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

2024
Views:

446

Pages:

2
Author:

Ruby Bates

Question 1 In calculus, it can be shown that the arctangent function can be approximated by the polynomial: arctan ⁡ 𝑥 ≈ 𝑥 − 𝑥 3 3 + 𝑥 5 5 − 𝑥 7 7 arctanx≈x− 3 x 3 + 5 x 5 − 7 x 7 where 𝑥 x is in radians. Use a graphing utility to graph the arctangent function and its polynomial approximation in the same viewing window. How do the graphs compare? Answer: The red graph represents arctan ⁡ ( 𝑥 ) arctan(x), and the blue graph represents its polynomial approximation. Observation: The graphs are very similar close to the origin. However, as 𝑥 x moves farther from the origin, the polynomial approximation diverges from the actual arctangent graph due to the limited terms included in the approximation. Question 2 Find the Taylor polynomial 𝑇 3 ( 𝑥 ) T 3 (x) for the function 𝑓 ( 𝑥 ) f(x) centered at the number 𝑎 a. Graph 𝑓 f and 𝑇 3 ( 𝑥 ) T 3 (x) on the same screen. 𝑓 ( 𝑥 ) = 𝑥 + 𝑒 − 𝑥 , 𝑎 = 0 f(x)=x+e −x ,a=0 Answer: The Taylor polynomial 𝑇 3 ( 𝑥 ) T 3 (x) is given by: 𝑇 3 ( 𝑥 ) = 𝑓 ( 𝑎 ) + 𝑓 ′ ( 𝑎 ) ( 𝑥 − 𝑎 ) + 𝑓 ′ ′ ( 𝑎 ) 2 ! ( 𝑥 − 𝑎 ) 2 + 𝑓 ′ ′ ′ ( 𝑎 ) 3 ! ( 𝑥 − 𝑎 ) 3 T 3 (x)=f(a)+f ′ (a)(x−a)+ 2! f ′′ (a) (x−a) 2 + 3! f ′′′ (a) (x−a) 3 From the calculations: 𝑇 3 ( 𝑥 ) = 1 + 𝑥 + 𝑥 2 2 − 𝑥 3 6 T 3 (x)=1+x+ 2 x 2 − 6 x 3 Graph Observation: The red curve represents 𝑓 ( 𝑥 ) = 𝑥 + 𝑒 − 𝑥 f(x)=x+e −x , and the blue curve represents its Taylor polynomial 𝑇 3 ( 𝑥 ) T 3 (x). The polynomial 𝑇 3 ( 𝑥 ) T 3 (x) closely approximates 𝑓 ( 𝑥 ) f(x) near 𝑥 = 0 x=0, but differences become more noticeable as 𝑥 x moves away from 𝑎 = 0 a=0. Question 3 For the following polynomial 𝑓 ( 𝑥 ) = − 3 𝑥 2 + 6 𝑥 f(x)=−3x 2 +6x: (a) Find the degree. (b) Find the zeros, if any. (c) Find the 𝑦 y-intercept, if any. (d) Use the leading coefficient to determine the graph’s end behavior. (e) Determine algebraically whether the polynomial is even, odd, or neither. Answer: (a) Degree: 2 (b) Zeros: 𝑥 = 2 , 𝑥 = 0 x=2,x=0 (c) 𝑦 y-intercept: 0 0 (d) End Behavior: The graph falls to the left and right because the leading coefficient − 3 −3 is negative. (e) Even/Odd: The function is neither even nor odd. This can be determined algebraically as 𝑓 ( − 𝑥 ) ≠ 𝑓 ( 𝑥 ) f(−x)  =f(x) and 𝑓 ( − 𝑥 ) ≠ − 𝑓 ( 𝑥 ) f(−x)  =−f(x).