MAT133Y1: Calculus and Linear Algebra for Commerce 2023-24
Term Test 3: Practice Test
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Instructions
• Keep this booklet closed, until an invigilator announces that the test has begun. However, you may fill out
your information in the box above at any time.
• Please place your student ID card on your desk, so that is easy for an invigilator to check without disturbing
you during the test.
• Turn your cell phone completely off (not just in vibrate mode) and store it with your other belongings
under your desk or on the back of your chair.
• One approved scientific calculator may be used. No other aids are permitted.
• For all multiple choice questions, fill in your answer using dark pencil or ink. (unfilled
answers will not receive credit.
filled
). Ambiguous
• For written response questions, please write clearly and concisely in a linear fashion. Explain your steps. Use
full sentences. Correct final answers must be well-justified in order to receive credit.
• Please write your answers into the boxes. Ample space is provided within each box, however, if you must use
additional space, please use the blank page at the end of this booklet, and clearly indicate in the given box that
your answer is continued on the last page.
• This test booklet contains 9 pages and 10 questions. If your booklet is missing a page, please raise your hand
to notify an invigilator as soon as possible.
• Do not remove any page from this booklet.
• Scrap paper will be distributed separately. Please raise your hand to ask an invigilator if you need more scrap
paper. Anything you write on the scrap paper will not be marked.
• The formulas below are provided in case you need them. You don’t have to use them if you don’t need them.
The sum of a finite geometric series is given by A normal distribution has a density function of the form
n−1
X
aRi =
i=0
a(1 − Rn )
,
1−R
p(x) =
2
2
1
√ e−(x−µ) /(2σ )
σ 2π
where µ is the mean of the distribution and σ is the standard deviation, with σ > 0.
where a and R are real numbers, with R 6= 1.
The discriminant for a function of two variables, f (x, y), is D(x, y) = fxx (x, y)fyy (x, y) − fxy (x, y)2 .
Page:
2
3
4
5
6
7
8
9
Total
Points:
12
12
10
7
9
10
8
10
78
Score:
1. (2 points) There are two competing brands of cat treats at a local store: Cat Snacks and Kitty Bits.
Consider the weekly quantity q1 of Cat Snacks sold as a function of the price p1 of Cat Snacks and the
price p2 of Kitty Bits. What do you expect of the signs of the partial derivatives of q1 with respect to
p2 ? Why? Please select only the answer that is both correct and well-justified (if any).
∂q1
∂p2
> 0; if the price of Kitty Bits increases, the demand for Cat Snacks will likely increase.
∂q1
∂p2
> 0; if the price of Kitty Bits increases, the demand for Cat Snacks will likely decrease.
∂q1
∂p2
< 0; if the price of Kitty Bits increases, the demand for Cat Snacks will likely increase.
∂q1
∂p2
< 0; if the price of Kitty Bits increases, the demand for Cat Snacks will likely decrease.
None of the above.
2. (4 points) The concentration C of a drug in a patient’s bloodstream is given by C = f (x, t), where x is
the amount (in mg) of drug injected and t is the time (in hours) since the injection. The contour map
for f is shown below. (Contours are included for each C = 0.1, 0.2, . . . . There are no concentrations
larger than 0.6.) Sketch the graphs of the two cross-sections indicated below. (Label your axes!)
(a) The graph of C = f (1, t)
(b) The graph of C = f (x, 2)
3. (6 points) Consider the following contour diagram of a linear function.
y
8
(a) Determine the values of f (1, 0), f (1, 3) and f (1, 2).
7
f (1, 0) =
, f (1, 3) =
, and f (1, 2) =
.
6
(b) Find the gradient of f (x, y). Then add ∇f (3, 3) to the figure,
placing its tail at (3, 3).
5
4
5
3
The gradient is ∇f (x, y) =
4
3
2
.
(c) Find a unit vector ~u in the direction of ∇f (3, 3).
2
1
~u =
1
.
x
1
2
3
4
5
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4. (12 points) The sensation of cold caused by the combined effect of temperature and wind is called wind
chill. The wind chill index W (in degrees Celsius) is a function of the temperature T (in degrees Celsius)
and the wind speed s (in kilometres per hour):
s→
T ↓
−5
−10
−15
−20
10
−9
−15
−21
−27
20
−12
−18
−24
−30
30
−13
−20
−27
−33
40
−14
−21
−28
−34
e.g., W (−10, 10) = −15 means that, although the
temperature is −10◦ C, if the wind is blowing at a
speed of 10 km/h, the cold sensation on your skin will
make it feel as though it is −15◦ C.
Using the above table: estimate each of the following quantities, show your reasoning, then, if prompted,
interpret its meaning in the form of a sentence. Your audience for the interpretation is the general public.
(a)
∂W
∂s
(T,s)=(−15,20)
≈
value
units
Estimate the value. Show your reasoning. Then write your answer (with units) in the box above.
Interpret the value for the public:
(b)
∂W
∂T
(T,s)=(−15,20)
≈
value
units
Estimate the value. Show your reasoning. Then write your answer (with units) in the box above.
Interpret the value for the public:
(c) If it’s −16◦ C and the wind speed is 22 km/h the wind chill W is ≈
value
Estimate the value. Show your reasoning. Then write your answer (with units) in the box above.
Page 3
units
5. Akiyo goes for a hike in a park, on a terrain modelled by the equation,
1
z = g(x, y) = 2 − x2 + xy − y 2
4
where x and y are distances east and north of the park entrance respectively and z is the elevation above
sea level; all are measured in kilometres.
(a) (3 points) Akiyo is standing at the point with (x, y) = (2, 3). In which direction would she step
in order to move upwards as quickly as possible? Please state your answer in the form of a vector
pointing in this direction; it doesn’t necessarily have to be a unit vector.
She should step in the direction of the vector
.
(b) (2 points) If Akiyo starts at the point (x, y) = (2, 3) and steps in the direction of the vector found
in (a), what slope will she experience?
She will experience a slope of
.
(c) (3 points) Akiyo is standing at the point with (x, y) = (2, 3). If she steps in the direction of the
vector 4~i + 3~j, what slope will she experience?
She will experience a slope of
.
(d) (2 points) Akiyo’s cousin doesn’t like steep paths. When they are at the point (x, y) = (2, 3), they
want to hike up a path with slope 1. Starting at the point (2, 3), in how many different directions
can Akiyo’s cousin step to experience a path of slope 1? Write 1-2 sentences explaining your answer.
Akiyo’s cousin can step in
One
Two
Page 4
Three
Other different directions.
6. (7 points) The contour map for a function z = f (x, y) is shown below.
(a) At which labelled point(s) does f have a local minimum?
2A 2B 2C 2D 2E 2F
(b) At which labelled point(s) does f have a saddle point?
2A 2B 2C 2D 2E 2F
(c) ∇f (J) points in which of the following directions?
~i
−~i
~j
−~j
2
2
∇f (J) = ~0
G
G
2
2
H
H
2
2
J
J
Other:
(d) ∇f (C) points in which of the following directions?
~i
−~i
~j
(e) Approximate ∇f (B). ∇f (B) ≈
−~j
∇f (C) = ~0
~i+
Other:
~j.
(f) The equation of the plane tangent to z = f (x, y) at the point E is z = ax + by + c. Determine the
signs of a and b.
a > 0, b > 0
a > 0, b < 0
a < 0, b > 0
a < 0, b < 0
Other
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7. (9 points) Consider the function f (x, y) = xy(x − 2)
(a) Find all critical points of f (x, y). To save you time: fx = 2y(x − 1), and fy = x(x − 2). Please
show your work and explain your logic carefully.
Critical points (list in order of increasing x component):
(b) Choose one (just one, your choice!) of the critical points you found and classify it (local max, local
min, saddle point). Please show your work.
The critical point
.
Page 6
is a
8. (10 points) Timo’s Tires has two factories making tires. Factory A produces x(x+4) grams of pollutant
when making x tires (in thousands). Factory B produces y(y + 2) grams of pollutant when making y
tires (in thousands).
(a) Timo wants to produce exactly 3025 tires. How many tires should Timo produce at each of his two
factories if he wants to minimize the total amount of pollutant, f (x, y) = x(x + 4) + y(y + 2)?
Please show and explain your work.
Timo should produce
tires at Factory A and
total pollutant produced will be
tires at Factory B. In doing this, the
g
(b) It turns out that Timo only needs to produce 2900 tires, rather than 3025 tires. Approximately
how much less (total) pollutant will be made?
Overall, approximately
g
less pollutant will be produced.
(c) Terrible Timo is Timo’s evil twin. He wants to maximize the amount of pollution when making
3025 tires. How many tires should Terrible Timo produce at each of his two factories?
Terrible Timo should produce
tires at Factory A and
this, the total pollutant produced will be
g
Page 7
tires at Factory B. In doing
162
153
144
135
126
117
108
99
90
81
72
63
54
45
N
9. (8 points)
18
16
Every month, Bob’s Bagels supplies local cafes
with bagels from their bakery in Mississauga and
their bakery in North York. Bob’s total monthly
costs C (in hundreds of dollars) of producing the
bagels is a function of the number of bagels M
(in hundreds) produced in Mississauga, and the
number of bagels N (in hundreds) produced in
North York. The contour diagram for C as a
function of M and N is shown to the right.
14
12
36
10
27
8
18
6
9
4
2
0
0
2
4
6
8
10
12
14
16
18
M
(a) Bob’s Bagels produces 800 bagels at their bakery in Mississauga and 400 bagels at their bakery in
North York. What are Bob’s total monthly costs?
$1200
$1800
$3200
$3600
$4500
$5400
(b) If Bob needs to produce a few more bagels, is it better for him to produce them at the bakery in
Mississauga or at the bakery in North York? Explain your answer.
Bob should produce the bagels at the bakery in
Mississuaga
North York
.
(c) Next month, Bob will need to produce 1800 bagels between the two bakeries to fill all of his orders.
What equation represents this constraint?
Carefully add the constraint curve to the contour map above.
(d) How many bagels should he produce at each bakery to minimize his total monthly costs? Explain
your answer.
Bob should produce
bagels in Mississauga and
(number)
bagels in North York.
(number)
Page 8
10. (10 points)
The tables in EX100 have adjustable legs (you can make them
longer and shorter). To model the table shown here, let the floor
of EX100 be the xy plane with x and y measured in meters.
The bottom of the 1.11m leg touches the floor at (0, 0), and the
bottom of the 1.15m leg touches the floor at (0, 0.5).
Assume that where the legs join the table there is enough flexibility at the joint so that all the legs of the table are perpendicular
to the floor. (The tabletop is not perpendicular to the legs.)
(a) What is the slope in the x direction of the tabletop?
value
(b) What is the slope in the y direction of the tabletop?
value
(c) What is the equation of the plane that contains the tabletop?
Write your answer in the box below. Write one or two sentences about why this is the answer.
equation of plane
(d) The fourth leg is 1.1m long and is not touching the floor. How much should it be extended so that
it just touches the floor, stopping the table from wobbling?
Write your answer in the box below. Write one or two sentences about why this is the answer.
Extend the leg by
m.
value
Page 9