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Business Calculus Study Guide: Derivatives, Function Behavior, and Applications

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Section 1.7: Derivatives and Higher-Order Derivatives

Partial Derivatives for Multivariable Functions

When a function depends on more than one variable, partial derivatives measure the rate of change with respect to one variable while keeping others constant.

  • Definition: The partial derivative of a function with respect to is denoted .

  • Example: For :

  • Application: Partial derivatives are used in optimization and modeling in business and economics.

Higher-Order Derivatives

Higher-order derivatives describe how the rate of change itself changes. The second derivative measures curvature, and the third derivative measures the rate of change of curvature.

  • Notation: is the second derivative, is the third derivative.

  • Example 1:

  • Example 2:

  • Example 3:

Evaluating Derivatives at Specific Points

To find the value of a derivative at a specific input, substitute the value into the derivative formula.

  • Example:

  • Example: , find at

Section 1.8: Rates of Change and Applications

Average vs. Instantaneous Rate of Change

The average rate of change measures the change over an interval, while the instantaneous rate of change (derivative) measures the change at a single point.

  • Average Speed: , where is distance and is time.

  • Instantaneous Velocity: , the derivative of distance with respect to time.

  • Example:

    • Average speed from to :

    • Instantaneous speed at :

Derivatives as Instantaneous Rates: Distance, Velocity, Acceleration

Derivatives relate position, velocity, and acceleration. In business calculus, these concepts model change in cost, revenue, or other variables.

  • Velocity:

  • Acceleration:

  • Example:

    • At :

      • ft

      • ft/sec (downward)

      • ft/sec2

    • Initial velocity and position: ft/sec, ft

    • Object on ground: sec and sec

    • Object at 160 ft: sec ( ft/sec), sec ( ft/sec)

    • Speed 16 ft/sec: sec (upward), sec (downward)

    • Maximum height: sec, ft

    • Velocity at impact: ft/sec

Derivative Approximations for Small Changes

Derivatives can estimate the change in a function's output for small changes in input using linear approximation.

  • Formula:

  • Example: ,

Section 2.1: Function Behavior Across Domain

Characteristics of Functions

Understanding how a function behaves across its domain is essential for analyzing business models and optimization problems.

  • Intervals of Increase/Decrease: Where the function is rising or falling.

  • Maxima/Minima: Highest or lowest points (relative or absolute).

  • Concavity: Upward (concave up) or downward (concave down) curvature.

  • Inflection Points: Where concavity changes.

  • Asymptotes: Lines the function approaches but never crosses.

  • Intercepts: Where the function crosses the axes.

Example Table: Function Characteristics

Characteristic

Value/Interval

Interval(s) of Increase

Interval(s) of Decrease

Relative Maxima

Relative Minima

None

Interval(s) of Upward Concavity

None

Interval(s) of Downward Concavity

Inflection Point(s)

None

x-intercept(s)

and

y-intercept(s)

Horizontal Asymptote(s)

None

Vertical Asymptote(s)

None

Section 2.2: First and Second Derivative Rules

First Derivative and Function Behavior

The first derivative indicates where a function is increasing or decreasing.

  • If : Function is increasing.

  • If : Function is decreasing.

  • If : Possible maximum or minimum (critical point).

  • Example: for means is decreasing on .

  • for means is increasing on .

  • If exists and at , then has a relative minimum at .

Second Derivative and Concavity

The second derivative describes the curvature of the function.

  • If : Function is concave up (shaped like a cup).

  • If : Function is concave down (shaped like a cap).

  • Inflection Point: Where changes sign, the function changes concavity.

  • Example: for means is concave up on .

  • for means is concave down on .

  • At , has an inflection point or possibly a vertical asymptote.

Sketching Functions Based on Derivative Information

Given derivative information, you can sketch the general shape of a function.

  • Example 1: has y-intercept , is increasing and concave up on .

  • Example 2: has zero slope at , is decreasing and concave up on , decreasing and concave down on .

Interpreting Values of Function and Derivatives at a Point

At a specific point, the sign and value of , , and indicate position, slope, and curvature.

  • If : Function value is negative.

  • If : Slope is negative (decreasing).

  • If : Concave down.

  • If : Function crosses the axis.

  • If : Slope is zero (possible extremum).

  • If : Possible inflection point.

  • If : Function value is positive.

  • If : Slope is positive (increasing).

  • If : Concave up.

Additional info: These concepts are foundational for business calculus, especially in optimization, marginal analysis, and modeling change in economic variables.

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