Department

MathematicsCourse Code

MATH 32AProfessor

AllStudy Guide

MidtermThis

**preview**shows half of the first page. to view the full**1 pages of the document.**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

−π/4≤t≤π/4.

2. Find the cumulative length function s(t) (starting from a= 1) of the parametric

curve hln t, √2t, 1

2t2i.

3. Sketch in the xy-plane the domain of f(x, y) = √4−y2

ln(y−x2).

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) = x3y−3xy2at (2,1).

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