Business Calculus: Sample Problems on Derivatives, Graphing, and Optimization

Derivatives and Their Applications

Second Derivative of Functions

The second derivative of a function provides information about the function's concavity and points of inflection. It is especially useful for analyzing exponential, logarithmic, rational, and radical functions.

• Definition: The second derivative of a function $f(x)$ is denoted as $f''(x)$ and is the derivative of the first derivative $f'(x)$.

• Concavity: If $f''(x) > 0$, the function is concave up; if $f''(x) < 0$, it is concave down.

• Inflection Points: Points where $f''(x) = 0$ and the concavity changes.

• Example: For $f(x) = x^3$, $f'(x) = 3x^2$, $f''(x) = 6x$. The inflection point is at $x = 0$.

First Derivative and Sign Charts

The first derivative $f'(x)$ indicates where a function is increasing or decreasing. A sign chart helps visualize these intervals.

• Increasing/Decreasing: If $f'(x) > 0$, the function increases; if $f'(x) < 0$, it decreases.

• Critical Points: Where $f'(x) = 0$ or is undefined; potential maxima, minima, or inflection points.

• Example: For $f(x) = x^2 - 4x + 3$, $f'(x) = 2x - 4$. Setting $f'(x) = 0$ gives $x = 2$ as a critical point.

Graphing and Optimization

Concavity and Inflection Points

Analyzing the second derivative allows us to determine where a function is concave up or down and to locate inflection points.

• Concave Up: $f''(x) > 0$

• Concave Down: $f''(x) < 0$

• Inflection Point: $f''(x) = 0$ and concavity changes

Absolute Maximum and Minimum on a Closed Interval

To find the absolute extrema of a function on a closed interval, evaluate the function at critical points and endpoints.

• Steps:

  1. Find $f'(x)$ and solve $f'(x) = 0$ for critical points.

  2. Evaluate $f(x)$ at critical points and endpoints.

  3. Compare values to determine absolute maximum and minimum.

• Example: For $f(x) = x^2$ on $[0,2]$, $f'(x) = 2x = 0$ at $x=0$. Evaluate $f(0)=0$, $f(2)=4$; minimum at $x=0$, maximum at $x=2$.

Tangent Line and Intervals of Increase/Decrease

The slope of the tangent line at a point $x_0$ is $f'(x_0)$. The sign of $f'(x)$ determines intervals of increase or decrease.

• Tangent Line Equation: $y = f(x_0) + f'(x_0)(x - x_0)$

• Intervals: Use sign chart for $f'(x)$ to find where function increases or decreases.

Special Functions: Exponential, Logarithmic, and Rational

Behavior of Exponential, Logarithmic, and Rational Functions

These functions have unique properties regarding their derivatives and graphs.

• Exponential: $f(x) = e^x$, $f'(x) = e^x$

• Logarithmic: $f(x) = ext{ln}(x)$, $f'(x) = 1/x$

• Rational: $f(x) = 1/x$, $f'(x) = -1/x^2$

• Example: For $f(x) = e^x$, both the function and its derivative are always positive, so the function is always increasing.

Graphing and Intersections

Graphing Functions and Their Intersections

To graph functions and find intersections, set the functions equal and solve for $x$.

• Intersection: Solve $f(x) = g(x)$ for $x$.

• Example: $e^x = x^2$; solve for $x$ numerically or graphically.

Optimization and Applied Problems

Finding Maximum and Minimum Points

Optimization involves finding the highest or lowest value of a function, often subject to constraints.

• Steps:

  1. Find $f'(x)$ and solve $f'(x) = 0$ for critical points.

  2. Use $f''(x)$ to determine if each critical point is a maximum or minimum.

  3. Check endpoints if the domain is restricted.

• Example: Maximize $f(x) = -x^2 + 4x$; $f'(x) = -2x + 4 = 0$ gives $x = 2$ as maximum.

Probability and Normal Distribution

Some problems involve finding probabilities using the normal distribution, which is a continuous probability distribution.

• Normal Distribution: $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

• Application: Find the probability that a value falls within a certain interval using the cumulative distribution function (CDF).

Applied Optimization Problems

Applied problems may ask for the maximum profit or minimum cost, given a function and constraints.

• Steps:

  1. Express the quantity to be optimized as a function.

  2. Find the derivative and solve for critical points.

  3. Interpret the solution in the context of the problem.

• Example: Given a profit function $P(x)$, find $P'(x)$ and solve $P'(x) = 0$ to find the production level that maximizes profit.

Summary Table: Derivative Properties of Common Functions

Function Type	Function	First Derivative	Second Derivative
Exponential	$f(x) = e^x$	$f'(x) = e^x$	$f''(x) = e^x$
Logarithmic	$f(x) = \ln(x)$	$f'(x) = 1/x$	$f''(x) = -1/x^2$
Rational	$f(x) = 1/x$	$f'(x) = -1/x^2$	$f''(x) = 2/x^3$
Radical	$f(x) = \sqrt{x}$	$f'(x) = 1/(2\sqrt{x})$	$f''(x) = -1/(4x^{3/2})$