21B-B11.FP - y being sin y and cos y 4(20 points Find the...

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. NAME ................................ STUDENT NUMBER ................................ Math 21B section B Winter 2011 Practice Final. Please do not use calculators, books or more than one sheet (both sides) of notes. You need not simplify your answers. 1. (20 points) A sample starts with 1 mg of each of two radioactive isotopes. One isotope has a half-life which is twice as long as the half-life of the other. How much of each will be left when there is ten times as much of one as of the other? 2. (40 points) The following table gives some values of a function f ( x ): x -4 -2 0 2 4 f(x) 0 3 3 3 0 Consider the region below the curve y = f ( x ), above the x -axis and between x = - 4 and x = 4. (a) Use Simpson’s rule to estimate the area of the region. (b) Use Simpson’s rule to estimate the centroid of the region by estimat- ing the moments and using your estimate of the area from above. 3. (20 points) Find the volume of an object with height π 2 and rectangular cross sections with side lengths at height y being sin( y ) and cos( y ).
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Unformatted text preview: y being sin( y ) and cos( y ). 4. (20 points) Find the surface area of the surface obtained by rotating the parameterized curve with x ( t ) = 1 + sin( t 2 ) and y ( t ) = 1 + cos( t 2 ) for t ∈ [0 , √ 2 π ] about the y axis. 5. (20 points) It requires half as much work to pull the end of a spring from 2 inches to 3 inches extenion as it does to pull it the next inch (from 3 inches to 4 inches extension). How many inches extended is the rest position of the spring? 6. (20 poimts) Find x ( t ) if ( t + 1) dx dt = x 2 + 1 with the initial condition that x (0) = 0. 7. (30 points) Find the following numbers: (a) d dx i t = x t =-1 e-t 2 dt at x = 0 (b) i x = ∞ x =1 2 dx x 2 + x (c) i x =1 x =0 √ xe √ x dx 8. (30 points) Find the following families of functions: (a) i dx x (ln( x )) 2 if x > (b) i x 2 dx ( x 2-1) 5 2 if x > 1 (c) i r 1-cos( x ) dx...
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