Business Calculus: Functions, Limits, Derivatives, and Applications Study Guide

Functions and Their Properties

Definition and Representation of Functions

A function is a relation that assigns exactly one output value for each input value from its domain. Functions can be represented algebraically, graphically, or verbally.

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

• Function Notation: $f(x)$ denotes the value of the function $f$ at input $x$.

• Example: If $f(-10) = -3$ and $f(2) = -3$, $f$ can still be a function as long as each input has only one output.

Equations and Functions

Not all equations represent functions. To determine if an equation is a function, check if each input yields only one output.

• Example: $4x^2 - \frac{y}{2} = 4$ is not a function of $x$ because for some $x$ values, there may be multiple $y$ values.

• Example: $2x^4 + 5y^2 = -4$ does not represent a function since $y$ is squared and the equation is not solved for $y$.

Limits and Continuity

Definition of a Limit

The limit of a function $f(x)$ as $x$ approaches $a$ is the value that $f(x)$ gets closer to as $x$ gets closer to $a$.

• Notation: $\lim_{x \to a} f(x)$

• Existence: The limit exists if both the left-hand and right-hand limits are equal.

Continuity

A function $f(x)$ is continuous at $x = a$ if:

• $f(a)$ is defined

• $\lim_{x \to a} f(x)$ exists

• $\lim_{x \to a} f(x) = f(a)$

If any of these conditions fail, the function is not continuous at $x = a$.

Limit Problems and Applications

• Example: $\lim_{x \to -4} H(x)$ asks for the value that $H(x)$ approaches as $x$ approaches $-4$.

• Continuity Check: To check if $H$ is continuous at $x = -4$, verify the three conditions above.

• Infinite Limits: $\lim_{x \to \infty} \frac{5x^2 - 9x + 8}{4x^2 - 8x + 1}$ can be evaluated by dividing numerator and denominator by the highest power of $x$.

Domain of Functions

Finding the Domain

The domain of a function is the set of all input values for which the function is defined.

• Rational Functions: For $f(x) = \frac{x-2}{6x-12}$, the domain excludes values that make the denominator zero.

• Example: Set $6x-12 = 0 \implies x = 2$. So, the domain is all real numbers except $x = 2$.

Quadratic Functions and Their Graphs

Vertex, Axis of Symmetry, and Intercepts

A quadratic function has the form $f(x) = ax^2 + bx + c$. Its graph is a parabola.

• Vertex: The vertex is the point $(h, k)$ where $h = -\frac{b}{2a}$ and $k = f(h)$.

• Axis of Symmetry: The line $x = h$ is the axis of symmetry.

• Opens Up/Down: If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.

• y-intercept: Set $x = 0$ to find $f(0)$.

• x-intercepts: Solve $f(x) = 0$ for $x$.

• Example: For $f(x) = x^2 - 8x + 7$:

  • Vertex: $x = 4$, $f(4) = -9$

  • Axis of symmetry: $x = 4$

  • Opens upward ($a = 1 > 0$)

  • y-intercept: $f(0) = 7$

  • x-intercepts: Solve $x^2 - 8x + 7 = 0$

Derivatives and Tangent Lines

Definition of the Derivative

The derivative of a function $f(x)$, denoted $f'(x)$, measures the rate of change of $f$ with respect to $x$.

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

• Power Rule: If $f(x) = ax^n$, then $f'(x) = a n x^{n-1}$

Tangent Line to a Curve

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

• $y = f(a) + f'(a)(x - a)$

• Example: For $f(x) = 3x^2 - 9x - 66$ at $(6, -12)$, find $f'(6)$ and use the point-slope form.

Applications in Business Calculus

Compound Interest

Continuous compounding uses the formula:

• $A = Pe^{rt}$, where $A$ is the amount, $P$ is the principal, $r$ is the rate, and $t$ is time.

• Example: $P = 9900$, $r = 0.09$, $t = 4$ years.

Profit, Revenue, and Cost Functions

In business calculus, profit is the difference between revenue and cost.

• Profit Function: $P(x) = R(x) - C(x)$

• Average Unit Profit: $\overline{P}(x) = \frac{P(x)}{x}$

• Example: If $C(x) = 77x + 1900$ and $R(x) = 146x$, then $P(x) = 146x - (77x + 1900)$

Average Rate of Change

The average rate of change of a function $f$ from $x = a$ to $x = b$ is:

• $\frac{f(b) - f(a)}{b - a}$

• Application: Used to measure changes in price, cost, or other business metrics over time.

Summary Table: Key Concepts

Concept	Definition	Formula/Example
Function	Relation with one output per input	$f(x)$
Limit	Value approached as $x$ nears $a$	$\lim_{x \to a} f(x)$
Continuity	No breaks or jumps at $x = a$	$\lim_{x \to a} f(x) = f(a)$
Derivative	Instantaneous rate of change	$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
Compound Interest	Interest compounded continuously	$A = Pe^{rt}$
Profit	Revenue minus cost	$P(x) = R(x) - C(x)$
Average Rate of Change	Change in value per unit interval	$\frac{f(b) - f(a)}{b - a}$

Additional info: Some explanations and examples have been expanded for clarity and completeness, based on standard business calculus curriculum.