Business Calculus: Differentiation, Continuity, and Tangent Lines Study Guide

Differentiability and Continuity

Identifying Non-Differentiable Points

In calculus, a function is differentiable at a point if its derivative exists at that point. Non-differentiable points often occur where the graph has a sharp corner, cusp, vertical tangent, or discontinuity.

• Sharp Corners or Cusps: The slope changes abruptly, so the derivative does not exist.

• Vertical Tangents: The slope approaches infinity, making the derivative undefined.

• Discontinuities: The function is not continuous, so it cannot be differentiable.

• Example: For a graph with labeled points $x_1, x_2, x_3, x_4$, check for corners, cusps, or breaks at these points to determine non-differentiability.

Continuity at a Point

A function is continuous at $x=a$ if:

• $f(a)$ is defined

• $\lim_{x \to a} f(x)$ exists

• $\lim_{x \to a} f(x) = f(a)$

Piecewise Functions: For $G(x) = \begin{cases} -x+5 & x \leq 2 \\ x+2 & x > 2 \end{cases}$, check continuity at $x=2$ by evaluating both pieces and their limits.

• Example: $G(2) = -2+5 = 3$; $\lim_{x \to 2^-} G(x) = 3$; $\lim_{x \to 2^+} G(x) = 4$. Since the left and right limits are not equal, $G(x)$ is not continuous at $x=2$.

Average Rate of Change

Definition and Application

The average rate of change of a function $C(x)$ from $x=a$ to $x=b$ is:

• $\frac{C(b) - C(a)}{b - a}$

Example: If $C(x) = 13x + 900$, the average rate of change from $x=2$ to $x=13$ is:

• $C(13) = 13 \times 13 + 900 = 1069$

• $C(2) = 13 \times 2 + 900 = 926$

• Average rate: $\frac{1069 - 926}{13 - 2} = \frac{143}{11} \approx 13$

Additional info: In business, this represents the average cost increase per unit sold.

Differentiation Techniques

Basic Derivative Rules

• Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$

• Sum Rule: $\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$

• Product Rule: $\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$

• Quotient Rule: $\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

Finding Derivatives

• Example: $f(x) = x$; $f'(x) = 1$

• Example: $f(x) = \frac{x^2 - 36}{x - 6}$; use quotient rule.

• Example: $F(x) = (x^2 - 2x)^3 / (x^2 - 36)$; use quotient and chain rules.

• Example: $y = \sqrt{u}$, $u = x^2 + 4$; use chain rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

Formal Definition of the Derivative

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

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

Example: For $f(x) = 3x^2 - 4x + 1$, find $f'(2)$ using the definition.

Non-Differentiable Points for Rational Functions

Identifying Points of Non-Differentiability

For rational functions, non-differentiability often occurs where the denominator is zero (vertical asymptotes).

• Example: $f(x) = \frac{x^2 - 36}{x - 6}$ is not differentiable at $x=6$ because the denominator is zero.

Tangent Lines and Slopes

Finding Horizontal Tangents

A tangent line is horizontal where the derivative is zero.

• Example: For $f(x) = x^3 - 3x^2$, set $f'(x) = 0$ to find horizontal tangents.

• $f'(x) = 3x^2 - 6x$; solve $3x^2 - 6x = 0$ for $x$.

Equation of Tangent Line

The equation of the tangent line to $f(x)$ at $x=a$ is:

• $y = f(a) + f'(a)(x - a)$

Example: For $y = \frac{8x}{x^2 + 1}$ at $(1,4)$, find $f'(1)$ and use the point-slope form.

Summary Table: Differentiation Rules

Rule	Formula	Example
Power Rule	$\frac{d}{dx} x^n = n x^{n-1}$	$\frac{d}{dx} x^3 = 3x^2$
Product Rule	$\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$	$\frac{d}{dx} [x^2 \cdot \sin x]$
Quotient Rule	$\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$	$\frac{d}{dx} \left( \frac{x^2}{x+1} \right)$
Chain Rule	$\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$	$\frac{d}{dx} \sin(x^2)$