This

**preview**shows half of the first page. to view the full**3 pages of the document.**Math 21a Old Exam One – Spring 2007 Spring, 2009

Part I: Multiple choice. Each problem has a unique correct answer. You do not need to justify your

answers in this part of the exam.

1The vectors A=−2i+ (t−1)j+ 2kand B=j+tkare parallel when:

(a) t= 0;

(b) t=1

3;

(c) t= 1 and t= 2;

(d) all values of t;

(e) no values of t.

2If Aand Bare vectors in space, then the expression (A+B)·(A×B) equals

(a) 0(the zero vector) because A+Band A×Bare always parallel;

(b) 0 (the number zero) because Aand Bare always perpendicular to A×B;

(c) a positive number since (A+B)·(A×B) is the volume of a parallelepiped;

(d) undeﬁned since we cannot take the dot product of a vector and a scalar;

(e) an arbitrary number, depending on the particular vectors Aand B.

3The curve r(t) = he√2tcos t, e√2tsin ti, 0 ≤t≤π, has length

(a) 1

√3e√2π−1;

(b) 1

√2e√2π−1;

(c) q2

3e√2π−1;

(d) q3

2e√2π−1;

(e) given by a deﬁnite integral which cannot be evaluated explicitly;

(f) given by an indeﬁnite integral.

The questions in problems 4-5 refer to the marked points in the diagram below, which shows the level

curves of a function f(x, y).

4At which of the labeled points is the partial derivative ∂f

∂x strictly positive?

(a) at Sand T(b) at Sand W

(c) at Tand W(d) at Vand W

(e) at Vand Y(f) only at V

(g) at Wand Y(h) only at T

5At the point Y, what can one say about the values of ∂f

∂y and ∂2f

∂y2?

(a) ∂f

∂y >0 and ∂2f

∂y2>0 (b) ∂f

∂y >0 and ∂2f

∂y2<0

(c) ∂f

∂y <0 and ∂2f

∂y2>0 (d) ∂f

∂y <0 and ∂2f

∂y2<0

(e) ∂f

∂y = 0 and ∂2f

∂y2>0 (f) ∂f

∂y = 0 and ∂2f

∂y2<0

(g) ∂f

∂y >0 and ∂2f

∂y2= 0 (h) ∂f

∂y <0 and ∂2f

∂y2= 0

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