Name: __________________________________
Date: _______________________
Calculus and Vectors (MCV4U) – Chapter 2 SAMPLE TEST
K:
/14 I:
Communication
Rubric
Communication –
use of mathematical
language, symbols,
visuals and
conventions
/5
C:
/3
A:
/10
Total:
/32
Exemplary
Proficient
Developing
Infrequent
Always uses mathematical
language, symbols, visuals, and
conventions correctly and
efficiently. (Zero (0) errors)
Uses mathematical
language, symbols,
visuals, and conventions
correctly most of the
time. (One (1) error)
Uses mathematical
language, symbols, visuals,
and conventions correctly
some of the time. (Two (2)
errors)
Uses mathematical
language, symbols, visuals,
and conventions correctly
occasionally (more than
two errors).
2
1
0.5
0
1. Differentiate the following functions. Do not leave negative or fraction exponents in your final answers.
a)
(2K)
b)
c) Use the product rule to differentiate
possible.
(2K)
. Simplify your answer as much as
(3K)
1
Name: __________________________________
Date: _______________________
2. Identify which curve or line in the graph represents each function:
Justify your choice.
3. The position of a moving particle is given by
in seconds.
.
(3I)
, where s is measured in metres and t is
a) Determine the velocity of the particle after 4 seconds. Round your answer to the nearest hundredth.
(2A)
b) When is the particle at rest? Round your answer to the nearest hundredth.
2
(2I)
Name: __________________________________
Date: _______________________
4. Use Leibniz notation to determine the derivative of the following function when
answer to the nearest hundredth.
,
. Round your
(2K)
(3A)
5. Use the quotient rule to differentiate
. Simplify your answer as much as possible, similar to
what was shown in class.
(5K)
3
Name: __________________________________
Date: _______________________
6. The power, P, in watts, produced by a certain engine is given by
, where R, is the
resistance, in ohms ().
a) Determine the rate of change of power. Simplify as much as you can, similar to what was shown in
class.
(2A)
b) Determine the rate of change of power when the resistance is 5 . Round your answer to the nearest
hundredth.
(1A)
c) Determine the value of R for which the rate of change of the power is 0. Explain the meaning of this
value for this situation. Round your answer to the nearest tenth.
(2A)
(1C)
4
Read and follow all the instructions and questions carefully.
Part 1: Wonderland is accepting proposals for a new roller coaster idea! In this section, you
will go through some design ideas and evaluate them.
a) First, you need to draw a sketch of your roller coaster.
There are a few requirements: to keep the ride exciting, your rollercoaster must have a
minimum of three local extrema and two points of inflections.
(2I)
Complete your sketch on the graph below. Be sure to label each of your extrema, your
POIs and to classify the extrema. (Note: Your graph may cross over the x-axis)
(2C)
Page 2
b) Your friend comes up with the following function, 𝑦 =
#&'
(# )
, for her roller coaster
prototype, and does not understand why it does not meet the requirements shown in
part (a). Show that both the IP and the CP criteria are not fulfilled. If necessary, round
your answers to the nearest hundredth.
(6A)
(1C)
Page 3
c) One of the submissions for the roller coaster proposal contains the following
information:
- There are two local extrema at 𝑥 ≅ −1.721 and 𝑥 ≅ 0.387
- There is one possible point of inflection at 𝑥 ≅ −0.667
- The x-intercept is at 𝑥 ≅ −2.831
- The y-intercept is at 𝑦 = 3
- The function representing the roller coaster is 𝑦 = 3𝑥 6 + 6𝑥 ' − 6𝑥 + 3, where the
derivatives are as follows: 𝑦 8 = 9𝑥 ' + 12𝑥 − 6 and 𝑦 88 = 18𝑥 + 12
Complete the first and second derivative tests in the chart below to get a better idea of
what the roller coaster would look like.
(6K)
(2A)
Use the table below to perform a COMBINED 1st AND 2nd Derivative Test.
(OR, do the two derivatives in separate tables if you prefer.)
Page 4
d) Draw a sketch of the roller coaster on the graph below (from the previous chart). Be
sure to graph all necessary points, label your CPs and POIs and classify all of your CPs on
the graph. Choose an appropriate scale for the graph.
Note: You will need to calculate for the necessary y-values.
(4I)
(2C)
Page 5
MCV4U – 4.3 Practice
Date: ___________________________________
4.3 – Practice
1. Find each derivative with respect to x.
b) y = (sin x)
a) f ( x) = sin(3x + 5)
−1
c) y = 6 x cos x − 3x
2
d) f ( x) = sin(πx )
3
2. Determine each derivative with respect to t.
 2π 
t
 7 
a) x = −3 sin
2
2
c) x = t cos (2t − t )
b) y = sin(sint )
d) y =
4t 2 − 5t + 6
cos(2t )
3. Determine the derivative of each function with respect to the independent variable.
a) y = 5(sin θ + cosθ )
2
c) t =
cos(x 2 )
sin x
b) x = 3θ sin(θ )
4
2
d) f (θ ) = sin θ cos(2θ ) + cosθ sin(2θ )
4. Determine the slope of the line tangent to y =
1
π
sin x cos x at x = .
2
4
5. Calculate the slope of each function at the indicated value of x.
a) y = 3 sin2 x + x , when x =
π
3
b) y = 2 sin2 x cos x , when x = π
6. Determine the equation of the line tangent to the graph of each function.
π
1
π
b) y =
, when x =
a) y = −2 x cos(2 x) , at x =
2
4
sin x
4
1 3
π
7. Find the equation of the line tangent to y = − sin (2 x) at x = .
2
6
8. a)Determine the derivative of y = cos2 x − sin2 x
b) Use a double angle formula to re-write y = cos2 x − sin2 x . Find the derivative of the new
function.
c) Which derivative was easier to determine? Why?
1
Name: ____________________________________
Date: _______________________
Calculus and Vectors (MCV4U) – Chapter 5 Practice Test
K:
/13
I:
/8
C:
/2
A:
/12 Total:
/35
Communication
Always uses mathematical
Uses mathematical
Uses mathematical
Uses mathematical
language, symbols, visuals,
language, symbols, visuals, language, symbols, visuals, language, symbols, visuals,
use of mathematical
conventions, rounding and units conventions, rounding and conventions, rounding and conventions, rounding and
language, symbols, visuals,
correctly and efficiently.
units correctly most of the units correctly some of the units correctly occasionally.
conventions, rounding and
(Zero-one errors.)
time. (two-three errors)
time. (four-five errors.)
(More than five errors.)
units
2
1.5
1
0.5
1. Solve for x. Round answers to three decimal places, where needed.
x
a) - 5 = ln x
(2 K)
b) 12e = 48
(2 K)
2. Differentiate each of the following with respect to x. Simplify your answers as much as was
demonstrated in class. Express answers fully factored using positive exponents only.
+
a) 𝑦 = 𝑥 $ − 5'
(1 K)
b) 𝑦 = −2𝑒 *,'
(2K)
1
Name: ____________________________________
Date: _______________________
c) 𝑦 = (𝑥 . + 2𝑥)𝑒 *$'
d) 𝑓 𝑥 =
(3 K)
2 34
(3 K)
5' 6
2
Name: ____________________________________
Date: _______________________
3. Find the intervals of concavity for a function, given that the second derivative is
𝑓 7 ′ 𝑥 = 2𝑥𝑒 *9' − 𝑥 9 𝑒 *9' . Round any final answers to 2 decimal places, where necessary.
*You may or may not need to use all columns in the chart.
(6A)
Interval
Test Value
Intervals of concave up: ___________________________________
Intervals of concave down: ___________________________________
3
Name: ____________________________________
Date: _______________________
4. A virus is spreading through Crestwood! Anyone infected by this virus suddenly becomes an
expert in calculus J The number of people infected after t days is given by:
t
3
N (t ) = 5(2) .
For each part, write conclusions in words, quoting proper units. All final answers should be
rounded to the nearest whole number.
a) How many people were infected initially?
(1 A)
b) If the spread of virus continues, how long will it take for the entire school population
(approximately 400 people) to become experts in calculus (i.e. infected)?
(3 A)
c) What is the rate of the virus infection after 7 days?
4
(2 A)
Name: ____________________________________
Date: _______________________
t
3
d) Rewrite the equation N (t ) = 5(2) in base e . Be exact and leave your answer as a
simplified fraction (no decimals) if needed.
5. True or False: Write True or False in the blank beside each statement.
a) 𝑒 ' is increasing only when 𝑥 < 0.
b) The domain of ln 𝑥 is 𝑥 > 0.
c) If 𝑥 < 0, then 0 < 𝑒 ' < 1.
d) The range of 𝑒 ' is {𝑦 ∈ ℝ}.
e) For f x = ln 𝑥 , 𝑓 𝑥 $ = 3𝑓 𝑥
5
(3 I)
(5 I)