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MATH 32A Midterm: MATH32A Midterm 2 2010 SpringExam

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MATH 32A (Butler)
Practice for Midterm II
Try to answer the following questions without the use of book, notes or calculator.
Time yourself and try to finish the test in less than 50 minutes.
1. A particle moves through three dimensional space with velocity
v(t) = hsec2t, 2 sec ttan t, tan2ti.
At time t= 0 the particle is at h0,1,2i, find the position function of the particle for
2. Find the cumulative length function s(t) (starting from a= 1) of the parametric
curve hln t, 2t, 1
3. Sketch in the xy-plane the domain of f(x, y) = 4y2
4. Given the implicitly defined surface z+ sin z=xy find 2z/∂x∂y only in terms
of z. (Hint: find both z/∂x and z/∂y then take the derivative of one of these with
respect to the appropriate variable to find 2z/∂x∂y, at the end a substitution from
the original relationship defining the surface then gives the desired form.)
5. Show that u(x, t) = sin (x+ sin t) is a solution to the partial differential equation
utuxx =uxutx.
6. Find the tangent plane to f(x, y) = x3y3xy2at (2,1).
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