Definition of Average Rate of Change
The expression
is called the difference quotient for f at a and represents the average rate of change of y = f(x) from a to a + h .

Geometrically, the rate of change of f from a to a+h the slope of the secant line through the point P(a, f(a)) and Q(a+h, f(a+h)).
If f(t) is the position function of a particle that is moving on a straight line, then in the time interval from t = a to t = a+h, the change in position is f(a+h) − f(a), and the average velocity of the particle over the time interval is
Example 1. The displacement of a particle moving in a straight line is given by the equation of motion f(t) = t3 − 4t + 3. Find the average velocity of the particle over the interval 0 ≤ t ≤ 4.
Solution :
Exercises – Rate of Change
Multiple Choice Questions
1. The traffic flow at a particular intersection is modeled by the function f defined by f(t) = 25+6cos(x/3) for 0 ≤ t ≤ 120. What is the average rate of change of the traffic flow over the time interval 30 ≤ t ≤ 40.
(A) 0.743
(B) 0.851
(c) 0.935
(d) 1.176
2. The rate of change of the altitude of a hot air balloon rising from the ground is given by y(t) = t3−3t2 + 3t for 0 ≤ t ≤ 10. What is the average rate of change in altitude of the balloon over the time interval 0 ≤ t ≤ 10.
(A) 56
(B) 73
(c) 85
(d) 94
Free Response Questions
t (sec) | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 |
f(t) (ft/sec) | 0 | 28 | 43 | 67 | 82 | 85 | 74 | 58 | 42 | 35 |
3. The table above shows the velocity of a car moving on a straight road. The car’s velocity v is measured in feet per second.
(a) Find the average velocity of the car from t = 60 to t = 90.
(b) The instantaneous rate of change of f with respect to x at x = a can be approximated by finding the average rate of change of f near x = a. Approximate the instantaneous rate of change of f at x = 40 using two points, x = 30 and x = 50.
Rate of Change in Business FAQs
What is the rate of change in business?
In a general sense, the rate of change in business refers to how quickly a specific metric, variable, or aspect of the business (like sales, costs, market trends, or technology) is increasing or decreasing over a period of time. It can describe internal operational changes or external market dynamics.
How do you define the rate of change in a business metric?
The rate of change for a specific business metric is calculated as the change in the value of that metric divided by the change in time.
Formula: Rate of Change = (Change in Metric Value) ÷ (Change in Time)
For example, if sales increased by $10,000 over one year, the rate of change in sales is $10,000/year. In calculus, it refers to the instantaneous rate of change (the derivative).
Why is understanding the rate of change important in business?
Understanding the rate of change is crucial for:
- **Forecasting & Planning:** Predicting future sales, costs, or market size.
- **Performance Evaluation:** Assessing growth or decline trends in key metrics.
- **Adaptation:** Responding effectively to rapid changes in the market, technology, or competitive landscape.
- **Risk Management:** Identifying accelerating negative trends early on.
- **Opportunity Identification:** Spotting areas of rapid positive change.
A high rate of change (positive or negative) often indicates a dynamic situation requiring attention.
What are examples of "rate of change" in a business context?
Examples include:
- **Sales Growth Rate:** The percentage increase in sales revenue over a period.
- **Cost Increase Rate:** The rate at which operating expenses are rising.
- **Market Growth Rate:** How quickly the total size of the market for a product or service is expanding.
- **Technological Change Rate:** The speed at which new technologies are emerging or becoming obsolete.
- **Interest Rate Changes:** How quickly borrowing costs are increasing or decreasing.
What drives the rate of change in the business environment today?
Several forces contribute to the accelerating rate of change in the modern business environment:
- **Technological Advancement:** Rapid development and adoption of new technologies (AI, digitalization, automation).
- **Globalization:** Increased interconnectedness of markets and economies.
- **Changing Consumer Behavior:** Evolving preferences, digital literacy, and demand for personalization.
- **Increased Competition:** New entrants, disruptive business models, and global players.
- **Regulatory Shifts:** Changes in government policies, trade agreements, and compliance requirements.
- **Environmental & Social Factors:** Growing awareness and impact of climate change, social equity, and sustainability.
Businesses must be agile and adaptable to thrive in such dynamic conditions.