Introduction to Derivatives: Rate of Change and Slopes

Motivation: Why Study Rate of Change?

Imagine you're driving a car. Your average speed from one town to another might be 60 km/h. But at a given moment — say, when a police radar checks you — your instantaneous speed might be 72 km/h.

This difference is key in calculus:

Derivatives measure how fast something is changing at a single moment — not just over an interval.

The Big Idea: From Average to Instantaneous Rate of Change

In algebra, we learned how to compute the slope between two points:

$$\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$$

But what if we want the rate of change at exactly one point?
We need a new idea — a limit — which leads to the derivative.

Connecting to Graphs: Secant and Tangent Lines

• A secant line passes through two points on a curve.
• A tangent line touches the curve at a single point — representing the instantaneous rate of change.

As the two points of the secant get closer together, the secant line becomes the tangent line.

Interactive Example: Average Speed Between Two Times

We define a position function:

$$s(t) = t^3 - 2t^2 - t + 3$$

It models a moving object's position over time.

Use the sliders below to choose a starting and ending time ($t_0$ and $t_1$) to see:

• The curve $s(t)$
• A secant line connecting $s(t_0)$ and $s(t_1)$
• The average speed (slope) over $[t_0, t_1]$

As you bring $t_0$ and $t_1$ closer, the red secant line approaches the tangent line.

Interactive Graph

Move the sliders to explore how the average rate of change behaves!

Additional Example: A Leaking Water Tank

Scenario 1: Constant Leak

Suppose a tank leaks water at a steady rate of 2 liters per minute.

• If the tank starts with 100 liters, the water volume after $t$ minutes is:

$$V(t) = 100 - 2t$$

• The rate of change is constant:

  $$\frac{V(t_2) - V(t_1)}{t_2 - t_1} = -2 \text{ liters/minute}$$

This is a linear function. The slope (rate) is always the same. We don’t even need calculus to describe how fast it's leaking — the rate of change is already constant.

Scenario 2: Increasing Leak (Crack Worsens Over Time)

Now suppose the leak gets worse over time — the more water that escapes, the faster it leaks.

• Let’s say the volume behaves like:

$$V(t) = 100 - t^2$$

Now the leak starts slowly, but over time the rate of leakage increases.

Let’s compute average rates:

• From $t = 0$ to $t = 2$:

  $$\frac{V(2) - V(0)}{2 - 0} = \frac{100 - 4 - 100}{2} = -2 \text{ L/min}$$

• From $t = 2$ to $t = 4$:

  $$\frac{V(4) - V(2)}{2} = \frac{84 - 96}{2} = -6 \text{ L/min}$$

Even though both intervals are 2 minutes long, the tank is losing water more quickly later on.
This is where derivatives shine — they tell us exactly how fast the leak is worsening at any moment.

Summary of Concepts

Practice Exercises

1. Estimating Speed of a Thrown Ball

A ball’s height after $t$ seconds is:

$$h(t) = -5t^2 + 20t$$

• (a) Find the average speed from $t = 1$ to $t = 2$
• (b) Find the average speed from $t = 2$ to $t = 3$
• (c) Describe what is happening to the speed over time

2. Table of Values (Function: $f(x) = x^2$)

• (a) Find average rate of change from $x = 1$ to $x = 2$
• (b) Find average rate of change from $x = 1$ to $x = 1.5$
• (c) Estimate the slope at $x = 1$

3. Graph-Based Estimation

A function $f(x)$ (e.g. $x^2$) is plotted.

• (a) Sketch secant lines from $x = 1$ to $x = 3$, then $x = 1.5$ to $x = 2.5$
• (b) Visually estimate the slope at $x = 2$

4. Real-Life Scenario

A runner covers:

• 8 meters from $t = 1$ to $t = 3$

• 3 meters from $t = 3$ to $t = 4$

• (a) Compute average speed on both intervals

• (b) Is her speed increasing or decreasing? Explain

• (c) Sketch a possible distance-time graph

5. Conceptual Challenge: Constant Acceleration

A car’s speed increases by 3 m/s every second.

• (a) What type of position function describes this?
• (b) Sketch the position-time and speed-time graphs
• (c) What does the slope of the speed-time graph represent?

Extra Advanced Exercises

1. Approaching the Instantaneous Rate with Limits

Let the position of an object be given by:

$$s(t) = \sqrt{t + 1}$$

• (a) Compute the average rate of change from $t = 3$ to $t = 4$
• (b) Then compute it from $t = 3$ to $t = 3.1$, and from $t = 3$ to $t = 3.01$
• (c) Estimate the instantaneous rate of change at $t = 3$

Challenge: Try computing

$$\frac{s(3+h) - s(3)}{h}$$

for small values of $ h$ like $0.001, 0.0001$, and guess the limit as $h \to 0$.

2. Sharp Corner Behavior

Let $f(x) = |x - 2|$

• (a) Compute average rate of change from $x = 1.9$ to $x = 2.1$
• (b) Find the rate from left ($x = 1.99$ to 2) and from right ($x = 2$ to $2.01$)
• (c) Can we define a single tangent line at $x = 2$? Why or why not?

3. Multidimensional Rate

A balloon moves vertically and horizontally:

• Vertical position: $h(t) = 2t^2$
• Eastward position: $x(t) = 5t$

(a) Find the average vertical rate of change from $t = 2$ to $t = 4$.
(b) Compute total distance traveled.
(c) Estimate how fast the total distance is changing at $t = 3$.

4. Instantaneous Rate from a Rational Function

Let the position of a car be given by:

$$s(t) = \frac{t^2 + 3t + 1}{t + 1}$$

• (a) Simplify the function (if possible)
• (b) Compute the average rate of change from $t = 1$ to $t = 2$
• (c) Compute average rates for intervals approaching $t = 1$ from both sides
• (d) Estimate the instantaneous rate at $t = 1$ and explain your reasoning

5. Function with Unpredictable Slope

Consider the function:

$$f(x) = x \cdot \sin\left(\frac{1}{x}\right), \quad \text{for } x \neq 0,\quad f(0) = 0$$

(a) Plot or sketch this function (use Desmos or another graphing tool)
(b) What happens to the slope of the secant line as $x \to 0$?
(c) Can we estimate the slope at $x = 0$? Why is this difficult?

Hint: This function is continuous at 0 but wildly oscillatory nearby.

Conclusion

• The rate of change is how fast something changes with respect to something else.
• The derivative is the mathematical tool that tells us this rate at a single instant.
• In this lecture, we saw how slopes of curves connect to real-world change.

In our next session, we’ll formally define the derivative using limits!

Recommendation:
Watch 3Blue1Brown's video on introdcution to derivatives and his series on calculus.