Chapter 2: The Concept of Slope

Q: Can a tangent line not be drawn at every point on a curve?

That is correct. At a point such as x=0 on y=|x|, where the slopes approaching from the left and right do not agree, a tangent line cannot be defined in the usual differentiable sense. Such a point is called a point of nondifferentiability.

Q: Why does taking h to 0 give the tangent line?

As the distance between two points (Δx=h) approaches zero, the two points on the curve approach the same point. In this limit, the secant line connecting the two points approaches the tangent line at that point.

From the Slope of a Line to the Slope of a Curve

Beginner

2.1 Slope of a Line

First, let's review the concept of the slope of a straight line, which you learned in middle school.

Definition: Slope of a Line

In the equation of a straight line, $y = mx + n$, the coefficient $m$ is called the slope.

The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$$m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{\text{vertical change}}{\text{horizontal change}}$$

Figure 1: Slope of a line. The slope of the line through $(1, 1)$ and $(4, 7)$ is $\dfrac{6}{3} = 2$.

What Does Slope Mean?

A slope of $m = 2$ means that "when $x$ increases by $1$, $y$ increases by $2$."

Figure 2: Slope magnitude and direction. If $m \gt 0$ the line rises to the right, if $m = 0$ it is horizontal, and if $m \lt 0$ it falls to the right.

2.2 The Problem with Curves

The slope of a straight line is the same no matter which two points you choose. But what about a curve?

Figure 3: A curve has a different slope at each point (tangent lines to $y = x^2$ at points A, B, C).

The Problem

How can we define the "slope at a particular point" on a curve?

We can calculate the slope by choosing two points, but this gives the average slope over an interval, not the slope at a single point.

2.3 Secant Lines and Tangent Lines

To solve this problem, we introduce the concepts of secant lines and tangent lines.

Definition: Secant Line

A straight line connecting two points on a curve is called a secant line.

Definition: Tangent Line

A tangent line is a straight line that represents the direction of a curve near a particular point on the curve. When the derivative exists at that point, the slope of the tangent line is equal to the derivative. A tangent line may intersect the curve at other points as well; it does not necessarily share exactly one point with the curve.

Figure 4: Secant line (left) and tangent line (right). A secant line connects two points on the curve; a tangent line shows the direction of the curve at that point.

The Idea: Approaching a Tangent Line with Secant Lines

Here is an important idea.

Key Idea

As point Q approaches point P arbitrarily closely, if the slope of the secant line converges to a finite limit, that limit is the slope of the tangent line at point P.

Therefore, the statement that a secant line approaches a tangent line requires the condition that the limit of the slopes exists.

Figure 5: As Q moves closer to P, the secant line PQ (colored lines) approaches the tangent line at P (red), and its slope approaches the slope of the tangent line.

2.4 Finding the Slope of a Tangent Line

Let's express this idea mathematically.

We want to find the slope of the tangent line at point P$(a, f(a))$ on the curve $y = f(x)$.

Step 1: Calculate the Slope of the Secant Line

Consider point P$(a, f(a))$ and another point Q$(a+h, f(a+h))$, which is $h$ units away from P in the horizontal direction.

Figure 6: The slope of the secant line PQ is $\dfrac{f(a+h)-f(a)}{h}$. The horizontal change is $h$ and the vertical change is $f(a+h)-f(a)$.

The slope of secant line PQ is:

$$\text{Slope of secant line} = \dfrac{f(a+h) - f(a)}{(a+h) - a} = \dfrac{f(a+h) - f(a)}{h}$$

Step 2: Let h Approach 0

As $h$ approaches $0$, point Q approaches point P.

At this point, the slope of the secant line approaches the slope of the tangent line:

$$\text{Slope of tangent line} = \lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$$

Important Formula

The slope of the tangent line at the point $(a, f(a))$ on the curve $y = f(x)$ is:

$$\lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$$

When this limit exists, its value is called the derivative at the point.

2.5 A Worked Example

Example: For $f(x) = x^2$, find the slope of the tangent line at $x = 2$.

Solution (Detailed Calculation)

Step 1: Calculate the slope of the secant line.

Let $a = 2$. The slope of the secant line is:

$$\dfrac{f(2+h) - f(2)}{h}$$

Step 2: Expand $f(2+h)$.

$$f(2+h) = (2+h)^2$$

Expanding $(2+h)^2$ gives:

$$(2+h)^2 = 2^2 + 2 \cdot 2 \cdot h + h^2 = 4 + 4h + h^2$$

Step 3: Calculate $f(2)$.

$$f(2) = 2^2 = 4$$

Step 4: Calculate the difference.

$$f(2+h) - f(2) = (4 + 4h + h^2) - 4 = 4h + h^2$$

Step 5: Divide by $h$.

$$\dfrac{f(2+h) - f(2)}{h} = \dfrac{4h + h^2}{h}$$

Factor the numerator:

$$\dfrac{h(4 + h)}{h} = 4 + h \quad (h \neq 0)$$

Step 6: Take the limit as $h \to 0$.

$$\lim_{h \to 0} (4 + h) = 4 + 0 = 4$$

Answer: The slope of the tangent line is 4.

Figure 7: The tangent line $y = 4x - 4$ to $y = x^2$ at the point $(2, 4)$. Its slope is $4$.

Chapter Summary

The slope of a straight line can be calculated as $\dfrac{\Delta y}{\Delta x}$.
For a curve, the slope varies from point to point.
Secant line: A straight line connecting two points on a curve.
Tangent line: A straight line representing the direction of a curve at a particular point.
Slope of a secant line → Slope of a tangent line (as $h \to 0$).
Slope of a tangent line $= \displaystyle\lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$.

Frequently Asked Questions

What is the difference between a secant line and a tangent line?

A secant line is a straight line connecting two points on a curve. A tangent line represents the direction of the curve near a particular point, and it may intersect the curve at other points as well. A secant line represents the "average slope" between two points, while a tangent line represents the "instantaneous slope" (the derivative) at a particular point.

Can a tangent line not be drawn at every point on a curve?

That is correct. At a point such as $x=0$ on $y = |x|$, where the slopes approaching from the left and right do not agree, a tangent line cannot be defined in the usual differentiable sense. The slope from the left is $-1$, while the slope from the right is $+1$. Since they are different, the function is said to be "not differentiable" at this point.

Why does taking $h \to 0$ give the tangent line?

As point Q approaches point P, if the slope of the secant line converges to a finite limit, that limit becomes the slope of the tangent line at point P. This is the fundamental idea behind differentiation.

Exercises

Problem 1 (Basic): Find the slope of $y = 3x - 2$.

Hint

Slope $= m$ (the coefficient in $y = mx + n$). Thus, $m = 3$.

Problem 2 (Standard): For $f(x) = x^2$, find the slope of the tangent line at $x = 3$ using the limit as $h \to 0$.

Hint

$\dfrac{(3+h)^2 - 9}{h} = \dfrac{6h + h^2}{h} = 6 + h \to 6$.

Problem 3 (Advanced): For $f(x) = x^3$, find the slope of the tangent line at $x = a$ using a limit.

Hint

Expand $(a+h)^3 = a^3 + 3a^2h + 3ah^2 + h^3$. Taking the limit as $h \to 0$ gives $3a^2$.