If the value of a definite integral has a pi in it, we can use the Taylor series
ID: 2851872 • Letter: I
Question
If the value of a definite integral has a pi in it, we can use the Taylor series expansion of the integrand to find an infinite series that may or may not provide useful approximations to pi. Recall that the infinite series 4 - 4/3 + 4/5 - 4/7 + . . . converges to pi, but is the absolute worst way possible to approximate pi. How many terms of this series would one have to add to get an approximation for pi with error less than 10^-4? Use trig substitution to verify that I_1 = 1/1 + 3x^2 dx has a pi in it. Use the Taylor series expansion of 1/1 + 3x^2 to find a series approximating I_1. This series does not converge. Why do bad things happen? Verify that I_2 = integral_0^1/3 1/1 + 3x^2 dx also has a pi in it. Use the Taylor series expansion of 1/1 + 3x^2 to find a series approximating I_2. Write this series in sigma notation. How many terms of this series would one have to add in order to get an approximation for pi with error less than 10^-4?Explanation / Answer
converges the so
y(x, t) =A cos(at) sin(bx)
Let x represent some position and t represent time.
Describe a physical situation that could be represented by this equation. As part of your description include a sketch and a written description.
Indicate what y and x correspond to in the situation you describe.<_>
How, if at all, would the physical situation you described in part A be different if a were twice as large? Explain how you determined your answer.
How, if at all, would the physical situation you described in part A be different if b were twice as large? Explain how you determined your answer.