An individuаl with muscle tоne оf the left bicep, wrist аnd finger flexоrs grаded as 2 on the Modified Ashworth Scale would exhibit which of the following in the left upper extremity:
Rectаngulаr Cооrdinаtes 1.) Find the arc length оf the curve (y=frac{x^{4}}{16}+frac{1}{2x^{2}}) along the interval (xinleft[1,2right]). Parametric Equations Curve #1 Let (x=5t^{2}-2) and (y=3t^{3}+t). 2.) Find (frac{d^{2}y}{dx^{2}}). (Getting the first derivative correct is worth 5 points.) 3.) Find the area under the curve from (t=1) to (t=3). Curve #2 Let (x=4t^{3}+10) and (y=3t^{2}). 4.) Find the arc length of the curve, from (t=0) to (t=2). Polar Coordinates 5a.) Convert the point (left(x,yright)=left(-5,-10right)) to polar coordinates, (left(r,thetaright)). Choose your coordinates so that (rgeq 0) and (thetainleft[0,2piright)). Give your (theta) value to the nearest (0.001). 5b.) Convert the point (left(r,thetaright)=left(6,-frac{pi}{3}right)) to rectangular coordinates (left(x,yright)). 6.) Along the polar curve (r=frac{1}{2}costheta), find the value of the derivative (frac{dy}{dx}) at the point (left(r,thetaright)=left(frac{1}{4},frac{pi}{3}right)). 7.) Find the area enclosed by the curve (r=sqrt{1+3costheta}) in the first quadrant. Integral Setups ONLY In each part, your answer should still be an integral. 8a.) Set up and simplify, but do not complete, the integral that would give the area of the surface formed by rotating (x=y^{2}+1) for (1leq y leq 3) about the (y)-axis. [5 points] 8b.) Set up and simplify, but do not complete, the integral that would give the arc length of the curve (r=3theta^{2}) through the interval (left[0,2piright]). [5 points] 8c.) Set up and simplify, but do not complete, the integral that would give the area of the surface formed by rotating (x=sin^{3}left(tright)), (y=cos^{3}left(tright)) about the (x)-axis, from (t=0) to (t=frac{pi}{2}). [5 points] Conic Sections 9.) Find an equation of the ellipse whose vertices are (left(3,7right)) and (left(3,13right)) and whose foci are (left(3,9right)) and (left(3,11right)). 10.) Given the hyperbola [frac{x^{2}}{25}-frac{y^{2}}{81}=1] Sketch a graph of this hyperbola, and list its foci, vertices, and the equations of its asymptotes. Extra Credit EC.) Find the arc length in #8b by finishing the integral. (Note: This only applies if the integral you create in #8b is fairly close to the correct one.)
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