find the position function s(t)describing the motion of the object.

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base of a solid in the xy-plane is the 1st quadrant region bounded y=x&y=x^2cross section of solid perpendicular to x-axis are equilateral triangles. whats the volume,in cubic units of solid? -Determine if Mean Value Theorem for integrals applies to the function f(x)=x^3-9x on the interval [-1,1]if so find x-coordinates of points guaranteed to exist by theorem.-object has a constant acceleration of 40 ft/sec^2,an inital velocity of -20ft/sec,&inital posit of10ft find function s(t)descrbe motion.

Apr 23rd, 2015

The graphs y = x and y = x2 intersect at the points (0, 0) and (1, 1). So, V = ∫01 A(x)dx, where A(x) is the area of the cross-section of the solid by a plane perpendicular to the x-axis. Since the area of an equilateral triangle with the side a equals A = (√(3)/4)a2, the function A(x) = (√(3)/4)(x – x2)2 and the volume V = ∫01 (√(3)/4)(x – x2)2dx = (√(3)/4)∫01(x2 – 2x3 + x4)dx =(√(3)/4)(x3/3 – 2x4/4 + x5/5)]01 (√(3)/4)(1/3 – 1/2 + 1/5) = √(3)/120.


According to the Mean Value Theorem ∫–11(x3 – 9x) dx = 2f(x*), where x* lies between –1 and 1. However, the integral equals 0 (the integrand is an odd function and the interval of integration is symmetric with respect to 0), thus, f(x*) = 0. The only possible value of x* is 0 (x3 – 9x = 0 has also solutions ±3 but they do not lie between –1 and 1).


The law of motion can be described as s(t) = s0 + v0t + at2/2 = 10 – 20t + 20t2, where time t is expressed in seconds and the position s in feet.


Apr 24th, 2015

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