[考題] 基礎數學/1012期中考/余屹正
1. Find following intergrals.
a) ∫ sin(2x)cos(3x) dx
b) ∫ (x-1)^2 * e^(2x) dx
c) ∫ e^(2x+1) * sin(πx/2) dx
d) ∫ [ x^2 / (4x^2 + 4x + 1)] dx
2. Find a solution of dy/dx = y^2 - 4 that passes through (0,4)
3. Let L(t) be the length of a fish at time t. Suppose that the fish grows
according to the von Bertalanffy equaton dL/dt = k(34 - L) with L(0)=2
a) Solve the differential equation
b) Determine k under the assumption that L(4)=10
c) Find the length of the fish when t=10
d) Find lim(t→∞)L(t)
4.
a) Find the Taylor polynomial for f(x) = arctan x about x=0 for │x│≦1
b) Explain why the following holds: (π/4) = 1 - (1/3) + (1/5) - (1/7) + ...
5. Discuss the convergence and divergence of
∫(0到∞) (x^p) * (e^(-x)) dx for 0＜p＜∞
6. Determine and explain whether integral is convergent
a) ∫(1到∞) [1/√(1 + 6^x)] dx
b) ∫(1到∞) [1/√(x + ln x)] dx
7.
a) How large should n be so that the midpoint rule approximation of
∫(0到2) x^2 dx is accurate to within 10^(-4)
b) How large should n be so that the trapezoidal rule approximation of
∫(0到2) x^2 dx is accurate to within 10^(-4)
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