Business Calculus Study Guide: Functions, Derivatives, Applications, and Problem Solving

Business Calculus: Core Concepts and Applications

Functions in Business Calculus

Functions are mathematical relationships that assign each input exactly one output. In business calculus, functions are used to model cost, revenue, profit, and other economic quantities.

• Cost Function (C(x)): Represents the total cost of producing x units.

• Revenue Function (R(x)): Represents the total income from selling x units.

• Profit Function (P(x)): Defined as $P(x) = R(x) - C(x)$ or, if selling price per unit is p(x), $P(x) = p(x) imes x - C(x)$.

Example: Given $C(x) = -0.03x^2 + 70x + 75$ and $p(x) = 80$, the profit function is $P(x) = 80x - (-0.03x^2 + 70x + 75) = 0.03x^2 + 10x + 75$.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain. They are useful for modeling situations where a rule changes based on the input value.

• Definition: A function $f(x)$ is piecewise if it is defined by multiple sub-functions, each applying to a certain interval.

• Example: $y = egin{cases} x + 3, & -1 ext{ to } x ext{ to } 2 \\ 3x - 1, & 1 ext{ to } x ext{ to } 3 \\ ext{Additional info: The graph in Problem 2 shows two linear segments, each with its own formula.}$

Intervals of Increase and Decrease

To determine where a function is increasing or decreasing, analyze its derivative.

• Increasing Interval: Where $g'(x) > 0$.

• Example: For $g(x) = 3x - 6 + 2$, $g'(x) = 3$, so the function is increasing for all $x$.

Optimization Problems

Optimization involves finding the maximum or minimum value of a function, often subject to constraints. In business, this is used to minimize costs or maximize profits.

• Example: A company must lay cable across a lake and along a road, with different costs per foot. The total cost function is minimized by choosing the optimal point to switch from lake to road.

• Formula: If $C = 250y + 150x$, where $y$ is distance under lake and $x$ is distance along road, use calculus to minimize $C$ given constraints.

Transformations of Functions

Transformations change the shape, position, or orientation of a graph.

• Vertical Stretch/Compression: $y = af(x)$ stretches by $a$.

• Horizontal Stretch/Compression: $y = f(bx)$ compresses by $1/b$.

• Shifts: $y = f(x-h) + k$ shifts right by $h$, up by $k$.

• Example: $h(x) = 4 oot{3}{x} - 6$ is a vertical stretch by 4, right shift by 3, down by 6.

Composite Functions

Composite functions combine two or more functions, applying one after the other.

• Notation: $(a ig(b ig(c(x)ig)ig))$ means apply $c$ to $x$, then $b$ to the result, then $a$.

• Example: If $a(x) = -x^3$, $b(x) = ext{cos}(x)$, $c(x) = oot{x}$, then $a(b(c(x))) = -( ext{cos}( oot{x}))^3$.

Logarithms and Natural Logarithms

Logarithms are the inverse of exponentials. The natural logarithm, $ext{ln}(x)$, uses base $e$.

• Change of Base Formula: $ext{log}_b(a) = rac{ ext{ln}(a)}{ ext{ln}(b)}$

• Example: $ext{log}_5(120) = rac{ ext{ln}(120)}{ ext{ln}(5)} hickapprox 2.9746$

Limits and Continuity

Limits describe the behavior of functions as inputs approach a value. Continuity means a function has no breaks or jumps.

• Limit Definition: $ext{lim}_{x o a} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$.

• Intervals of Continuity: A function is continuous on intervals where its graph is unbroken.

• Example: For a step graph, intervals of continuity are the ranges where the function does not jump.

Inverse Functions

The inverse of a function reverses its input and output.

• Finding the Inverse: Solve $y = f(x)$ for $x$ in terms of $y$.

• Example: For $f(x) = -5x + 3$, the inverse is $f^{-1}(x) = rac{3-x}{5}$.

Average Rate of Change and Velocity

The average rate of change measures how a quantity changes over an interval. In motion problems, this is average velocity.

• Formula: $ext{Average velocity} = rac{s(t_2) - s(t_1)}{t_2 - t_1}$

• Example: If $s(0) = 0$, $s(2) = 28$, then average velocity from $0$to $2$ is $rac{28-0}{2-0} = 14$ m/s.

Derivatives: Definition and Applications

The derivative measures the instantaneous rate of change of a function. It is foundational in calculus for finding slopes, rates, and optimization.

• Definition: $f'(x) = ext{lim}_{h o 0} rac{f(x+h) - f(x)}{h}$

• Example: For $f(x) = 2x^3$, $f'(x) = 6x^2$

• Higher Order Derivatives: The nth derivative is found by differentiating n times.

• Example: The fourth derivative of $f(x) = 5 ext{cos}(x)$ is $f^{(4)}(x) = 5 ext{cos}(x)$

Piecewise Differentiability

A piecewise function is differentiable at a point if both pieces are continuous and have matching derivatives at that point.

• Example: For $g(x) = egin{cases} 4x + ext{sin}(x), x < 0 \\ 2x^2, x ext{≥} 0 \\ ext{Check continuity and derivative at } x = 0$

Product and Chain Rule

These rules are used to differentiate products and compositions of functions.

• Product Rule: $(fg)' = f'g + fg'$

• Chain Rule: If $y = f(g(x))$, then $y' = f'(g(x))g'(x)$

Applications: Business and Economics

Calculus is used in business to optimize profit, minimize cost, and analyze rates of change.

• Profit Maximization: Set $P'(x) = 0$ to find critical points.

• Cost Minimization: Use derivatives to find minimum cost points.

• Marginal Analysis: Marginal cost, revenue, and profit are derivatives of their respective functions.

Tables: Data Analysis and Interpretation

Tables are used to present data for analysis, such as position over time or cost over quantity.

t (s)	s(t) (m)
0	0
1	20
2	28
4	62
5	82

Purpose: To calculate average velocity or rate of change over intervals.

Summary Table: Key Formulas and Concepts

Concept	Formula	Application
Profit	$P(x) = R(x) - C(x)$	Business profit calculation
Derivative	$f'(x) = ext{lim}_{h o 0} rac{f(x+h) - f(x)}{h}$	Rate of change
Average Velocity	$rac{s(t_2) - s(t_1)}{t_2 - t_1}$	Motion problems
Logarithm (change of base)	$ext{log}_b(a) = rac{ ext{ln}(a)}{ ext{ln}(b)}$	Logarithmic calculations
Product Rule	$(fg)' = f'g + fg'$	Differentiation
Chain Rule	$y' = f'(g(x))g'(x)$	Differentiation

Additional info: These problems cover essential business calculus topics including functions, derivatives, optimization, logarithms, piecewise functions, and applications to business scenarios.