Related Rates, Linear Approximations & Differentials
MAT1241: Calculus 1
University of Eswatini
In this comprehensive guide, we’ll delve into the fascinating topics of related rates, linear
approximations, and differentials. These concepts are essential in calculus and have practical
applications in various fields. Let’s explore each topic in detail and provide step-by-step
explanations along with examples.
1. Related Rates
a. Definition
Related rates involve finding the rate of change of one quantity with respect to another related
quantity. Typically, we analyze how the rates of change of two variables are connected.
b. Steps for Solving Related Rates Problems
1. Identify the variables involved.
2. Write down the given information and the relationship between the variables.
3. Differentiate both sides of the equation with respect to time.
4. Substitute known values and solve for the desired rate.
Example 1: Water Flow
Suppose water is pouring into a conical tank. The height of the water is increasing at a rate of 2
cm/min, and the radius of the base is 10 cm. Find the rate at which the water volume is
increasing when the water is 8 cm deep.
Solution:
1. Variables: Let (h) be the height of water, (r) be the radius of the base, and (V) be the
volume.
2. Relationship: .
3. Differentiate both sides with respect to time:
4. Given: cm/min, (r = 10) cm, and (h = 8) cm. Solve for .
MAT1241: Calculus 1
University of Eswatini
In this comprehensive guide, we’ll delve into the fascinating topics of related rates, linear
approximations, and differentials. These concepts are essential in calculus and have practical
applications in various fields. Let’s explore each topic in detail and provide step-by-step
explanations along with examples.
1. Related Rates
a. Definition
Related rates involve finding the rate of change of one quantity with respect to another related
quantity. Typically, we analyze how the rates of change of two variables are connected.
b. Steps for Solving Related Rates Problems
1. Identify the variables involved.
2. Write down the given information and the relationship between the variables.
3. Differentiate both sides of the equation with respect to time.
4. Substitute known values and solve for the desired rate.
Example 1: Water Flow
Suppose water is pouring into a conical tank. The height of the water is increasing at a rate of 2
cm/min, and the radius of the base is 10 cm. Find the rate at which the water volume is
increasing when the water is 8 cm deep.
Solution:
1. Variables: Let (h) be the height of water, (r) be the radius of the base, and (V) be the
volume.
2. Relationship: .
3. Differentiate both sides with respect to time:
4. Given: cm/min, (r = 10) cm, and (h = 8) cm. Solve for .
