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MCV4U Practice Final Exam
Calculus and Vector (Carleton University)
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Grade 12 Calculus & Vectors
FINAL EXAM
Name _________________________
Ms. Garcia
Scheduled date:
Scheduled time: 3 hours max
INSTRUCTIONS:
1.
2.
3.
4.
5.
6.
A Non-graphing non-programmable calculator may be used.
Answer all of the questions directly on the exam paper in the space provided.
The entire exam package must be handed in.
Ensure that you have a complete exam package of 13 pages in total.
Be sure to put your name on this page and any scrap paper you are submitting
The mark breakdown is below
Part A: Calculus
50 marks
Part B: Vectors
50 marks
Total
100 marks
Useful formulas:
a  b = (a2b3 − a3b2 , a3b1 − a1b3 , a1b2 − a2b1 )
n1 • n2  n3 = 0
SinA SinB
=
a
b
lim
x→0
d Pt − plane =
n • PQ
d l1−l 2 =
n
c 2 = a 2 + b 2 − 2abCosC
f (x + h ) − f (x )
h
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P1 P2 • n
n
lOMoARcPSD|20052073
PART 1: CALCULUS
1. Use the first principles of derivatives to find the derivative of
.
[5]
2. Evaluate the following limits:
a.
[1]
b.
[2]
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[3]
c.
3. Differentiate the following. Simplify your answers.
a.
b. f (x) =
(x 2 +1)3
(
)
x +1
[3]
[3]
2
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c. y = ecos x sin x
[3]
d.
[3]
e.
[4]
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4. For the function whose graph is given below, find the following:
a. lim− f ( x) =
x →2
b. lim+ f (x) =
x®2
c. lim f (x) =
x®2
d. lim− f ( x) =
x →−1
e. lim+ f ( x) =
x →−1
f. lim f ( x) =
x →−1
g. f (2) =
h. f (−1) =
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5. A cylindrical Pepsi can hold 350 cm 3 of pop. How should it be constructed so that minimum
amount of material is used?
[5]
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6. For the function.
f ( x) =
2 + x − x2
(x − 1)
2
, f ' ( x) =
x−5
(x − 1)3
,
a. Find domain.
[1]
b. Find vertical and horizontal asymptotes.
Examine vertical asymptote on either side of discontinuity
[3]
c. Find all intercepts.
[1]
d. Find critical points.
[1]
e. Find any local extrema.
[2]
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f. Find points inflection.
[3]
g. Sketch.
[3]
Label:
•
Intercepts
•
Asymptotes
•
Critical Point(s)
•
Point of Inflection(s)
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PART B: VECTORS
7. If , , and
are the standard unit basic vectors in 3-space, determine the value of.
[3]
8. The speed of a plane is 420 km/hr and its heading is 140°. A wind of 40km/hr is blowing on a
bearing of 040°. Determine the plane’s resultant velocity relative to the ground.
[6]
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are coplanar.
9. Determine if the vectors
→
[6]
→
10. Given a = 3,4,3 and b = 1,−2,5
a. Calculate the angle between a and b .
→
[3]
→
b. If c = [1,−3, t ] , find t so that a and c are perpendicular.
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is applied over a distance of
11. A force,
amount of work, measured in Joules, done in this situation.
12.
. Determine the
[3]
Find the magnitude of the torque vector, measured in Newton-metres, produced by a
cyclist exerting a force of F = [70,30,160]N on the shaft- pedal
long.
[4]
13. Calculate the area of the parallelogram whose adjacent sides are
→
→
a = 1,1,−2 and b = 3,3,−1 .
[4]
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14. The vertices of a quadrilateral ABCD are A 0,3,5 , B − 3,−1,17, C 4,−3,15 and D [ 7,1, 3]
Prove that ABCD is a parallelogram.
[5]
15. Determine the following:
a. Determine the vector equation of the plane that contains the following two lines
→
L1:
→
r = ( 4,−3,5) + t ( 2,0,3) , tR and L2: r = (4,−3,5) + s (5,1,−1) , sR
b. Determine the corresponding Cartesian equation.
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[3]
[5]
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16. Determine the value of k for which the direction vectors of the lines
L1:
y−2
x+3
z
x −1
z +1
=
=
and L2:
=
, y = −1 are perpendicular.
k
2
k −1
−2
1
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[5]