2. 2
Assessment
A. Read and answer each question.Write your answer on a separate sheet
of paper.
1. Which of the following does NOT define the slope of the tangent
line to the curve?
A. It is constant.
B. It is not constant and must be determined by a point.
C. It is equal the derivative of the function.
D. It is derived from the concept of the slope of a second line.
2. On which of the following conditions will a tangent line exist?
A. Function continuous at P.
B. Function discontinuousat P.
C. Curve with cusp at P.
D. Curve with corner at P.
3. Which of the following describes a line tangent to a given curve drawn at its
maximum or minimum point?
A. has a positive slope C. horizontal
B. has a negative slope D. vertical
4. Which is the line perpendicular to the tangent line at the point of tangency?
A. secant C. parallel
B. skew D. normal
5. Which of the following is equal to the slope of the tangent line?
A. Average rate of change
B. Instantaneous rate of change
C. Slope of the secant line
D. Slope of the line perpendicular to the given tangent line.
5. 5
A. – 2 B. 2 C. 1 D. – 1
C. Solve what is asked in the following problems. Write your solutions and final
answers on a separate sheet.
Tangent and normal lines are drawn to the curve 𝐲 = 𝐱𝟑 at 𝐱 = 𝟐.
11. What is the equation of the line tangent to the curve at the given point?
A. 𝑦 = 12𝑥 − 16 C. 𝑦 = 𝑥 − 12
B.𝑦 = 𝑥 − 16 D. 𝑦 = 12𝑥
𝑓′(𝑥) = lim
ℎ→0
𝑓(𝑥 + ℎ) − 𝑓(𝑥)
ℎ
= lim
ℎ→0
2𝑥
𝑥 + ℎ − 1
−
2𝑥
𝑥 − 1
ℎ
= lim
ℎ→0
2𝑥(𝑥 − 1) − 2𝑥(𝑥 + ℎ − 1)
ℎ(𝑥 + ℎ − 1)(𝑥 − 1)
= lim
ℎ→0
2𝑥2 − 2𝑥 − 2𝑥2 − 2ℎ − 2𝑥
ℎ(𝑥 + ℎ − 1)(𝑥 − 1)
= lim
ℎ→0
2𝑥2 − 2𝑥 − 2𝑥2 − 2ℎ − 2𝑥
ℎ(𝑥 + ℎ − 1)(𝑥 − 1)
= lim
ℎ→0
−2ℎ
ℎ(𝑥 + ℎ − 1)(𝑥 − 1)
= lim
ℎ→0
ℎ(−2)
ℎ(𝑥 + ℎ − 1)(𝑥 − 1)
= lim
ℎ→0
−2
(𝑥 + ℎ − 1)(𝑥 − 1)
=
−2
(𝑥 + 0 − 1)(𝑥 − 1)
=
−2
(𝑥 − 1)(𝑥 − 1)
𝑓′(2)=
−2
(𝑥 − 1)(𝑥 − 1)
=
−2
(2 − 1)(2 − 1)
=
−2
(1)(1)
= =
−2
1
= −2
𝑥 = 2, 𝑦 = 23 = 𝑦 = 8 𝑎𝑛𝑑
𝑑𝑦
𝑑𝑥
= 3𝑥2
𝑠𝑙𝑜𝑝𝑒 𝑎𝑡 𝑥 = 2,
𝑑𝑦
𝑑𝑥
= 3 × 22 = 𝑚 = 12
𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡,
𝑦 − 8 = 12(𝑥 − 2)
𝑦 = 12𝑥 − 24 + 8
𝑦 = 12𝑥 − 16
6. 6
12. What is the equation of the normal line drawn to the curve at the given point?
A. C.
B. D. 𝑦 = 𝑥 + 98
Mr. Dela Cruz encourageshis students to solve Math problems fast and with
accuracy. He observed in his Math class that the time it takes a student to
solve x word problems is defined by the function (𝒙) = 𝟑𝒙𝟐 − 𝒙 where f(x) is in
minutes.
13. Find the average rate of change in solving time from 1 to 3-word problems.
A. 6 minutes/problem C. 11 minutes/problem
B. 21 minutes/problem D. 25 minutes/problem
𝑥 = 2, 𝑦 = 23 = 𝑦 = 8 𝑎𝑛𝑑
𝑑𝑦
𝑑𝑥
= 3𝑥2
𝑠𝑙𝑜𝑝𝑒 𝑎𝑡 𝑥 = 2,
𝑑𝑦
𝑑𝑥
= 3 × 22 = 𝑚 = 12
𝑚1𝑚2 = 1
𝑚2 =
1
12
𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑛𝑜𝑟𝑚𝑎𝑙
𝑦 − 𝑦1 = 𝑚2(𝑥 + 𝑥1)
𝑦 − 8 =
1
2
(𝑥 + 2)
𝑦 =
𝑥 + 2 + 96
12
𝑦 =
𝑥 + 98
12
=
∆𝑓(𝑥)
∆𝑥
=
𝑓(3) − 𝑓(1)
3 − 1
=
3𝑥32 − 3𝑥2 − 1
2
=
22
2
= 11 (11 𝑚𝑖𝑛𝑠 𝑝𝑒𝑟 𝑤𝑜𝑟𝑑 𝑝𝑟𝑜𝑏𝑙𝑒𝑚)
7. 7
14. Find the average rate of change in solving time from 3 to 5-word problems.
A. 5 minutes/problem C. 13 minutes/problem
B. 20 minutes/problem D. 23 minutes/problem
15. Solve for the instantaneous rate of change in solving time at 2-word problems.
A. 6 minutes/problem C. 11 minutes/problem
B. 21 minutes/problem D. 26 minutes/problem
3 𝑤𝑜𝑟𝑑 𝑝𝑟𝑜𝑏𝑙𝑒𝑚:𝑓(3) = 3 × 32 − 3 = 27 − 3 = 24 𝑚𝑖𝑛
5 𝑤𝑜𝑟𝑑 𝑝𝑟𝑜𝑏𝑙𝑒𝑚:𝑓(5) = 3 × 52 − 5 = 75 − 5 = 70 𝑚𝑖𝑛
𝑟𝑎𝑡𝑒 =
70 − 24
5 − 3
=
46
2
= 23 ( min 𝑝𝑒𝑟 𝑝𝑟𝑜𝑏𝑙𝑒𝑚)
𝑓′(𝑥) = (3𝑥2 − 𝑥)′
= 6𝑥 − 1
𝑠𝑜 𝑓′(2) = 6(2) − 1
= 12 − 1
= 11(min 𝑝𝑒𝑟 𝑤𝑜𝑟𝑑 𝑝𝑟𝑜𝑏𝑙𝑒𝑚)
8. 8
Additional Activities
Solve the following problems:
1. Determine the values of x where a curve 𝑦 = 𝑥3 − 3𝑥2 − 9𝑥 + 12 has horizontal
tangent lines.
Bataan is known for its mountains and trails such as the Dambana ng Kagitingan and Duhat Trail which
are ideal for history and nature lovers. The slopes of the hills in Duhat trail represent a curve. (Photo
credits: Bataan Weather Page)
𝑑𝑦
𝑑𝑥
=
𝑑
𝑑𝑥
(𝑥3 − 3𝑥2 − 9𝑥 + 12)
=
𝑑
𝑑𝑥
(𝑥3)−
𝑑
𝑑𝑥
(3𝑥2) −
𝑑
𝑑𝑥
(9𝑥) +
𝑑
𝑑𝑥
(12)
=
𝑑
𝑑𝑥
(𝑥3)− 3
𝑑
𝑑𝑥
(𝑥2) − 9
𝑑
𝑑𝑥
(𝑥) + 0
=
𝑑
𝑑𝑥
(𝑥3)− 3
𝑑
𝑑𝑥
(𝑥2) − 9
𝑑
𝑑𝑥
(𝑥) + 0
𝑑𝑦
𝑑𝑥
= 3𝑥3−1 − (3 × 2𝑥2−1) − 9𝑥1−1
= 3𝑥2 − 6𝑥 − 9
= 3(𝑥2 − 2𝑥 − 3)
3(𝑥2 − 2𝑥 − 3) = 0
𝑥2 − 2𝑥 − 3 = 0
9. 9
2. Aside from mountain ranges and trails, Bataan is known for its beautiful
beaches.
White beach in Barangay Paysawan, Bagac, Bataan
A tourist threw a pebble in the sea, causing water ripples. The shape formed
were circles increasing in area. The formula for finding the area of a circle is
given by 𝐴 = 𝜋𝑟2.
a. Find the average rate at which the area of a circle changes with r as the
radius increases from 2 to 4 units.
𝑥2 − 3𝑥 + 𝑥 − 3 = 0
𝑥(𝑥 − 3) − (𝑥 − 3) = 0
(𝑥 − 3)(𝑥 + 1) = 0
𝑥 − 3 = 0 | 𝑥 + 1 = 0
𝑥 = 3 | 𝑥 = −1
𝑟
1 = 2: 𝐴1 = 𝜋𝑟2 = 𝜋 × 22 = 4𝜋
𝑟2 = 4: 𝐴2 = 𝜋𝑟2 = 𝜋 × 42 = 16𝜋
𝑣 =
𝐴2 − 𝐴1
𝑟2 − 𝑟
1
=
16𝜋 − 4𝜋
4 − 2
=
12𝜋
2
= 6𝜋
10. 10
b. Solve for the instantaneous rate at which the area changes with r, when r
= 5.
𝐴 = 𝜋𝑟2
𝐴′ = 2𝜋𝑟
𝑟 = 5 𝑣 = 𝐴′ = 2𝜋 × 5 = 10𝜋
