MCV4U Grade 12 Calculus and Vectors FINAL
EXAM (With Solutions)
Instructions
The test has 5 parts. An approximate time is given for each part. Look over the test before you
begin, and leave some time to review your work at the end.
Part A: Multiple Choice (30 marks)
1. Evaluate the following limit:
lim𝑥→ −1 𝑥2+ 9𝑥−4
𝑥3
a) 12 b) – 6 c) – 12 d) 6
Solution
2 2
𝑥 + 9𝑥−4 (−1) + 9(−1)− 4 1−9−4
lim𝑥→ = (−1)3
= −1
= 12.
−1 𝑥3
Answer: a)
2. What is the limit of f (x) = 𝑥2− 2𝑥 − 8
0 as x approaches – 2?
𝑥+2
a) 0 b) c) – 6 d) –16
0
Solution
lim 𝑥2− 2𝑥 − 8 = lim (𝑥 + 2)(𝑥 − 4)
= lim (x – 4) = – 2 – 4 = – 6.
𝑥→ 𝑥+2 𝑥→ −2 𝑥+2 𝑥→ −2
−2
Answer: c)
3. What is the average rate of change between x = 1 and x = 3 for the function g(x) = – 3(x – 1)2
+ 11?
a) – 6 b) 5 c) – 2.5 d) 0.4
Solution
g(1) = – 3(1 – 1)2 + 11 = 11,
g(3) = – 3(3 – 1)2 + 11 = –1,
rav [1, 3] 𝑔(3) − 𝑔(1) = −1−11 = – 6.
= 3−1 2
Answer: a)
,4. For the function g(x) = 8x2 – x + 4, at what tangent point is the instantaneous rate of change
equals –1?
a) (1, 11) b) (–1, 13) c) (0, 4) d) (2, 34)
Solution
, g’ (x) = 16x – 1
g’ (x) = –1 ⟹16x – 1= –1, 16x = 0, x = 0 ∴ (0, 4).
Answer: c)
5. Which of the following is the derivative of h(x) = 40x?
a) h’ (x) = 40xx – 1
b) h’ (x) = 40x (ln 40)
c) h’ (x) = 40x (ln 1)
d) h’ (x) = 40x (ln
x)
Answer: b)
Which of the following is the derivative of f(x) = 72𝑥2+ 𝑥?
f’ (x) = (72𝑥2+ 𝑥) (4x + 1)
f’ (x) = (2x2 + x) 72𝑥2+ 𝑥 −1
f’ (x) = 72𝑥2+ 𝑥 (ln 7)
f’ (x) = (ln 7) (72𝑥2+ 𝑥) (4x + 1)
Answer: d)
Which of the following is the derivative of h(x) = 4x3 + e16x – 32x – 1?
h’ (x) = 12x2 + 16e16x – 2(ln 3)32x – 1
h’ (x) = 12x2 + 16e16x – (ln 3)62x – 1
h’ (x) = 12x + 16ex – 62x – 1
h’ (x) = 12x + 16ex – (2)32x – 1
Answer: a)
2𝑥2
8. Which of the following is the derivative of f(x) = cos 𝑥 ?
4𝑥
a) cos 𝑥 +=
f’ (x) 2𝑥2 sin 𝑥
cos2𝑥
b) f’ (x) 4𝑥 sin 𝑥 − 2𝑥2 cos 𝑥
= cos2𝑥
4𝑥
c) f’ (x) = − 2𝑥2
cos 𝑥
cos2𝑥
EXAM (With Solutions)
Instructions
The test has 5 parts. An approximate time is given for each part. Look over the test before you
begin, and leave some time to review your work at the end.
Part A: Multiple Choice (30 marks)
1. Evaluate the following limit:
lim𝑥→ −1 𝑥2+ 9𝑥−4
𝑥3
a) 12 b) – 6 c) – 12 d) 6
Solution
2 2
𝑥 + 9𝑥−4 (−1) + 9(−1)− 4 1−9−4
lim𝑥→ = (−1)3
= −1
= 12.
−1 𝑥3
Answer: a)
2. What is the limit of f (x) = 𝑥2− 2𝑥 − 8
0 as x approaches – 2?
𝑥+2
a) 0 b) c) – 6 d) –16
0
Solution
lim 𝑥2− 2𝑥 − 8 = lim (𝑥 + 2)(𝑥 − 4)
= lim (x – 4) = – 2 – 4 = – 6.
𝑥→ 𝑥+2 𝑥→ −2 𝑥+2 𝑥→ −2
−2
Answer: c)
3. What is the average rate of change between x = 1 and x = 3 for the function g(x) = – 3(x – 1)2
+ 11?
a) – 6 b) 5 c) – 2.5 d) 0.4
Solution
g(1) = – 3(1 – 1)2 + 11 = 11,
g(3) = – 3(3 – 1)2 + 11 = –1,
rav [1, 3] 𝑔(3) − 𝑔(1) = −1−11 = – 6.
= 3−1 2
Answer: a)
,4. For the function g(x) = 8x2 – x + 4, at what tangent point is the instantaneous rate of change
equals –1?
a) (1, 11) b) (–1, 13) c) (0, 4) d) (2, 34)
Solution
, g’ (x) = 16x – 1
g’ (x) = –1 ⟹16x – 1= –1, 16x = 0, x = 0 ∴ (0, 4).
Answer: c)
5. Which of the following is the derivative of h(x) = 40x?
a) h’ (x) = 40xx – 1
b) h’ (x) = 40x (ln 40)
c) h’ (x) = 40x (ln 1)
d) h’ (x) = 40x (ln
x)
Answer: b)
Which of the following is the derivative of f(x) = 72𝑥2+ 𝑥?
f’ (x) = (72𝑥2+ 𝑥) (4x + 1)
f’ (x) = (2x2 + x) 72𝑥2+ 𝑥 −1
f’ (x) = 72𝑥2+ 𝑥 (ln 7)
f’ (x) = (ln 7) (72𝑥2+ 𝑥) (4x + 1)
Answer: d)
Which of the following is the derivative of h(x) = 4x3 + e16x – 32x – 1?
h’ (x) = 12x2 + 16e16x – 2(ln 3)32x – 1
h’ (x) = 12x2 + 16e16x – (ln 3)62x – 1
h’ (x) = 12x + 16ex – 62x – 1
h’ (x) = 12x + 16ex – (2)32x – 1
Answer: a)
2𝑥2
8. Which of the following is the derivative of f(x) = cos 𝑥 ?
4𝑥
a) cos 𝑥 +=
f’ (x) 2𝑥2 sin 𝑥
cos2𝑥
b) f’ (x) 4𝑥 sin 𝑥 − 2𝑥2 cos 𝑥
= cos2𝑥
4𝑥
c) f’ (x) = − 2𝑥2
cos 𝑥
cos2𝑥
