MCV 4U1 Grade 12 Calculus and Vectors
Application of Derivatives Test 5
K/U
/17
App
/17
Th/I
/12
Comm
/12
Total
/54
Multiple Choice (12, K/U)
1 mark for the correct answer and 1 mark for showing your work.
1. What is the second derivative of 𝑓 (𝑥 ) =
3
5
3
3
𝑥 3 − 𝑥 2 + 6?
3
a. 𝑓 ′′ (𝑥) = 5x2 – √𝑥
2
b. 𝑓 ′′ (𝑥) = 10x + √𝑥 + 6
4
c. 𝑓 ′′ (𝑥) = 10x +
d. 𝑓 ′′ (𝑥) = 5x2 – √𝑥 + 6x
2
3
4 √𝑥
3
2. Determine the maximum value of f(x) = – 3x3 + 2x2 + 2 , –2 ≤ 𝑥 ≤ 3.
a. 2
b. 3.185
c. 18
d. –7
3. Let the graph below represent f(x) on the interval [– 2, 2]. For which
values of x is 𝑓 ′ (𝑥 ) = 0?
a. – 2, –1, 0, 1, 2
b. – 2, –1, 1, 2
c. –1, 1
d. 0
4. Determine the maximum value of the function f(x) = – (4x – 2)2 + 5
on the interval [–1, 2].
a. 5
c. –31
b. 0.5
d. 1
2
5. Determine the critical values of the function f(x) = – 𝑥 3 + 5𝑥 2 + 27
3
on the interval [–2, 2].
a. 0
c. –2, 2
b. –2, 0, 2
d. –l
6. Suppose the cost, in dollars, of manufacturing x items is approximated
by the function C(x) = 0.2x2 + 5x +500, for 1 ≤ 𝑥 ≤ 1000. If the cost of
𝐶(𝑥)
manufacturing one item can be represented by the function U(x) =
,
𝑥
determine how many items should be manufactured to minimize the unit
cost.
a. 50
c 13
b. 1000
d. 500
Problems
7. Determine 𝑓 ′′ (𝑥) if 𝑓 (𝑥 ) = √𝑥 3 + 2 . [2, K/U]
8. For y =
𝑥
2+
𝑥2
, determine 𝑦 ′′ (1). [3, K/U]
9. Let s (t) = – 5t2 + 40t + I describe a toy rocket's motion as it is
launched vertically into the air. Determine the maximum height, in
metres, that it reaches over the time interval 1 ≤ 𝑡 ≤ 5. [3, App.]
10. A rectangular garden is to be planted against the side of a barn. The
gardener has 80 metres of fencing, and will construct a three-sided
fence; the side of the barn will be the fourth side. Determine the
dimensions of the fence that will give the garden the maximum area. [4,
App.]
11. A box is to be constructed from a sheet of cardboard that measure 3
m × 5 m. A square from each corner of the sheet will be cut and then the
sides will be folded up to create the box. What is the greatest volume
this box can be if it is constructed in this manner? [6. App.]
12. The cost of producing an ordinary cylindrical tin can is determined
by the materials used for the wall and the end pieces. If the end pieces
are twice as expensive per square centimeter as the wall, find the
dimensions (to the nearest millimeter) to make a 900 cm3 can at minimal
cost. [4, App.]
13. Two particles start at the origin on the s-axis at time t = 0 s and move
along the s-axis. Their movement can be expressed by the following
1
equations: s1(t) = t2 – 3t and s2(t) = 9t – t2
2
a. When will the two points have the same speed? [Comm, 3]
b. When will the two points have the same position? [Comm, 3]
c. What are the respective velocities of the points when they have the
same position? [Comm, 2]
14. A movie theater wants to determine the price per ticket that should
be charged in order to maximize its revenue. The theater can
accommodate 2000 movie goers per night, but the average attendance is
about 1400 people per night. The average ticket price is about $8.00.
Market research shows that, for each $0.50 reduction in the ticket price,
attendance increases by 250 spectators. What should the theater charge
per ticket to maximize the revenue? [6, Th/I]
15. A landlord manages l00 apartments in a complex downtown and
charges his tenants monthly. When the rent is $800 a month, all of the
apartments are rented. For every $15 increase in rent, one apartment
becomes empty. On average, each apartment needs $60 invested in
maintenance and repairs each month. The landlord wants to know how
much he should charge per month in order to have the greatest profit. [6.
Th/l]