Calculus Practice Problems: Functions, Limits, and Applications

Functions and Their Graphs

Understanding and Graphing Functions

Functions are mathematical relationships that assign each input value to exactly one output value. In calculus, analyzing and graphing functions is fundamental for understanding their behavior and properties.

• Definition: A function is a rule that assigns to each element x in a set called the domain exactly one element, f(x), in a set called the codomain.

• Graphing: To graph a function, plot points (x, f(x)) for various values of x and connect them smoothly if the function is continuous.

• Examples:

  • For $f(x) = \frac{1}{x^2 + 4}$, the graph is always positive and decreases as |x| increases.

  • For $h(x) = -x^2 + 2x + 8$, the graph is a downward-opening parabola.

Applications of Functions: Price-Demand and Revenue Models

Price-Demand Function

In economics, the price-demand function models how the price of a product affects the quantity demanded. This is often used to optimize pricing strategies and predict sales.

• Definition: The price-demand function p(x) gives the price at which x units of a product can be sold.

• Example Table: The following table shows sample price-demand data for microchips:

x (millions)	p(x)
5	75
7	63
9	54
11	47
13	41

• Graphing: Plot the data points and sketch the price-demand curve.

• Estimating Price: Use the function or table to estimate price for a given demand.

Revenue Function

The revenue function calculates total income from sales, given by multiplying price by quantity.

• Formula: $R(x) = x \cdot p(x)$

• Example Table: Revenue for each demand level:

x (millions)	R(x) (million dollars)
5	375
7	441
9	486
11	517
13	533

• Graphing: Plot revenue points and sketch the revenue curve.

• Application: Used to determine optimal production levels for maximum revenue.

Piecewise Functions and Applications

Piecewise-Defined 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 input value.

• Example: The cost function for renting a car for x days:

$C(x) = \begin{cases} 50x & \text{if } 0 < x \leq 3 \\ 150 + 40(x-3) & \text{if } 3 < x \leq 7 \\ 310 + 30(x-7) & \text{if } x > 7 \end{cases}$

• Application: Used to calculate costs that change based on quantity or time.

• Graphing: Plot each piece on its interval, noting jumps or changes in slope.

Linear Supply and Demand Models

Supply and Demand Equations

Linear models are often used to approximate supply and demand relationships in economics.

• Supply Equation: $p = m x + b$ where p is price, x is quantity, m is slope, b is intercept.

• Demand Equation: Similar form, but slope is typically negative.

• Equilibrium Point: The intersection of supply and demand curves, representing market balance.

Cost Functions and Average Cost

Total and Average Cost

Cost functions model the expenses associated with producing goods. Average cost is total cost divided by number of units produced.

• Total Cost Function: $C(x)$ gives total cost for producing x units.

• Average Cost Function: $\bar{C}(x) = \frac{C(x)}{x}$

• Application: Used to analyze efficiency and optimize production.

Compound Interest and Exponential Growth

Compound Interest Formula

Compound interest is calculated on both the initial principal and the accumulated interest from previous periods.

• Formula: $A = P \left(1 + \frac{r}{n}\right)^{nt}$

• P = principal, r = annual interest rate, n = number of compounding periods per year, t = number of years.

• Application: Used to determine future value of investments.

Doubling Time

The time required for an investment to double in value under compound interest can be found using logarithms.

• Formula: $t = \frac{\ln 2}{\ln(1 + r)}$

Limits and Continuity

Evaluating Limits

Limits describe the behavior of a function as the input approaches a particular value. They are foundational for calculus concepts such as continuity and derivatives.

• Definition: $\lim_{x \to a} f(x)$ is the value f(x) approaches as x gets close to a.

• One-Sided Limits: $\lim_{x \to a^-} f(x)$ (from the left), $\lim_{x \to a^+} f(x)$ (from the right).

• Graphical Evaluation: Use the graph to determine the value approached from each side.

• Examples:

  • Given a graph of g(x), find $g(3)$, $\lim_{x \to 3^-} g(x)$, $\lim_{x \to 3^+} g(x)$, etc.

Algebraic Limits

Limits can also be evaluated algebraically, especially for rational, radical, and piecewise functions.

• Examples:

  • $\lim_{x \to 5} \frac{x}{x^2 - 25}$

  • $\lim_{x \to 4} \sqrt{5x + 4}$

  • $\lim_{x \to 5} \frac{x^2 + 2x + 10}{x^2 - x - 5}$

  • $\lim_{x \to 5} \frac{x^2 - 4x - 21}{x^2 - x - 5}$

• Techniques: Direct substitution, factoring, rationalizing, and recognizing indeterminate forms.

Summary Table: Key Formulas

Concept	Formula
Revenue	$R(x) = x \cdot p(x)$
Average Cost	$\bar{C}(x) = \frac{C(x)}{x}$
Compound Interest	$A = P \left(1 + \frac{r}{n}\right)^{nt}$
Doubling Time	$t = \frac{\ln 2}{\ln(1 + r)}$
Limit	$\lim_{x \to a} f(x)$

Additional info: These problems cover essential calculus topics including functions, graphing, piecewise functions, economic applications, limits, and compound interest. The notes expand on brief questions to provide full academic context and formulas for exam preparation.