MATH 16B - MIDTERM 1
September. 19, 2016
Name:......................................................
Student ID:................................................
Section number or time, and GSI’s name: ..........................................
Exam Rules:
• The exam starts 3:10pm and ends 3:55pm.
• Draw a box around your answers.
• No cell phones, or calculators, or electronics of any kind.
• No “notecards,” or “cheat sheets,” or any kind of aid material.
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MATH 16B - MIDTERM 1
(1) (5 points) Find the correct answer. You don’t need to justify your answers, just circle either the
A, the B, the C or the D.
(a) To find
A:
B:
C:
D:
the level curve of a function f (x, y) going through a point (a, b) you must:
Graph of the function f (a, b).
Graph the set of points (x, y, z) such that z = f (x, y).
Find c = f (a, b), and graph the set of points (x, y) such that f (x, y) = c.
Find the gradient of f (x, y) at the point (a, b).
(b) Suppose you have a function f (x, y) that gives you the volume of a certain container depending
on two parameters x and y. You want to find the parameters x, y that maximize this volume.
What technique do you use?
A : First derivative test.
B : Lagrange Multipliers.
C : Double integral.
D : Level curves.
(c) Which of the following corresponds to (9/4)⇡ when written in degrees:
A : 14 ⇡.
B : 60 .
C : 45 .
D : 90 .
⇡
(d) What is the value of sin( 3⇡
2 ) · cos(0) + sin( 2 )?
A : -1
B : 0.
C : 1.
D : ⇡.
(e) Find which of the following is incorrect:
A : sin(t) · csc(t) = 1
B : tan( t) = tan(t)
C : csc(t + 2⇡) = csc(t)
D : tan(t) · csc(t) = sec(t).
MATH 16B - MIDTERM 1
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Name:......................................................
(2) (5 points) Let f (x, y) = sin(x · sin y).
Calculate
@f
@x =
Calculate
@2f
=
@x2
Calculate
@2f
@y@x =
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MATH 16B - MIDTERM 1
(3) (5 points) Calculate
Show your work.
Z
0
1Z 1
0
ex + ey dydx
MATH 16B - MIDTERM 1
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Name:......................................................
(4) (5 points) Use the method of Lagrange multipliers to solve the following problem: Four hundred
and eighty dollars are available to fence in a garden. The fencing for the north and south sides of
the garden cost $10 per foot and the fencing for the east and west sides cost $15 per foot. Find
the dimensions of the largest posible garden.
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MATH 16B - MIDTERM 1
(5) (5 points)
Consider the function f (x, y) = sin x·sin y. Suppose that you’re told that both partial derivatives
@f
@f ⇡ ⇡
become 0 at (0, 0) and at ( ⇡2 , ⇡2 ). That is, that @f
@x (0, 0) = 0, @y (0, 0) = 0, @x ( 2 , 2 ) = 0 and
@f ⇡ ⇡
@y ( 2 , 2 )
= 0.
Here are the partial derivatives of f :
@f
= cos x · sin y
@x
Here are its second partial derivatives:
@2f
=
@x2
sin x · sin y,
@2f
=
@y 2
and
@f
= sin x · cos y.
@y
sin x · sin y
and
@2f
= cos x · cos y
@y@x
(a) Decide whether the point (0, 0) is a relative minimum, relative maximum, or saddle point.
(b) Do the same for ( ⇡2 , ⇡2 ). Show your work.
MATH 16B - MIDTERM 1
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Name:......................................................
(6) (5 points)
Suppose you are given three data points (0, 6), (3, 0), and (6, 0), and you want to find line of
the form y = Ax + B that best fits them in terms of minimizing the least-squares error. If you
had to choose from one of the following two options, which one is better? Justify your answer.
(a) A = 1 and B = 5.
(b) A = 0 and B = 2.
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MATH 16B - MIDTERM 1
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