Calculus Practice Problems: Functions, Limits, and Applications

Functions and Their Graphs

Graphing Functions

Understanding how to graph functions is a foundational skill in calculus. The graph of a function visually represents the relationship between the input variable (usually x) and the output variable (usually f(x) or y).

• Example 1: For $f(x) = \frac{3}{x+4}$, plot points for several values of x and observe the behavior as x approaches -4 (vertical asymptote).

• Example 2: For $h(x) = x^2 - 2x + 8$, recognize this as a quadratic function (parabola opening upwards). The vertex can be found by completing the square or using $x = -\frac{b}{2a}$.

Key Points:

• Identify the type of function (rational, quadratic, etc.).

• Locate intercepts, asymptotes, and vertex (for quadratics).

• Sketch the graph based on calculated points and features.

Applications: Price-Demand and Revenue Functions

Modeling with Functions

Functions are used to model real-world relationships, such as price-demand and revenue in economics. The price-demand function relates the price of a product to the quantity demanded.

• Price-Demand Function: $p(x) = 75 - 3x$ where p is price per chip and x is the number of millions of chips demanded.

• Revenue Function: $R(x) = x \cdot p(x)$, representing total revenue for selling x million chips.

Example Table: Price-Demand Data

x (millions)	p(x)
10	45
15	30
18	21
20	15

Example Table: Revenue Data

x (millions)	R(x)
10	450
15	450
18	378
20	300

Key Steps:

• Plot data points and sketch the function graph.

• Calculate revenue for given demand values.

• Interpret the meaning of maximum revenue and its corresponding demand.

Piecewise Functions and Cost Analysis

Understanding Piecewise Functions

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

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

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

• Calculate the cost for specific values of x by determining which interval x falls into.

• Piecewise functions often appear in cost, tax, and rate problems.

Linear Supply and Demand Models

Supply and Demand Equations

Linear equations can model supply and demand in economics. The general form is $p = m x + b$, where p is price, x is quantity, m is slope, and b is intercept.

• Supply Equation: Relates price to quantity supplied.

• Demand Equation: Relates price to quantity demanded.

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

Example: If supply is 7,500,000 bushels at $2.82 per bushel, and demand is 7,900,000 bushels at $2.57 per bushel, set up equations and solve for equilibrium.

Cost Functions and Average Cost

Fixed and Variable Costs

Cost functions model the total cost of producing goods, including fixed and variable costs. The average cost function divides total cost by the number of units produced.

• Total Cost Function: $C(x)$, where x is the number of units produced.

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

Example: If fixed costs are $300 per day and total costs are $5,100 per day for 200 boards, then$C(x)$is linear in$x$.

Compound Interest and Exponential Growth

Compound Interest Formula

Compound interest is calculated using the formula:

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

• P: Principal (initial amount)

• r: Annual interest rate (decimal)

• n: Number of compounding periods per year

• t: Number of years

Example: Find the future value of $24,000 invested for 7 years at 4.35% compounded continuously:

$A = Pe^{rt}$

For continuous compounding, use the exponential function.

Limits and Continuity

Evaluating Limits

Limits are a central concept in calculus, describing the behavior of functions as the input approaches a particular value.

• Notation: $\lim_{x \to a} f(x)$

• One-sided limits: $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$

• Graphical Interpretation: Use the graph to determine the value the function approaches from the left and right.

Example: For the given graph of $g(x)$, evaluate $g(3)$, $\lim_{x \to 2^-} g(x)$, $\lim_{x \to 2^+} g(x)$, etc.

Algebraic Limits

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

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

• Factor and simplify if possible before substituting the value.

• Check for indeterminate forms (e.g., $\frac{0}{0}$) and use algebraic techniques to resolve.

Sample Limit Problems:

• $\lim_{x \to 3} \frac{x}{x^2-25}$

• $\lim_{x \to 4} \sqrt{5x+4}$

• $\lim_{x \to 5} \frac{x^2+2x+10}{x^2-4x-5}$

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

Key Steps:

• Substitute the value directly if the function is continuous at that point.

• If direct substitution yields an indeterminate form, factor or rationalize as needed.