Business Calculus Test 1 Review: Limits, Derivatives, Tangents, and Applications

Limits and Asymptotes

Understanding Limits and Asymptotes

Limits are fundamental in calculus, describing the behavior of functions as inputs approach specific values. Asymptotes are lines that a graph approaches but never touches, indicating the function's behavior at infinity or near undefined points.

• Horizontal Asymptote: A line $y = c$ that the graph approaches as $x \to \infty$ or $x \to -\infty$.

• Vertical Asymptote: A line $x = a$ where the function grows without bound as $x$ approaches $a$.

• Limit Notation: $\lim_{x \to a} f(x)$ denotes the value $f(x)$ approaches as $x$ nears $a$.

Example: The provided graph has a horizontal asymptote at $y=0$ and a vertical asymptote at $x=4$.

Evaluating Limits from Graphs

Limit Problems and Non-Differentiability

Limits can be estimated from graphs by observing the function's behavior near the point of interest. Non-differentiability occurs at points where the graph has sharp corners, cusps, or discontinuities.

• Finding Limits: Use the graph to determine $\lim_{x \to a} f(x)$ by checking the value approached from both sides.

• Non-Differentiability: Points where the function is not smooth or has jumps.

• Continuity: A function is continuous over an interval if there are no breaks, jumps, or holes in the graph.

Example: The function is not differentiable at $x = -4, 3, 1, 2, 4$ and is continuous over $(-2,2)$.

Derivatives and the Limit Definition

Calculating Derivatives

The derivative measures the rate of change of a function. The limit definition of the derivative is:

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

• Second Derivative: Measures the rate of change of the rate of change (concavity).

Example: For $f(x) = x^4$, the first derivative is $f'(x) = 4x^3$ and the second derivative is $f''(x) = 12x^2$.

Finding Tangent Lines

Equation of the Tangent Line

The tangent line to a function at a point $x=a$ has the equation:

• Formula: $y = f'(a)(x - a) + f(a)$

• Application: Used to approximate function values near $x=a$.

Example: For $f(x) = 4x^2 - 2x + 5$ at $x=3$, find $f'(3)$ and $f(3)$, then substitute into the formula.

Critical Points and Slope of Zero

Finding Where the Derivative is Zero

Critical points occur where the derivative equals zero, indicating possible maxima, minima, or points of inflection.

• Set $f'(x) = 0$ and solve for $x$.

• Interpretation: These points are where the function's slope is horizontal.

Example: For $f(x) = 3x^2 - 5x + 2$, set $f'(x) = 6x - 5 = 0$ to find $x = 5/6$.

Applications: Word Problems

Pool Draining Problem

Word problems in business calculus often involve rates of change and interpreting function values in context.

• Given: $g(x) = .15x^2 - 50x + 4000$ (water left in pool, $x$ in minutes)

• Average Rate of Change: $\frac{g(b) - g(a)}{b - a}$ over interval $[a, b]$

• Instantaneous Rate: $g'(x)$ at a specific $x$

Example:

• Amount after 50 min: $g(50)$

• Average rate between 50 and 100 min: $\frac{g(100) - g(50)}{100 - 50}$

• Instantaneous rate at 80 min: $g'(80)$

Hot Air Balloon Problem

Height of a balloon given by $h(x) = .0001x^3 - .024x^2 + 1.44x$ (feet, $x$ in minutes).

• Find height at $x=20$: $h(20)$

• Average speed between $x=20$ and $x=50$: $\frac{h(50) - h(20)}{50 - 20}$

• Velocity at $x=80$: $h'(80)$

• Interpretation: If $h'(80) < 0$, the balloon is descending.

Derivatives of Functions

Common Derivative Rules

Use power, product, and chain rules to find derivatives.

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

• Product Rule: $\frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

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

Example: For $g(x) = x^4 - 2x^3 + 2x - 10$, $g'(x) = 4x^3 - 6x^2 + 2$.

Table: Summary of Key Problems and Answers

Comparison of Problem Types and Solutions

Problem Type	Example	Solution
Limit from Graph	$\lim_{x \to 0} f(x)$	0
Derivative (Power Rule)	$f(x) = x^4$	$f'(x) = 4x^3$
Second Derivative	$f(x) = x^4$	$f''(x) = 12x^2$
Tangent Line at $x=3$	$f(x) = 4x^2 - 2x + 5$	$y = f'(3)(x-3) + f(3)$
Critical Point	$f(x) = 3x^2 - 5x + 2$	$x = 5/6$
Word Problem (Pool)	$g(x) = .15x^2 - 50x + 4000$	Average rate: $-27.5$ gal/min
Word Problem (Balloon)	$h(x) = .0001x^3 - .024x^2 + 1.44x$	Velocity at $x=80$: $-25.6$ ft/min (descending)

Additional info:

• Some function forms and answers were inferred from context and standard calculus practice.

• All equations are provided in LaTeX format for clarity.