Function End Behavior

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Answer Key & Explanations

Function End Behavior · Grade 12 · Worksheet 1

1. A civil engineer is designing a suspension bridge where the cable's shape is modeled by the function f(x) = (4x³ - 3x² + 7)/(2x³ + 5x - 1), with x representing the horizontal distance from the center in meters. As the distance from the center increases indefinitely in either direction, what height does the cable approach above the roadway? Answer: 2 Solution: Identify the degrees of the numerator and denominator. Both are degree 3 polynomials. For rational functions where the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
   Full step-by-step solution

   Step 1: Identify the degrees of the numerator and denominator. Both are degree 3 polynomials. Step 2: For rational functions where the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. Step 3: The leading coefficient of the numerator is 4, and the leading coefficient of the denominator is 2. Step 4: Calculate the ratio: 4/2 = 2. Step 5: Therefore, as x approaches positive or negative infinity, f(x) approaches 2. The answer is 2.

2. lim(x→∞) (3x⁵ - 2x³ + 7)/(4x⁵ - x² + 9) = ? Answer: 3/4 Solution: Identify the highest degree terms in numerator and denominator Numerator: 3x⁵ Denominator: 4x⁵ (3x⁵/x⁵ - 2x³/x⁵ + 7/x⁵)/(4x⁵/x⁵ - x²/x⁵ + 9/x⁵) = (3 - 2/x² + 7/x⁵)/(4 - 1/x³ + 9/x⁵) As x → ∞, 2/x² → 0, 7/x⁵ → 0, 1/x³ → 0, 9/x⁵ → 0 So the limit becomes (3 - 0 + 0)/(4 - 0 + 0) = 3/4 The answer is 3/4.
   Full step-by-step solution

   Step 1: Identify the highest degree terms in numerator and denominator Numerator: 3x⁵ Denominator: 4x⁵ Step 2: Divide both numerator and denominator by x⁵ (3x⁵/x⁵ - 2x³/x⁵ + 7/x⁵)/(4x⁵/x⁵ - x²/x⁵ + 9/x⁵) = (3 - 2/x² + 7/x⁵)/(4 - 1/x³ + 9/x⁵) Step 3: Evaluate the limit as x approaches infinity As x → ∞, 2/x² → 0, 7/x⁵ → 0, 1/x³ → 0, 9/x⁵ → 0 So the limit becomes (3 - 0 + 0)/(4 - 0 + 0) = 3/4 The answer is 3/4.

3. lim(x→∞) (4x⁵ - 2x³ + 8)/(3x⁵ + 5x² - 1) = ? Answer: 4/3 Solution: Identify the highest degree terms in numerator and denominator Numerator: 4x⁵ Denominator: 3x⁵ For large x values, the lower degree terms become insignificant lim(x→∞) (4x⁵ - 2x³ + 8)/(3x⁵ + 5x² - 1) ≈ lim(x→∞) (4x⁵)/(3x⁵) (4x⁵)/(3x⁵) = 4/3 The limit equals the ratio of the leading coefficients…
   Full step-by-step solution

   Step 1: Identify the highest degree terms in numerator and denominator Numerator: 4x⁵ Denominator: 3x⁵ Step 2: For large x values, the lower degree terms become insignificant lim(x→∞) (4x⁵ - 2x³ + 8)/(3x⁵ + 5x² - 1) ≈ lim(x→∞) (4x⁵)/(3x⁵) Step 3: Simplify the ratio of the leading terms (4x⁵)/(3x⁵) = 4/3 Step 4: The limit equals the ratio of the leading coefficients lim(x→∞) (4x⁵ - 2x³ + 8)/(3x⁵ + 5x² - 1) = 4/3 The answer is 4/3.

4. A biomedical engineer is modeling the concentration of a new medication in a patient's bloodstream over time using the function C(t) = (3t^3 - 2t^2 + 5)/(t^3 - 4), where C is measured in milligrams per liter and t is time in hours. As time approaches infinity, what concentration level does the medication approach in the patient's bloodstream? Answer: 3 mg/L Solution: To find the concentration as time approaches infinity, we analyze the function C(t) = (3t^3 - 2t^2 + 5)/(t^3 - 4). Identify the degrees of the numerator and denominator. - The numerator is 3t^3 - 2t^2 + 5.
   Full step-by-step solution

   To find the concentration as time approaches infinity, we analyze the function C(t) = (3t^3 - 2t^2 + 5)/(t^3 - 4). Step 1: Identify the degrees of the numerator and denominator. - The numerator is 3t^3 - 2t^2 + 5. The highest power of t is t^3, so the degree is 3. - The denominator is t^3 - 4. The highest power of t is t^3, so the degree is also 3. Step 2: Since the degrees of the numerator and denominator are equal, the limit as t approaches infinity is determined by the ratio of the leading coefficients. - The leading coefficient of the numerator is 3 (from 3t^3). - The leading coefficient of the denominator is 1 (from t^3). Step 3: Calculate the limit. - The limit as t approaches infinity is (leading coefficient of numerator) / (leading coefficient of denominator) = 3 / 1 = 3. Step 4: Interpret the result. - Therefore, as time approaches infinity, the concentration of the medication in the bloodstream approaches 3 mg/L. Final Answer: 3 mg/L

5. lim(x→-∞) (6x⁶ - 11x⁴ + 1)/(3x⁶ + 7x² - 6) = ? Answer: 2 Solution: Identify the highest degree terms in numerator and denominator. Numerator: 6x⁶ (degree 6). Denominator: 3x⁶ (degree 6).
   Full step-by-step solution

   Step 1: Identify the highest degree terms in numerator and denominator. Numerator: 6x⁶ (degree 6). Denominator: 3x⁶ (degree 6). Step 2: Since the degrees are equal, divide both numerator and denominator by x⁶: (6x⁶/x⁶ - 11x⁴/x⁶ + 1/x⁶) / (3x⁶/x⁶ + 7x²/x⁶ - 6/x⁶) = (6 - 11/x² + 1/x⁶) / (3 + 7/x⁴ - 6/x⁶) Step 3: Evaluate the limit as x→-∞. As x→-∞, 11/x² → 0, 1/x⁶ → 0, 7/x⁴ → 0, 6/x⁶ → 0. Step 4: The limit simplifies to (6 - 0 + 0) / (3 + 0 - 0) = 6/3 = 2. The answer is 2.

6. A solid is formed by rotating the region bounded by the curves y = x^2 and y = 4 about the line y = 6. This creates a three-dimensional shape resembling a bowl with a parabolic cross-section. Calculate the volume of this solid of revolution using the method of washers. Answer: 384π/5 Solution: When finding volumes of revolution using washers, we consider the area between two curves rotated around an external axis.
   Full step-by-step solution

   When finding volumes of revolution using washers, we consider the area between two curves rotated around an external axis. The washer method calculates the volume by summing cross-sectional areas perpendicular to the axis of rotation. For regions bounded by curves, we determine the outer and inner radii based on the distance from each curve to the axis of rotation.

7. Consider the function f(x) = (3x^4 - 2x^3 + 5x)/(x^4 - 7x^2 + 1). As x approaches positive infinity, what value does f(x) approach? Answer: 3 Solution: f(x) = (3x^4 - 2x^3 + 5x) / (x^4 - 7x^2 + 1) Identify the highest power of x in the denominator. The denominator is x^4 - 7x^2 + 1. The highest power is x^4.
   Full step-by-step solution

   Let's find the limit of f(x) as x approaches positive infinity. We have: f(x) = (3x^4 - 2x^3 + 5x) / (x^4 - 7x^2 + 1) Step 1: Identify the highest power of x in the denominator. The denominator is x^4 - 7x^2 + 1. The highest power is x^4. Step 2: Divide every term in the numerator and denominator by x^4. This is the standard method for finding limits at infinity of rational functions. Numerator: (3x^4 - 2x^3 + 5x) / x^4 = 3x^4/x^4 - 2x^3/x^4 + 5x/x^4 = 3 - 2/x + 5/x^3 Denominator: (x^4 - 7x^2 + 1) / x^4 = x^4/x^4 - 7x^2/x^4 + 1/x^4 = 1 - 7/x^2 + 1/x^4 So f(x) = [3 - 2/x + 5/x^3] / [1 - 7/x^2 + 1/x^4] Step 3: Take the limit as x → ∞. As x → ∞: - 2/x → 0 - 5/x^3 → 0 - 7/x^2 → 0 - 1/x^4 → 0 So we get: lim(x→∞) f(x) = (3 - 0 + 0) / (1 - 0 + 0) = 3/1 = 3 Step 4: Conclusion. The function approaches 3 as x approaches positive infinity. Answer: 3

8. lim_{x→∞} (2x⁴ - 5x³ + 7)/(3x⁴ + x² - 9) = ? Answer: 2/3 Solution: Identify the degrees of numerator and denominator. Both are degree 4. For rational functions where degrees are equal, the limit at infinity equals the ratio of leading coefficients.
   Full step-by-step solution

   Step 1: Identify the degrees of numerator and denominator. Both are degree 4. Step 2: For rational functions where degrees are equal, the limit at infinity equals the ratio of leading coefficients. Step 3: Leading coefficient of numerator is 2, leading coefficient of denominator is 3. Step 4: The limit is 2/3. The answer is 2/3.