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Mr. Smith’s CALC CALC Assignment 4.2: Constant of Integration Recall that working with anti-derivatives was simply our way of “working backwards” In determining anti-derivatives, we were simply looking to find out what equation we started with in order to produce the derivative that was before us Ex. Find the anti-derivative of a(t) = 3t - 6e2t Definitions: An anti-derivative of f(x) is any function F(x) such that F`(x) = f(x) If F(x) is any antiderivative of f(x) then the most general anti-derivative of f(x) is called an indefinite integral and denoted f(x)dx = F(x) + C where C is any constant In this definition the is called the integral symbol, f(x) is called the integrand, x is called the integration variable and the “C” is called the constant of integration So we can interpret the statement f(x)dx as “determine the integral of f(x) with respect to x” The process of finding an indefinite integral (or simply an integral) is called integration. (x4 + 3x – 9)dx = (x2x)dx = (x4 + 3x – 9)dx = (cos + 2sin3)d = e2xdx = (8x + sec2x)dx = sin(2x)dx = (2 - x)2dx = Now for that Pesky Constant of Integration “C”. Given that f(x)dx = F(x) + C, we can determine a specific function if we knew what C was equal to so if we knew something about the function F(x), then we could solve for C Evaluate (x3 – 3x + 1)dx if F(0) = -2 F(x) = ¼x4 – 3/2x2 + x + C Since F(0) = -2 = ¼(0)4 – 3/2(0)2 + (0) + C So C = -2 and F(x) = ¼x4 – 3/2x2 + x - 2 Position, Velocity, Acceleration Mr. Smith’s CALC —An object moves along a co-ordinate line with a velocity v(t) = 2 - 3t + t2 meters/sec. Its initial position is 2 m to the right of the origin. —(a) Determine the position of the object 4 seconds later —(b) Determine the total distance traveled in the first 4 seconds