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MATH 1260 - Test # 2 Sample Questions

Formula Sheet

The dot product of two plane vectors u = u1, u2 and v = v1, v2 isu v = u1v1 + u2v2.

The dot product of two space vectors u = u1, u2, u3 and v = v1, v2, v3 isu v = u1v1 + u2v2 + u3v3.

The cross product of two vectors u = u1, u2, u3 and v = v1, v2, v3 isu v = u2v3 u3v2, u3v1 u1v3, u1v2 u2v1.

The projection of a vector u onto a vector v isu v||v ||2

v .

The determinant of a 2 by 2 matrix is

det

a bc d

= ad bc.

The determinant of a 3 by 3 matrix is

det

a

bc

= a (b c ).

The gradient of a function f(x, y) isf = fx, fy.

The gradient of a function f(x,y,z) isf = fx, fy, fz.

The tangent plane of a function f(x, y) at a point P = (a, b) isz = fx(P) (x a) + fy(P) (y b) + f(P).

Chain RuleD(g f)P = Dgf(P) DfP.

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(1) Calculate the following limits or show that they do not exist:

(a) lim(x,y)(0,0)

2xy2

x2 + y2 .

(b) lim(x,y)(0,0)

x2y

x3 + y4.

(c) lim(x,y)(0,0)

x + 2y2

3x2 + 2y2.

(d) lim(x,y)(0,0)

2x3y2

x6 + y4.

(e) lim(x,y)(0,0)

x2y

3x2 + 2y2.

(f) lim(x,y)(0,0)

x2y

x4 + 3y2.

(g) lim(x,y)(0,0)

3x3 + 2

y

x2 + y2.

(h) lim(x,y)(0,0)

3x3 y23x3 + 2y2

.

(i) lim(x,y)(0,0)

x2y5

2x4 + 3y10.

(j) lim(x,y)(0,0)

x2y2

2x2 + 3y4.

(2) Find a function f: R3

R

2 such that f(v ) =

0 for every vectorv in the plane of equation

2x 3y + z = 0.(3) Let f: R2 R be the function f(x, y) = 3x 2y. Show that

(a) for every number c, the set of vectors in R2 mapping to c is a line;(b) all the lines in part (a) are parallel.

(4) Let f: R2 R2 be defined by f(x, y) = (2x y, x + 2y). Find all vectors v such thatf(

v ) = c

v for some number c.

(5) Let A be a 2 by 2 matrix. Show that, if there exists a non-zero vectora such that

a A = 0 ,

then det(A) = 0.

(6) Find all the numbers a and all non-zero vectorsv such that

v

1 10 2

= a

v .

(7) Let f(x, y) = 2x2y3 3xey + x ln(x) sin(xy).(a) Find the partial derivatives fx and fy.(b) Find the gradient of f at the point (1, 0).

(8) Find gx, gy, and gz for the function g(x,y,z) = (x + z2)cos(y) sin(2y).

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(9) Show that, if f: R2 R is a function and P is a point, then the gradient f(P) of f at P isequal to

Df1,0(P), Df0,1(P)

. (Remember that Df

u(P) denotes the directional derivative of

f at the point P in the direction u)

(10) Consider the graph of the function f(x, y) = 2x2 ln(xy) + 3

y. Find the tangent plane atthe point (1, 1).

(11) Consider the functions

f: R3 R2 f(x1, x2, x3) =

x21x3, 3x1 +x2x3

g : R2

R

2 g(y1, y2) =

y1, 3y21

2y1e

y1y2

.Find the derivative of g f at the point (1,3, 1).

(12) Let f(x,y,z) =

x2 + y2 + z2,

9x

x + y + z

and g(x, y) = (y x, 2xy2). Find the derivative

of the composition g f at the point (1, 1, 1).

(13) Let z = f(x, y) be a function of two variables and let x = (u2 v2)/2 and y = uv bechange-of-variable equations. Calculate zu and zv in terms of fx, fy, u, and v.

(14) Let f(x, y) = y + 3x2. Verify by direct calculation that the gradient vector f(1,1) isperpendicular to the level curve f(x, y) = 2 at (1,

1). Sketch the level curve near (1,

1) and

the gradient vector at (1,1).

(15) (a) Find the stationary points of the function f(x, y) = xy and classify them.(b) Find the global maximum and minimum of f(x, y) on the region x2 + y2 8.

(16) (a) Find the stationary points of the function f(x, y) = 2x2 + x + y2 2 and classify them.(b) Find the global maximum and minimum of f(x, y) in the region x2 + y2 4.

(17) Let f(x, y) = xy2 5x2y. Find the maximum and minimum value of f on the region abovethe graph of y = x2 and below the line y = 4.

(18) Your boss wants you to build a rectangular box. He gives you $9 to buy the materials tobuild the box, and he wants the bottom to be stronger. The material for the bottom costs $2/ft,and the material for the sides and the top costs $1/ft. What are the dimensions of the box withthe largest volume that you can build with the $9 your boss gave you?

(19) A shipping box with two dividers is to be constructed from 48 m2 of cardboard. What arethe dimensions of the box with largest volume that can be constructed?

(20) Find the maximum volume of a rectangular box with no top that can be built with 1200square feet of material.

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