = 12.5π Cancel π from both sides: (1/3) * (25/144) * h^3 = 12.5 Multiply both sides by 3: (25/144) * h^3 = 37.5 Multiply both sides by 144: 25 * h^3 = 37.5 * 144 --- **Step 6: Solve for h^3** 37.5 * 144 = (75/2) * 144 = 75 * 72 = 5400 So: 25 * h^3 = 5400 h^3 = 5400 / 25 = 216 --- **Step 7: Find h** h^3 = 216 h = cube root of 216 = 6 --- **Step 8: Conclusion** The height of the smaller cone is 6 cm. --- **Final answer:** 6 cm