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05:58{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
b. Expand sin t² and cos t² in a Maclaurin series, and then integrate to find the first four nonzero terms of the Maclaurin series for S and C.
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
b. Estimate f(0.2) and give a bound on the error in the approximation.
f(x) = ln (1 + x) ≈ x − x²/2
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
b. Estimate f(0.2) and give a bound on the error in the approximation.
f(x) = sin x ≈ x
Probability: sudden−death playoff Teams A and B go into suddendeath overtime after playing to a tie. The teams alternate possession of the ball, and the first team to score wins. Assume each team has a 1/6 chance of scoring when it has the ball, and Team A has the ball first.
b. The expected number of rounds (possessions by either team) required for the overtime to end is (1/6) ∑ₖ₌₁∞ k(5/6)ᵏ⁻¹. Evaluate this series.
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x)=3ˣ, a=0
Taylor series
b. Write the power series using summation notation.
f(x) = ln x, a = 3
