MATH 32A Midterm: MATH32A Midterm 2 2010 SpringExam
Course CodeMATH 32A
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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) = √4−y2
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
6. Find the tangent plane to f(x, y) = x3y−3xy2at (2,1).
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