Chapter 2: Applications of the Derivative – Business Calculus Study Notes

Applications of the Derivative

Chapter Outline

• Describing Graphs of Functions

• The First- and Second-Derivative Rules

• The First- and Second-Derivative Tests and Curve Sketching

• Curve Sketching (Conclusion)

• Optimization Problems

• Further Optimization Problems

• Applications of Derivatives to Business and Economics

Describing Graphs of Functions

Increasing and Decreasing Functions

Understanding how a function behaves over intervals is fundamental in calculus. A function can be classified as increasing or decreasing based on the direction of its graph.

• Increasing Function: A function $f(x)$ is increasing on an interval if, for any $x_1 < x_2$ in the interval, $f(x_1) < f(x_2)$. This means the graph rises as $x$ increases.

• Decreasing Function: A function $f(x)$ is decreasing on an interval if, for any $x_1 < x_2$ in the interval, $f(x_1) > f(x_2)$. The graph falls as $x$ increases.

• Derivative Test: $f'(x) > 0$ implies $f(x)$ is increasing; $f'(x) < 0$ implies $f(x)$ is decreasing.

Relative and Absolute Extrema

Extrema are points where a function reaches its highest or lowest values, either locally or globally.

• Relative Maximum: A point $x=a$ where $f(x)$ changes from increasing to decreasing.

• Relative Minimum: A point $x=a$ where $f(x)$ changes from decreasing to increasing.

• Absolute Maximum: The largest value of $f(x)$ on its domain.

• Absolute Minimum: The smallest value of $f(x)$ on its domain.

Changing Slope

The slope of a function's graph can change, indicating acceleration or deceleration in the rate of change. For example, if the average annual income is rising at an increasing rate, the graph's slope becomes steeper as time progresses.

Concavity

Concavity describes the direction in which a graph bends.

• Concave Up: $f''(x) > 0$; the graph lies above its tangent line and opens upward.

• Concave Down: $f''(x) < 0$; the graph lies below its tangent line and opens downward.

Inflection Points

An inflection point is where the graph changes concavity (from up to down or vice versa). It is not necessarily where the slope changes from increasing to decreasing.

Intercepts

Definition	Definition
x-Intercept: A point at which a graph crosses the x-axis.	y-Intercept: A point at which a graph crosses the y-axis.

Asymptotes

Horizontal Asymptote	Vertical Asymptote
A straight, horizontal line that a graph follows indefinitely as $x$ increases without bound. Occurs when $\lim_{x \to \infty} f(x)$ exists.	A straight, vertical line that a graph follows indefinitely as $y$ increases without bound. Occurs when the denominator equals zero, $x=a$.

6-Point Graph Description

• Intervals of increase/decrease, relative maxima/minima

• Maximum and minimum values

• Intervals of concavity, inflection points

• x-intercepts, y-intercept

• Undefined points

• Asymptotes

The First- and Second-Derivative Rules

First Derivative Rule

The first derivative indicates the slope of the function.

• If $f'(a) > 0$, $f(x)$ is increasing at $x=a$.

• If $f'(a) < 0$, $f(x)$ is decreasing at $x=a$.

Second Derivative Rule

The second derivative indicates the concavity of the function.

• If $f''(a) > 0$, $f(x)$ is concave up at $x=a$.

• If $f''(a) < 0$, $f(x)$ is concave down at $x=a$.

First & Second Derivative Scenarios

Conditions	Description	Graph
$f'(a)>0$, $f''(a)>0$	Increasing, concave up	Upward curve
$f'(a)>0$, $f''(a)<0$	Increasing, concave down	Upward but bending down
$f'(a)<0$, $f''(a)>0$	Decreasing, concave up	Downward but bending up
$f'(a)<0$, $f''(a)<0$	Decreasing, concave down	Downward curve

The First- and Second-Derivative Tests and Curve Sketching

Curve Sketching

Curve sketching involves analyzing the function and its derivatives to understand its behavior.

1. Compute $f(x)$, $f'(x)$, and $f''(x)$.

2. Locate all relative maxima and minima.

3. Study concavity and locate inflection points.

4. Consider intercepts and other properties to complete the sketch.

Critical Values

Critical values are points where $f'(x)=0$ or $f'(x)$ is undefined. These are candidates for local extrema.

First Derivative Test

• If $f'(x)$ changes from positive to negative at $x=a$, $f(x)$ has a local maximum at $a$.

• If $f'(x)$ changes from negative to positive at $x=a$, $f(x)$ has a local minimum at $a$.

• If $f'(x)$ does not change sign, there is no local extremum at $a$.

Second Derivative Test

• If $f'(a)=0$ and $f''(a)<0$, $f(x)$ has a local maximum at $a$.

• If $f'(a)=0$ and $f''(a)>0$, $f(x)$ has a local minimum at $a$.

Test for Inflection Points

Set $f''(x)=0$ and solve for $x$. Check if the concavity changes at these points to confirm inflection points.

Examples and Applications

Example: Increasing/Decreasing Functions

• Given $f(x)$, find intervals where $f'(x)>0$ (increasing) and $f'(x)<0$ (decreasing).

Example: Curve Sketching

• Given $f(x)=6x^3 + \frac{3}{2}x^2 - 3x + 3$, find critical values by solving $f'(x)=0$.

• Use the first and second derivative tests to classify extrema and sketch the graph.

Example: Drug in Bloodstream

• Use the first derivative to determine when the drug level is increasing or decreasing.

• Use the second derivative to determine concavity at a given time.

Summary Table: Key Terms

Term	Definition
Increasing Function	$f'(x)>0$
Decreasing Function	$f'(x)<0$
Relative Maximum	Change from increasing to decreasing
Relative Minimum	Change from decreasing to increasing
Concave Up	$f''(x)>0$
Concave Down	$f''(x)<0$
Inflection Point	Change in concavity
Critical Value	$f'(x)=0$ or $f'(x)$ undefined
Asymptote	Horizontal: $y=\lim_{x\to\infty}f(x)$; Vertical: $x=a$ where denominator is zero

Conclusion

Understanding the applications of the derivative is essential for analyzing and sketching graphs, identifying extrema, and solving optimization problems in business calculus. Mastery of these concepts enables students to interpret and predict the behavior of functions in various economic and business contexts.