MA113_2010SPRING_EXAM1_PROFSOLN_[0]

MA113_2010SPRING_EXAM1_PROFSOLN_[0] - MA113 Calculus II The...

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MA113 Calculus II The Cooper Union, Spring 2010 Prof. Alex Casti Midterm Exam Solutions Problem 1 (25 points) Give brief answers to the following questions, which may involve short mathematical calculations. (a) Suppose a spring exerts a force F ( x ) = 3 x 3 (Newtons). Calculate the work done in stretching the spring 2 meters (assume the equilibrium position is x = 0 ). W = Z 2 0 F ( x ) dx = Z 2 0 3 x 3 dx = 12 joules . (b) Consider the parametric curve described by x ( t ) = 2 t and y ( t ) = e t 2 . Calculate dy dx and d 2 y dx 2 , but do not begin by writing y = y ( x ) to do it. Use the formulas dy dx = dy dt dx dt dx dt = 2 dy dt = 2 te t 2 dy dx = te t 2 t = x 2 = 1 2 xe x 2 4 . and d 2 y dx 2 = d dt dy dx dx dt d dt dy dx = ( 1 + 2 t 2 ) e t 2 d 2 y dx 2 = 1 2 ( 1 + 2 t 2 ) e t 2 = 1 2 1 + 1 2 x 2 e x 2 4 . (c) Calculate the area of the polar curve r = sin 2 θ over θ 0 , π 2 . The area A is given by A = 1 2 Z π 2 0 r 2 d θ = 1 2 Z π 2 0 sin 4 θ d θ To obtain the antiderivative it helps to simplify the integrand using standard trigonometric identities:
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sin 2 θ = 1 2 ( 1 - cos2 θ ) sin 4 θ = 1 4 ( 1 - cos2 θ ) 2 = 1 4 ( 1 - 2cos2 θ + cos 2 2 θ ) = 1 4 ( 1 - 2cos2 θ + 1 - sin 2 2 θ ) = 1 4 2 - 2cos2 θ - 1 2 ( 1 - cos4 θ ) = 1 4 3 2 - 2cos2 θ + 1 2 cos4 θ = 1 8 ( 3 - 4cos2 θ + cos4 θ ) A = 1 16 Z π 2 0 [ 3 - 4cos2 θ + cos4 θ ] d θ = 1 16 3 θ - 2sin2 θ + 1 4 sin4 θ π 2 θ = 0 = 1 16 3 π 2 - 2sin π + 1 4 sin2 π A = 3 π 32 (d) Prove that a time-dependent vector r ( t ) with constant length is orthogonal to its derivative d r dt . If k r ( t ) k 2 = c for some constant c , it then follows that 0 = d dt k r k 2 = d dt ( r · r ) = r · d r dt + d r dt · r = 2 r · d r dt , and so r · d r dt = 0, which means that r is orthogonal to d r dt . (e) State whether the following assertions are true or false : (1) The centripetal force does no work. TRUE. (2) The improper integral R 0 f ( x ) dx may diverge even if R 0 | f ( x ) | dx is finite. FALSE. The absolute convergence test states that R 0 f ( x ) dx must converge if R 0 | f ( x ) | dx converges. (3) The position vector r ( t ) for a trajectory is always orthogonal to the velocity vector v ( t ) = d r dt . FALSE. Consider the example r ( t ) = t ˆ i + ˆ j , for which v ( t ) = ˆ i , and thus r ( t ) · v ( t ) = t . It is true that v ( t ) is tangent to the curve traced out by r ( t ) , but this does not imply that r ( t ) is orthogonal to v ( t ) . (4) The trajectory described by the polar curve r = k 1 + e cos θ , with k > 0 , is unbounded if 0 < e < 1 . FALSE. The trajectory is an ellipse, which is a bounded curve, unlike a parabola ( e = 1 ) or a hyperbola ( e 1 ) . (5) The curvature of a cycloid can be infinite somewhere along the curve. TRUE. The curvature κ ( θ ) of the cycloid defined parametrically by x ( θ ) = θ - sin θ and y ( θ ) = 1 - cos θ is κ ( θ ) = 1 2 3 2 1 - cos θ , and clearly the curvature is infinite whenever cos θ = 1 ( θ = 0 , 2 π , 4 π ,... ) .
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Problem 2 (10 points) Consider a particle traveling a trajectory described by r ( t ) = t 2 ˆ i + t 2 ˆ j + 2 3 t 3 ˆ k, with t [ 0 , ) .
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