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04:2245–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.
46. ∫(0 to 2) x⁴ dx; n = 30
c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.
2. Give an example of each of the following.
b. A repeated linear factor
Prove the following orthogonality relations (which are used to generate Fourier series). Assume m and n are integers with m ≠ n.
c.
π
∫ sin(mx) cos(nx) dx = 0, when |m + n| is even
0
94. [Use of Tech] Skydiving A skydiver has a downward velocity given by v(t) = V_T [(1 - e^(-2gt/V_T))/(1 + e^(-2gt/V_T))],
where t = 0 is the instant the skydiver starts falling, g = 9.8 m/s² is the acceleration due to gravity, and V_T is the terminal velocity of the skydiver.
c. Verify by integration that the position function is given by
s(t) = V_T t + (V_T²/g) ln[(1 + e^(-2gt/V_T))/2],
where s'(t) = v(t) and s(0) = 0.
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
67. Let f(x) = √(x³ + 1).
b. Calculate f''(x).
82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.
b. Find the area of the region bounded by y = x * e^(-a * x) and the x-axis on the interval [0, 4], where a > 0.
