CUENCA INSTITUTE, INC.
Senior High School Department
BASIC CALCULUS
.
Name: ________________________________
Date: ______________
Worksheet 7 (Standard Methods of Differentiation)
A. Determine the differential coefficient with respect to the variable using the
standard methods of differentiation.
𝒇(𝒙) = 𝟓𝒙𝟐 − 𝟒𝒙 + 𝟏𝟑
𝒇(𝒙) =
𝟏
+ 𝟑𝟐
𝒙𝟐
𝒇(𝒙) = 𝟒𝒆𝟑𝒙
𝒇(𝒙) = 𝟒√𝒙
𝒇(𝜽) = 𝟏𝟐 𝐬𝐢𝐧 𝟓𝜽
𝒇(𝒙) = 𝟏𝟓 𝐥𝐧 𝟑𝒙
B. An alternating voltage is given by: 𝑣 = 100 sin 200𝑡 volts, where 𝑡 is the time in
seconds. Calculate the rate of change of voltage when:
(a) 𝒕 = 𝟎. 𝟎𝟎𝟏 𝒔.
(b) 𝒕 = 𝟎. 𝟎𝟓 𝒔.
1|STEM
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CUENCA INSTITUTE, INC.
Senior High School Department
BASIC CALCULUS
Name: ________________________________
Date: ______________
Worksheet 8 (Differentiation of a Product)
Directions: Differentiate the given functions with respect to the variable.
1. Find the differential coefficient of 𝒚 = 𝟒𝒙𝟐 𝐜𝐨𝐬 𝟐𝒙
2. Find the rate of change of 𝒚 with respect to 𝒙 given : 𝒚 = √𝒙𝟑 𝐥𝐧 𝟐𝒙
3. Evaluate
𝒅𝒊
𝒅𝒕
, correct to 3 decimal places, when 𝒕 = 𝟎. 𝟏, and 𝒊 = 𝟏𝟓𝒕 𝐬𝐢𝐧 𝟑𝒕.
4. Determine the rate of change of voltage, given 𝒗 = 𝟒𝒕 𝐬𝐢𝐧 𝟐𝒕 volts when 𝒕 = 𝟎. 𝟑𝒔.
2|STEM
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CUENCA INSTITUTE, INC.
Senior High School Department
BASIC CALCULUS
Name: ________________________________
Date: ______________
Worksheet 9 (Differentiation of a Quotient)
Directions: Differentiate the given functions with respect to the variable.
1. Find the differential coefficient of 𝒚 =
𝟐 𝐜𝐨𝐬 𝟑𝒙
𝒙𝟐
2. Find the rate of change of 𝒚 with respect to 𝒙 given : 𝒚 =
3. Evaluate
𝒅𝒚
𝒅𝒙
𝒆𝟐𝒙
𝟐 𝐬𝐢𝐧 𝒙
, correct to 3 decimal places, when 𝒙 = 𝟐, and 𝒚 =
𝟐𝒙
𝒙𝟐 +𝟏
.
3|STEM
.
CUENCA INSTITUTE, INC.
Senior High School Department
BASIC CALCULUS
Name: ________________________________
Date: ______________
Worksheet 10
(Function of a Function and Successive Differentiation)
Find the derivative of 𝒚 = (𝟐𝒙𝟑 + 𝟓𝒙)𝟑
Show that the differential equation
𝒅𝟐 𝒚
𝒅𝒚
−
𝟒
+ 𝟒𝒚 = 𝟎
𝒅𝒙𝟐
𝒅𝒙
is satisfied when 𝒚 = 𝒙𝒆𝟐𝒙
4|STEM
.
CUENCA INSTITUTE, INC.
Senior High School Department
BASIC CALCULUS
.
Name: ________________________________
Worksheet 11
Date: ______________
Directions: Solve the following problems. Show your COMPLETE SOLUTIONS
and BOX your final answer. Send a clear picture of your output to
my messenger account on or before May 20, 2022 (until 4:30pm).
Problem 1: The length ℓ meters of a certain metal rod at temperature 𝜃°𝐶 is
given by: 𝑙 = 1 + 0.0005𝜃 + 0.0000005𝜃 2 . Determine the rate of
change of length when the temperature is 100°C.
Problem 2: The distance x meters moved by a car in a time t seconds is given
by: 𝑥 = 4𝑡 3 − 3𝑡 2 + 4𝑡 − 2. Determine the velocity and
acceleration when 𝑡 = 1𝑠.
Problem 3: Locate the turning point on the curve 𝑦 = 3𝑥 2 − 8𝑥 and determine
its nature by determining the sign of the second derivative.
Problem 4: A rectangular area is formed having a perimeter of 50cm.
Determine the length and width of the rectangle if it is to enclose
the maximum possible area.
Problem 5: Find the equation of the equation of the tangent and the equation
of the normal to the curve 𝑦 = 1 + 𝑥 − 𝑥 2 at the point (−2, −5).
Problem 6: Given 𝑦 = 5𝑥 2 − 𝑥, determine the approximate change in 𝑦 if 𝑥
changes from 2 to 2.05.
5|STEM
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