formulas1 - Possibly helpful formulas and models 1 du = ln...

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Unformatted text preview: Possibly helpful formulas and models 1 du = ln |u| + C u du = arctan u + C 1 + u2 eu du = eu + C un−1 sin u du un cos u du = un sin u − n ∞ nu nu u e du = u e − n n −1 u u un e−u du = n! e du 0 Equation of motion: x(t) is position at time t, a is acceleration (constant), v0 is initial velocity, x0 is initial position: 1 x(t) = at2 + v0 t + x0 2 Cross-current travel: y (x) is vertical displacement by the time the swimmer is x across the channel spanning [−a, a], with constant swimming speed vs and midstream fluid velocity v0 : dy v0 = dx vs x2 1− 2 a Torricelli’s law: y (t) is the depth of the water in the tank at time t, A(y ) is the cross-sectional area of the tank at height y , a is the area of the drain hole, and g is gravitational acceleration: A(y ) dy = −a 2gy dt ...
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