Applications of Derivatives

MCV 4U0
Unit 6
Part I: Theory
1. True or false.
Applications of Derivative Practice Test
Date:
(a) . If f ′ (c) = 0, then f must have either a relative minimum or a
relative maximum at c.
(b) . Since function f(x) = − 1
is increasing on the intervals (−∞, 0)
and (0, ∞), it is impossible to find x1 and x2 in the domain of f such that x1 > x2 and
f(x1) > f(x2).
(c) . On the interval (−1, 1], the function f(x) = x 4 has an absolute
maximum and an absolute minimum.
2. Fill in the blank.
(a) cos 2 θ+ = 1
(b) 1+ = sec 2 θ
(c) If y = sin x, then y ′ = .
(d) If r = cos θ, then dr
=
dθ
.
(e) For y = x 9 − 7x 3 + 2, d2 y
= .
dx2 (f) Extreme Values: If a function f(x) has a derivative at every point in the interval a ≤
x ≤ b, calculuate f(x) at
• all points in the interval a ≤ x ≤ b, where f ′ (x) =
• the x = a and x = b
The value of f(x) on the interval a ≤ x ≤ b is the largest of these
values, and the value of f(x) on the interval is the smallest of
these values.
(g) If the position of an object, s(t), is a function of time, t, then the first derivative of
this function represents the of the object at time t. i.e., s ′ (t) =
.
(h) is the second derivative of the position function.
(i) a(t) < 0 indicates that the velocity is .
(j) a(t) = 0 indicates that the velocity is .
3. Multiple Choice.
(a) If f(x) = sec x + csc x, then f ′ (x) =
(a) sec 2 x + csc 2 x (b) csc x − sec x (c) sec x tan x − csc x cot x
(b) If f(x) = cos 2 x, then f ′′ (π) =
(a) −2 (b) 0 (c) 1 (d) 2π
(c) A particle’s position is given by s(t) = t 3 − 6t 2 + 9t. What is its acceleration at time
t = 4? (a) 0 (b) 9 (c) −9 (d) 12
1
x

MCV 4U0
Unit 6
Applications of Derivative Practice Test
Date:
(d) The maximum velocity attained on the interval 0 ≤ t ≤ 5 by the particle whose displacement
is given by s(t) = 2t 3 − 12t 2 + 16t + 2 is
(a) 286 (b) 46 (c) 16 (d) 0
(e) If f(x) = sec(4x), then f ′
π
16 is
(a) 4 √ 2 (b) √ 2 (c) 0 (d) 1
√ 2
(f) Using the graph below, on the interval [a, b] absolute minimum value is at
(a) point G (b) point H (c) point I (d) point J (e) point K
y
G
a
H
I
J
K c
(g) Using the graph from (3f), on the interval [a, b], the absolute maximum value is at
(a) point G (b) point H (c) point I (d) point J (e) point K
(h) Using the graph from (3f), on the interval [a, c], the absolute minimum is at
(a) point G (b) point H (c) point I (d) point J (e) point K
4. The critical numbers for f(x) = 3√ x − 1 are found at
(a) x = −1 (b) x = 0 (c) x = 1 (d) x = {−1, 0, 1}
Part II: Application
1. The position of a spring is described by the function s(t) = 3 cos(2x).
(a) The velocity function is v(t) = .
(b) The acceleration function is a(t) = .
2. Given f(x) = −2x 3 + 9x 2 + 4, find the extreme values of f(x) on [−1, 5].
2
b
x

MCV 4U0
Unit 6
Applications of Derivative Practice Test
Date:
3. The position function of a particle is given by
s(t) = √ t(3t 2 − 35t + 90)
where s is measured in metres and t is in seconds.
(a) Find the velocity and acceleration as a function of time.
(b) Find the velocity of the particle at t = 3.
(c) When is the particle at rest?
(d) When is the particle moving in a positive direction.
4. Mrs McCombs grows tomatoes along the side of her house. The squirrels are getting at them,
and she’s decided to put a fence around the tomatoes to save them. If she has 40 m of fencing
available to lay out using the straight side of her house as one line of the rectangle, what
dimensions will maximize the garden area?
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MCV 4U0
Unit 6
Applications of Derivative Practice Test
Date:
5. Rick’s American Cafe sells cheeseburgers with a yearly demand function of
and cost function
(a) State the revenue function.
p =
(b) Find the marginal revenue function.
800000 − x
200000
C(x) = 125000 + 0.42x
(c) What level of cheeseburger sales will maximize profits?
6. A piece of sheet metal, 60 cm by 30 cm, is to be used to make a rectangular box with an
open top. Determine the dimensions that will give the box with the largest volume.
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MCV 4U0
Unit 6
Applications of Derivative Practice Test
Date:
7. Ian and Ada are both training for a marathon. Ian’s house is located 20 km north of Ada’s
house. At 9:00 am one Saturday, Ian laves his house and jogs south at 8 km/h. At the same
time, Ada leaves her house and jogs east at 6 km/h. When are Ian and Ada closest together,
given that they both run for 2.5 h?
20
J
I
A
8. Determine the dy
dx
(a) y = tan x
(b) y =
s
sin x
1 + cos x
B
for each of the following
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MCV 4U0
Unit 6
(c) r = tan 3 (4θ + π)
(d) y = 6 cos( √ x)
Applications of Derivative Practice Test
Date:
9. The position of a particle as it moves horizontally is described by the function
s(t) = sin 2 t − 2 cos 2 t
where t ∈ [−π, π]. If s is the displacement in metres and t is the time in seconds, find the
absolute maximum and minimum displacements.
10. Find an equation of the tangent line to the curve y = cos2 x
sin 2 x at π
4 , 1 .
6