Introduction to Derivatives and Limits in Calculus

Derivatives: The Concept of Slope

Secant and Tangent Lines

The derivative of a function at a point describes the slope of the tangent line to the curve at that point. This concept generalizes the idea of slope from straight lines to curves.

• Secant Line: A line passing through two points on a curve. Its slope approximates the rate of change between those points.

• Tangent Line: A line that touches the curve at a single point and has the same slope as the curve at that point.

• Difference Quotient: The slope of the secant line between points $x$ and $x+h$ is given by: $\frac{f(x+h) - f(x)}{h}$

• As $h \to 0$, the secant line approaches the tangent line, and the difference quotient approaches the derivative.

Example: For $f(x) = x^2$, the slope at $x=1$ is:

• $\frac{f(1+h) - f(1)}{h} = \frac{(1+h)^2 - 1^2}{h} = \frac{1 + 2h + h^2 - 1}{h} = \frac{2h + h^2}{h} = 2 + h$

• As $h \to 0$, the slope is $2$.

Definition of the Derivative

Formal Definition

The derivative of $f(x)$ at $x=a$ is defined as:

• $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

• This limit, if it exists, gives the instantaneous rate of change of $f$ at $x=a$.

Notation: The derivative can be written as $f'(x)$, $\frac{df}{dx}$, or $\frac{d}{dx}f(x)$.

Examples

• Example 1: $f(x) = 2x - 1$

  • $f'(x) = \lim_{h \to 0} \frac{2(x+h) - 1 - (2x - 1)}{h} = \lim_{h \to 0} \frac{2h}{h} = 2$

  • The slope of the tangent line is always $2$.

• Example 2: $f(x) = 3x^2 + 2x + 1$

  • $f'(x) = \lim_{h \to 0} \frac{3(x+h)^2 + 2(x+h) + 1 - [3x^2 + 2x + 1]}{h}$

  • Expanding and simplifying gives $f'(x) = 6x + 2$.

  • At $x=2$, $f'(2) = 6 \times 2 + 2 = 14$.

Left and Right Derivatives

One-Sided Derivatives

Sometimes, the derivative from the left and right at a point may differ. The function is differentiable at a point only if both one-sided derivatives exist and are equal.

• Left Derivative: $\lim_{h \to 0^-} \frac{f(a+h) - f(a)}{h}$

• Right Derivative: $\lim_{h \to 0^+} \frac{f(a+h) - f(a)}{h}$

• If these limits are not equal, the function is not differentiable at $x=a$.

Example: For $f(x) = |x|$ at $x=0$:

• Left derivative: $\lim_{h \to 0^-} \frac{|h|}{h} = -1$

• Right derivative: $\lim_{h \to 0^+} \frac{|h|}{h} = 1$

• Since $-1 \neq 1$, $f(x)$ is not differentiable at $x=0$.

Limits

Definition and Properties

The concept of a limit is fundamental in calculus, describing the behavior of a function as the input approaches a particular value.

• Limit Notation: $\lim_{x \to a} f(x) = L$ means that as $x$ approaches $a$, $f(x)$ approaches $L$.

• Left Limit: $\lim_{x \to a^-} f(x)$

• Right Limit: $\lim_{x \to a^+} f(x)$

• If both left and right limits exist and are equal, the (two-sided) limit exists.

Graphical Interpretation: The limit describes the value that the function approaches, not necessarily the value at that point.

Equations of Tangent Lines

Finding the Tangent Line

To find the equation of the tangent line to $f(x)$ at $x=a$:

• Find $f(a)$ (the point of tangency).

• Find $f'(a)$ (the slope at $x=a$).

• Use the point-slope form: $y - f(a) = f'(a)(x - a)$

Example: For $f(x) = 3x^2 + 2x + 1$ at $x=2$:

• $f(2) = 3 \times 4 + 2 \times 2 + 1 = 15$

• $f'(2) = 14$

• Equation: $y - 15 = 14(x - 2)$

Summary Table: Derivative Existence

Function	Left Derivative	Right Derivative	Differentiable at Point?
$f(x) = x$ at $x=0$	1	1	Yes
$f(x) = |x|$ at $x=0$	-1	1	No
$f(x) = x^2$ at $x=0$	0	0	Yes

Key Terms and Definitions

• Derivative: The instantaneous rate of change of a function at a point.

• Limit: The value a function approaches as the input approaches a specific value.

• Differentiable: A function is differentiable at a point if its derivative exists there.

• Point-Slope Form: $y - y_1 = m(x - x_1)$, used for the equation of a line with slope $m$ through point $(x_1, y_1)$.

Additional info:

• These notes cover foundational concepts in differential calculus, including the definition and computation of derivatives, the geometric interpretation of tangent lines, and the formalism of limits.

• Examples and graphical sketches illustrate the process of finding derivatives and tangent lines, as well as the importance of one-sided limits for differentiability.