University:
Florida Gateway College
Course:
MAC 2312 | Calculus II
Academic year:

2024
Views:

118

Pages:

5
Author:

Ruby Bates

Question 1 What can you say about the graphs of polynomial functions with an even degree compared to the graphs of polynomial functions with an odd degree? Use graphs from the Polynomial Functions Investigation to justify your response. Answer: Step 1: A polynomial of even degree will have both ends pointing in the same direction. If the leading coefficient is positive, both ends rise to positive infinity. If the leading coefficient is negative, both ends fall to negative infinity. Step 2: A polynomial of odd degree will have ends pointing in opposite directions. If the leading coefficient is positive, the left end falls, and the right end rises. If the leading coefficient is negative, the left end rises, and the right end falls. Question 2 Using calculus, it can be shown that the tangent function can be approximated by the polynomial tan ⁡ 𝑥 ≈ 𝑥 + 𝑥 3 3 + 16 𝑥 5 15 tanx≈x+ 3 x 3 + 15 16x 5 , where 𝑥 x is in radians. Use a graphing utility to graph the tangent function and its polynomial approximation in the same viewing window. How do the graphs compare? Answer: The red graph represents the tangent function, and the blue graph represents the polynomial approximation. The graphs are very similar near 𝑥 = 0 x=0, but differences become apparent as 𝑥 x moves further away from 0 due to the limited terms in the approximation. Question 3 Using calculus, it can be shown that the secant function can be approximated by the polynomial sec ⁡ 𝑥 ≈ 1 + 𝑥 2 2 + 5 𝑥 4 24 secx≈1+ 2 x 2 + 24 5x 4 , where 𝑥 x is in radians. Use a graphing utility to graph the secant function and its polynomial approximation in the same viewing window. How do the graphs compare? Answer: The graphs of sec ⁡ 𝑥 secx (red) and its polynomial approximation (blue) are very close when 𝑥 x is near 0. However, as 𝑥 x increases, the polynomial approximation diverges from the true secant function due to the higher-order terms not included in the approximation. Question 4 Describe the similarities between: (a) The lines 𝑦 = 𝑥 y=x and 𝑦 = − 𝑥 y=−x and the graphs of other odd-degree polynomial functions. (b) The parabolas 𝑦 = 𝑥 2 y=x 2 and 𝑦 = − 𝑥 2 y=−x 2 and the graphs of other even-degree polynomial functions. Answer: Step 1 (Odd-degree polynomials): The graphs of odd-degree polynomials, like 𝑦 = 𝑥 3 y=x 3 , will have similar end behavior to 𝑦 = 𝑥 y=x and 𝑦 = − 𝑥 y=−x. If the leading coefficient is positive, the graph rises from quadrant III to quadrant I, similar to 𝑦 = 𝑥 y=x. If the leading coefficient is negative, the graph falls from quadrant II to quadrant IV, similar to 𝑦 = − 𝑥 y=−x. Step 2 (Even-degree polynomials): The graphs of even-degree polynomials, like 𝑦 = 𝑥 4 y=x 4 , will have similar end behavior to 𝑦 = 𝑥 2 y=x 2 and 𝑦 = − 𝑥 2 y=−x 2 . If the leading coefficient is positive, both ends rise to positive infinity, like 𝑦 = 𝑥 2 y=x 2 . If the leading coefficient is negative, both ends fall to negative infinity, like 𝑦 = − 𝑥 2 y=−x 2 .