Function End Behavior

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Answer Key & Explanations

Function End Behavior · Grade 12 · Worksheet 1

1. lim(x→∞) (4x³ - 2x² + 5)/(3x³ + 7x - 1) = ? Answer: 4/3 Solution: Identify the degrees of numerator and denominator. Both are degree 3. For rational functions where degrees are equal, the limit equals the ratio of leading coefficients.
   Full step-by-step solution

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

2. Tane is a materials engineer developing a new type of ceramic coating for high-temperature applications. He models the thermal conductivity K (in W/m·K) of the coating as a function of temperature T (in Kelvin) using the function K(T) = (9T^3 - 7T^2 + 5T - 3) / (3T^3 + 2T - 1). As the temperature approaches extremely high values (T → ∞), what limiting thermal conductivity value does the coating approach? Answer: 3 Solution: Identify the highest power terms in the numerator and denominator. In the numerator, 9T^3 is the highest power term with coefficient 9. In the denominator, 3T^3 is the highest power term with coefficient 3.
   Full step-by-step solution

   Step 1: Identify the highest power terms in the numerator and denominator. In the numerator, 9T^3 is the highest power term with coefficient 9. In the denominator, 3T^3 is the highest power term with coefficient 3. Step 2: For rational functions where the degrees of the numerator and denominator are equal (both degree 3), the horizontal asymptote is found by taking the ratio of the leading coefficients. Step 3: Calculate the ratio: 9 / 3 = 3. Step 4: As T approaches infinity, the lower-degree terms (-7T^2, 5T, -3 in the numerator; 2T, -1 in the denominator) become negligible compared to the T^3 terms. Step 5: Therefore, using limit notation: lim_(T→∞) K(T) = 9/3 = 3. The limiting thermal conductivity is 3 W/m·K.

3. A biomedical engineer 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 represent in terms of the drug's long-term behavior in the body? 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 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 numerator and denominator. - The numerator is 3t^3 - 2t^2 + 5. The highest power is t^3, so degree is 3. - The denominator is t^3 - 4. The highest power is t^3, so degree is 3. Step 2: Since degrees are equal, the limit as t → ∞ is the ratio of the leading coefficients. - Leading coefficient of numerator: 3 (from 3t^3) - Leading coefficient of denominator: 1 (from t^3) - Therefore, limit = 3/1 = 3. Step 3: Verify by dividing numerator and denominator by the highest power of t in the denominator (t^3): C(t) = [3t^3/t^3 - 2t^2/t^3 + 5/t^3] / [t^3/t^3 - 4/t^3] = [3 - 2/t + 5/t^3] / [1 - 4/t^3] Step 4: Take the limit as t → ∞: - As t → ∞, terms 2/t, 5/t^3, and 4/t^3 all approach 0. - So C(t) → (3 - 0 + 0) / (1 - 0) = 3/1 = 3. Step 5: Interpretation: The concentration approaches 3 mg/L in the long term. This represents the steady-state concentration of the medication in the bloodstream, where the rate of drug administration/distribution equals the rate of elimination/metabolism. ANSWER: 3 mg/L

4. A biomedical engineer is modeling the concentration of a new drug in a patient's bloodstream using the function C(t) = (3t^3 - 2t^2 + 5)/(t^3 + 1), where C is measured in milligrams per liter and t is time in hours. As time approaches infinity, what concentration level does the drug 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 + 1) Identify the highest power of t in the denominator. The denominator is t^3 + 1. The highest power is t^3.
   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 + 1) Step 1: Identify the highest power of t in the denominator. The denominator is t^3 + 1. The highest power is t^3. Step 2: Divide every term in the numerator and denominator by this highest power (t^3). Numerator: (3t^3)/(t^3) - (2t^2)/(t^3) + (5)/(t^3) = 3 - (2/t) + (5/t^3) Denominator: (t^3)/(t^3) + (1)/(t^3) = 1 + (1/t^3) So the function becomes: C(t) = [3 - (2/t) + (5/t^3)] / [1 + (1/t^3)] Step 3: Take the limit as t approaches infinity. As t becomes very large: - 2/t approaches 0 - 5/t^3 approaches 0 - 1/t^3 approaches 0 Substituting these values: C(t) = [3 - 0 + 0] / [1 + 0] = 3/1 = 3 Step 4: Interpret the result. As time approaches infinity, the drug concentration approaches 3 mg/L. This makes sense because for large values of t, the highest degree terms (3t^3 in numerator and t^3 in denominator) dominate the behavior of the function, and their ratio is 3/1 = 3. ANSWER: 3 mg/L

5. Sophia, an astrophysicist, is modeling the brightness of a distant quasar as it varies over time. The brightness B(t) (in arbitrary units) is given by the rational function B(t) = (7t^4 - 9t^3 + 12t - 5) / (14t^4 + 6t^2 - 11), where t represents time in millions of years. As t approaches positive infinity, what constant brightness level does the quasar's brightness approach? Answer: 0.5 Solution: Identify the highest power of t in the numerator and the denominator. The highest power in the numerator is t^4, and in the denominator it is also t^4. Find the leading coefficient of the numerator, which is 7.
   Full step-by-step solution

   Step 1: Identify the highest power of t in the numerator and the denominator. The highest power in the numerator is t^4, and in the denominator it is also t^4. Step 2: Find the leading coefficient of the numerator, which is 7. Find the leading coefficient of the denominator, which is 14. Step 3: For a rational function where 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 4: Calculate the ratio: 7 / 14 = 1/2 = 0.5. Step 5: Therefore, as time approaches infinity, the brightness of the quasar approaches 0.5 arbitrary units. The answer is 0.5.

6. A three-dimensional solid is formed by rotating the region bounded by the curve y = sqrt(x), the line y = 0, and the line x = 4 about the x-axis. This creates a curved solid that resembles a horn or trumpet shape. Calculate the volume of this solid of revolution. Answer: 8π Solution: Identify the method for finding volume of revolution. Since we're rotating about the x-axis, we use the disk method. The formula for volume using disk method is V = π∫[f(x)]² dx from a to b.
   Full step-by-step solution

   Step 1: Identify the method for finding volume of revolution. Since we're rotating about the x-axis, we use the disk method. Step 2: The formula for volume using disk method is V = π∫[f(x)]² dx from a to b. Step 3: Our function is y = sqrt(x), and we're integrating from x=0 to x=4. Step 4: Square the function: [sqrt(x)]² = x Step 5: Set up the integral: V = π∫(x) dx from 0 to 4 Step 6: Evaluate the integral: ∫x dx = (1/2)x² Step 7: Apply the limits: [(1/2)(4)² - (1/2)(0)²] = [(1/2)(16) - 0] = 8 Step 8: Multiply by π: V = π × 8 = 8π The answer is 8π.

7. Liam is analyzing the end behavior of the function f(x) = (3x^4 - 2x^3 + 5x - 1)/(2x^4 + x^2 - 7) as x approaches positive and negative infinity. He needs to determine the horizontal asymptote of this rational function. What is the equation of the horizontal asymptote? Answer: y = 3/2 Solution: To find the horizontal asymptote of a rational function, we compare the degrees of the numerator and denominator and examine the leading coefficients. Identify the degrees of the numerator and denominator. The function is f(x) = (3x^4 - 2x^3 + 5x - 1) / (2x^4 + x^2 - 7).
   Full step-by-step solution

   To find the horizontal asymptote of a rational function, we compare the degrees of the numerator and denominator and examine the leading coefficients. Step 1: Identify the degrees of the numerator and denominator. The function is f(x) = (3x^4 - 2x^3 + 5x - 1) / (2x^4 + x^2 - 7). The numerator is 3x^4 - 2x^3 + 5x - 1. The highest power of x is 4, so the degree is 4. The denominator is 2x^4 + x^2 - 7. The highest power of x is 4, so the degree is 4. Step 2: Determine the rule for horizontal asymptotes when degrees are equal. When the degrees of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients. Step 3: Identify the leading coefficients. The leading term of the numerator is 3x^4, so its leading coefficient is 3. The leading term of the denominator is 2x^4, so its leading coefficient is 2. Step 4: Calculate the ratio of the leading coefficients. Ratio = (Leading coefficient of numerator) / (Leading coefficient of denominator) = 3 / 2. Step 5: State the equation of the horizontal asymptote. The horizontal asymptote is the line y = 3/2. This result holds true for both x approaching positive infinity and x approaching negative infinity.