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Use a Taylor-Series expansion to estimate cos(π/4), with a=0.7 and using three terms (up to the h2 term).

Question Description


1. Use a Taylor-Series expansion to estimate cos(π/4), with a=0.7 and using three terms (up to the h2 term).

2. Write "LSU" using 8-bit binary code. When you enter the answer, do not use any spaces in the full string.

Use a Taylor-Series expansion to estimate cos(π/4), with a=0.7 and using three terms (up to the h2 term).
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Final Answer

The solution is attached in pdf and word format.Please, let me know if you need any clarification.

Question 1:
Use a Taylor-series expansion to estimate cos(𝜋/4), with 𝑎 = 0.7 and using three terms (up to
the ℎ2 term).
Solution
The Taylor series of a function 𝑓(𝑥) that is differentiable at a point 𝑎 is given by:
𝑓(𝑥) = 𝑓(𝑎) +

𝑓 ′ (𝑎)
𝑓 ′′ (𝑎)
𝑓 ′′′ (𝑎)
(𝑥 − 𝑎) +
(𝑥 − 𝑎)2 +
(𝑥 − 𝑎)3 + ⋯
1!
2!
3!

For the given problem:
𝑓(𝑥) = cos(𝑥)

𝑎 = 0.7

𝑓(𝑥) = cos(𝑥)

𝑓(𝑎) = cos(0.7) = 0.7648

𝑓 ′ (𝑥) = −sin(𝑥)

𝑓 ′ (𝑎) = − sin(0.7) = −0.6442

𝑓 ′′ (𝑥) = −cos(𝑥)

𝑓 ′′ (𝑎) = − cos(0.7) = ...

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