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03:4845–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.
48. ∫(0 to π/4) (1/(1 + x²)) dx; n = 64
c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.
Gaussians An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), f(x) = e^(-ax²).
c. Complete the square to evaluate ∫ from -∞ to ∞ of e^(-(ax² + bx + c)) dx, where a > 0, b, and c are real numbers.
43. A hot-air balloon is launched from an elevation of 5400 ft above sea level. As it rises, the vertical velocity is computed using a device (called a variometer) that measures the change in atmospheric pressure. The vertical velocities at selected times are shown in the table (with units of ft/min).
c. A polynomial that fits the data reasonably well is:
g(t) = 3.49t³ - 43.21t² + 142.43t - 1.75
Estimate the elevation of the balloon after five minutes using this polynomial.
45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.
45. ∫(0 to 1) e^(2x) dx; n = 25
c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.
75. Exploring powers of sine and cosine
c. Prove that ∫₀ᵖⁱ sin²(nx) dx has the same value for all positive integers n.
75. {Use of Tech} Oscillator displacements Suppose a mass on a spring that is slowed by friction has the position function:
s(t) = e⁻ᵗ sin t
c. Generalize part (b) and find the average value of the position on the interval [nπ, (n+1)π], for n = 0, 1, 2, ...
