Business Calculus Exam 1 Study Guide: Limits, Derivatives, and Marginal Analysis

Limits and One-Sided Limits

Understanding Limits from Graphs

Limits describe the behavior of a function as the input approaches a particular value. One-sided limits consider the approach from only one direction (left or right).

• Limit from the Right (Right-Hand Limit): $\lim_{x \to a^+} f(x)$ is the value that f(x) approaches as x approaches a from the right (values greater than a).

• Limit from the Left (Left-Hand Limit): $\lim_{x \to a^-} f(x)$ is the value that f(x) approaches as x approaches a from the left (values less than a).

• Two-Sided Limit: $\lim_{x \to a} f(x)$ exists if and only if both one-sided limits exist and are equal.

• Function Value: $f(a)$ is the actual value of the function at x = a.

Key Fact: If the left and right limits are different, the two-sided limit does not exist at that point.

• Example: Given a graph, find $f(-3)$, $\lim_{x \to -3^+} f(x)$, $\lim_{x \to -3^-} f(x)$, and $\lim_{x \to -3} f(x)$, and repeat for other points such as $x=2, 0, 4$.

Algebraic Limits and Indeterminate Forms

Finding Limits Algebraically

Limits can often be found by direct substitution. If substitution leads to an indeterminate form (such as $0/0$), algebraic manipulation is required.

• Indeterminate Form: An expression like $0/0$ or $\infty/\infty$ that requires further simplification.

• Algebraic Techniques: Factor numerator and denominator, cancel common factors, and then substitute the value.

• Example: For $f(x) = \frac{2x^2 - 3x - 2}{x^2 + x - 6}$, find:

  • $\lim_{x \to 2} f(x)$

  • $\lim_{x \to 0} f(x)$

  • $\lim_{x \to 1} f(x)$

Hint: Factor both numerator and denominator to simplify before substituting.

Polynomial and Rational Inequalities

Solving Inequalities Using Sign Charts

To solve inequalities involving polynomials or rational functions, identify where the function is positive or negative.

• Partition Numbers: Values where the numerator or denominator is zero. These divide the real line into intervals.

• Sign Chart: Test a value from each interval to determine the sign of the function in that interval.

• Interval Notation: Express the solution set using intervals, e.g., $(-\infty, -2) \cup (0, 3)$.

• Example: For $f(x) = \frac{x-4}{x^2 + 2x}$:

  1. Find partition numbers (set numerator and denominator to zero).

  2. Make a sign chart for $f(x)$.

  3. Find the solution for $f(x) < 0$ and $f(x) > 0$ using interval notation.

Table: Example Sign Chart

Interval	Test Value	Sign of $x-4$	Sign of $x^2+2x$	Sign of $f(x)$
$(-\infty, -2)$	-3	-	+	-
$(-2, 0)$	-1	-	-	+
$(0, 4)$	2	-	+	-
$(4, \infty)$	5	+	+	+

Additional info: Table entries inferred for illustration; actual test values should be calculated as shown in class.

Definition of the Derivative (4-Step Process)

Derivative as a Limit

The derivative of a function at a point measures the instantaneous rate of change. It is defined as a limit.

• Definition: The derivative of $f(x)$ at $x=a$ is

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

• 4-Step Process:

  1. Compute $f(x+h)$

  2. Form the difference $f(x+h) - f(x)$

  3. Divide by $h$

  4. Take the limit as $h \to 0$

• Example: Find the derivative of $f(x) = 2x^2 + 8$ using the 4-step process.

Additional info: The result should match the derivative found using differentiation formulas.

Differentiation Formulas

Basic Rules for Finding Derivatives

Once the definition is understood, use formulas for efficiency. Rewrite functions as sums of power functions before applying rules.

• Constant Rule: $\frac{d}{dx}(c) = 0$

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

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

• Examples:

  • $\frac{d}{dx}(x) = 1$

  • $\frac{d}{dx}(\frac{1}{x}) = \frac{d}{dx}(x^{-1}) = -x^{-2}$

  • $\frac{d}{dx}(\sqrt{x}) = \frac{d}{dx}(x^{1/2}) = \frac{1}{2} x^{-1/2}$

  • $\frac{d}{dx}(4x^2 - 7x + 5) = 8x - 7$

  • $\frac{d}{dx}(3x - \frac{5}{x} + 2\sqrt{x}) = 3 + 5x^{-2} + x^{-1/2}$

  • For $f(x) = 4\sqrt{x} + 2$, $f'(x) = 2x^{-1/2}$

  • For $y = \frac{5x^3}{4} - \frac{2}{5}x^2$, $y' = \frac{15}{4}x^2 - \frac{4}{5}x$

  • For $f(x) = \frac{2x^5 - 4x^3 + 2x}{x^3}$, rewrite as $2x^2 - 4x + 2x^{-2}$, then differentiate.

Marginal Analysis in Business Applications

Revenue and Marginal Revenue

Marginal analysis uses derivatives to study how revenue changes as the number of items sold changes.

• Price-Demand Equation: Relates price p to demand x. Example: $x = 4000 - 40p$

• Express Price in Terms of Demand: Solve for p in terms of x:

$p = \frac{4000 - x}{40}$

• Revenue Function: $R(x) = x \cdot p(x)$

• Domain: Both price and demand must be nonnegative. Find the range of x for which $p \geq 0$ and $x \geq 0$.

• Marginal Revenue: The derivative $R'(x)$ gives the rate of change of revenue with respect to sales volume.

• Interpretation: $R'(a)$ estimates the additional revenue from selling one more unit at $x = a$.

• Example:

  1. Given $x = 4000 - 40p$, solve for $p$ in terms of $x$.

  2. Find $R(x)$ and its domain.

  3. Compute $R'(x)$ and evaluate at $x = 1600$ and $x = 2500$.

  4. Interpret the results: Is revenue increasing or decreasing? What does the marginal revenue mean in context?

Additional info: Marginal analysis is a key application of derivatives in business calculus, helping firms optimize production and pricing decisions.