Applications of Derivatives

Using Derivatives to Understand Functions

Derivatives turn geometric and numerical information about a function into powerful tools for analyzing its behavior. In this chapter, we focus on what you can do with derivatives, assuming you already know what a derivative is and how to compute basic derivatives from earlier chapters.

The main themes here are:

• Understanding how a function increases or decreases.
• Locating and classifying maxima and minima.
• Understanding how a graph bends (concavity and inflection).
• Connecting these ideas to simple real-world questions (like maximizing profit or minimizing cost) in the later subsections “Optimization” and “Related rates” and “Curve sketching”.

This chapter sets up the general toolbox; the named subsections will use it for specific kinds of problems.

Critical Points and Extrema

Derivatives help you find where a function reaches “peaks” and “valleys,” called extrema.

Local vs Global Extrema

Let $f$ be a function.

• A local (or relative) maximum at $x = c$ means that $f(c)$ is greater than or equal to nearby values:
  There exists an interval around $c$ such that for all $x$ in that interval,
  $$f(x) \le f(c).$$
• A local (or relative) minimum at $x = c$ means $f(c)$ is less than or equal to nearby values:
  There exists an interval around $c$ such that for all $x$ in that interval,
  $$f(x) \ge f(c).$$
• A global (or absolute) maximum on a set (for example on an interval $[a,b]$) means:
  For every $x$ in that set,
  $$f(x) \le f(c).$$
• A global (or absolute) minimum is defined similarly:
  For every $x$ in the set,
  $$f(x) \ge f(c).$$

A point may be both local and global (for example, the lowest point of a bowl-shaped graph on a closed interval).

Critical Points

A point where an extremum could occur is called a critical point.

For a differentiable function $f$:

• A number $c$ in the domain of $f$ is a critical number if
  $$f'(c) = 0 \quad \text{or} \quad f'(c) \text{ does not exist,}$$
  but $c$ is still in the domain of $f$.

The corresponding point $(c,f(c))$ on the graph is called a critical point.

Important facts:

• Local maxima or minima can occur only at critical points or at endpoints of the interval.
• Not every critical point is a maximum or minimum; it might be a “flat” point where the graph just passes through.

Often, critical points are found by:

1. Computing $f'(x)$.
2. Solving $f'(x) = 0$.
3. Checking where $f'(x)$ does not exist but $f(x)$ does.

Endpoints (if you are on a closed interval $[a,b]$) must also be checked separately.

Increasing and Decreasing Functions

Derivatives encode how a function moves: up or down.

Connection Between Sign of Derivative and Monotonicity

Let $f$ be differentiable on an interval.

• If $f'(x) > 0$ for all $x$ in the interval, then $f$ is increasing on that interval:
  $$x_1 < x_2 \implies f(x_1) < f(x_2).$$
• If $f'(x) < 0$ for all $x$ in the interval, then $f$ is decreasing on that interval:
  $$x_1 < x_2 \implies f(x_1) > f(x_2).$$
• If $f'(x) = 0$ for all $x$ in the interval, then $f$ is constant on that interval.

To analyze where $f$ is increasing or decreasing, a standard approach is:

1. Find the critical numbers of $f$.
2. Use these critical numbers to divide the real line into intervals.
3. Choose a test point in each interval and determine the sign of $f'$ there:
• $f'(x) > 0$: $f$ increases on that interval.
• $f'(x) < 0$: $f$ decreases on that interval.

This approach will be used repeatedly in optimization and curve sketching.

First Derivative Test for Local Extrema

The first derivative test uses the sign changes of $f'(x)$ around a critical point to identify local maxima and minima.

Suppose $c$ is a critical number.

• If $f'$ changes from positive to negative at $c$, then $f$ has a local maximum at $x = c$.
  (The function goes up, then down.)
• If $f'$ changes from negative to positive at $c$, then $f$ has a local minimum at $x = c$.
  (The function goes down, then up.)
• If $f'$ does not change sign (stays positive on both sides or negative on both sides), then $f$ has no local extremum at $c$.
  The point may be flat or a point of inflection, but not a peak or valley.

This test is particularly useful when $f'$ is easy to sign-check on intervals but $f''$ (second derivative) is more complicated.

Concavity and Inflection Points

Derivatives also help describe how a graph bends.

Second Derivative and Concavity

The second derivative $f''(x)$ measures how the rate of change itself is changing.

On an interval where $f''$ exists:

• If $f''(x) > 0$ for all $x$ in the interval, the graph is concave up there.
  Intuitively, the graph looks like a cup opening upwards.
• If $f''(x) < 0$ for all $x$ in the interval, the graph is concave down there.
  Intuitively, the graph looks like a cap or frown.

Concavity relates to the behavior of $f'$:

• If the graph is concave up, then $f'(x)$ is increasing on that interval.
• If the graph is concave down, then $f'(x)$ is decreasing on that interval.

This connection is helpful: to understand the shape of $f$, you can study $f'$, and to understand how $f'$ changes, you can study $f''$.

Points of Inflection

A point of inflection (or inflection point) is a point on the graph of $f$ where concavity changes:

• The graph changes from concave up to concave down, or from concave down to concave up, at that point.

At $x = c$:

• One often has $f''(c) = 0$ or $f''(c)$ undefined.
• But $f''(c) = 0$ by itself is not enough to guarantee an inflection point.
  You must also check that concavity actually changes sign: $f''$ changes from positive to negative or from negative to positive around $c$.

To test for an inflection point at $x = c$:

1. Ensure $f$ is defined at $x = c$.
2. Check that $f''$ changes sign (from $+$ to $-$ or $-$ to $+$) as $x$ passes through $c$.
3. If it does, then $(c, f(c))$ is an inflection point.

Inflection points often correspond to places where the behavior of the function changes in a meaningful way (for example, transition from slowing growth to speeding growth, or vice versa).

Second Derivative Test for Local Extrema

Sometimes you can use the second derivative directly to classify a critical point.

Assume:

• $f'(c) = 0$, so $c$ is a critical number.
• $f''(c)$ exists.

Then:

• If $f''(c) > 0$, $f$ has a local minimum at $x = c$.
  The graph is concave up and has a horizontal tangent; you are at the bottom of a bowl.
• If $f''(c) < 0$, $f$ has a local maximum at $x = c$.
  The graph is concave down and has a horizontal tangent; you are at the top of a hill.
• If $f''(c) = 0$ (or does not exist), the test is inconclusive:
  You cannot determine the type of critical point from this test alone.
  You must fall back on the first derivative test or other information.

This test is especially convenient if $f''(x)$ has a simple form and is easy to evaluate at critical points.

Using Derivatives on Closed Intervals

In many applied problems, variables are restricted to a range, often a closed interval $[a,b]$. To find absolute maxima and minima of $f$ on $[a,b]$:

1. Find all critical numbers of $f$ inside $(a,b)$ (where $f'(x) = 0$ or $f'$ does not exist, but $f$ does).
2. Evaluate $f$ at:
• Each critical number in $(a,b)$,
• The endpoints $x = a$ and $x = b$.
3. Compare the resulting values:
• The largest value is the absolute maximum on $[a,b]$.
• The smallest value is the absolute minimum on $[a,b]$.

This procedure will be used frequently in optimization problems where the domain is naturally bounded (for example, physical dimensions that must be nonnegative and not exceed some limit).

Tangent and Normal Lines

The derivative at a point also gives information about specific lines attached to the graph.

Tangent Line

At a point $x = c$ where $f$ is differentiable, the tangent line to the curve $y = f(x)$ has slope $f'(c)$ and passes through $(c, f(c))$.

An equation of the tangent line (point-slope form) is:
$$
y - f(c) = f'(c)\,(x - c).
$$

This line is the best linear approximation to $f$ near $x = c$ and plays a central role in approximations and numerical methods (covered elsewhere).

Normal Line

The normal line at $x = c$ is the line perpendicular to the tangent line at $(c,f(c))$.

• If $f'(c) \ne 0$, the slope of the tangent line is $m_{\text{tan}} = f'(c)$, so the slope of the normal line is
  $$m_{\text{norm}} = -\frac{1}{f'(c)}.$$

An equation of the normal line is:
$$
y - f(c) = -\frac{1}{f'(c)}\,(x - c).
$$

If $f'(c) = 0$, then the tangent line is horizontal and the normal line is vertical, with equation $x = c$.

These constructions are often used in geometric applications, and sometimes in optimization problems where lines must satisfy certain conditions relative to curves.

Using Derivatives to Approximate Values

While a full treatment of approximation methods is typically developed elsewhere, it is useful to note one basic application here: local linear approximation.

Near $x = a$, the function $f(x)$ can be approximated by its tangent line:

$$
f(x) \approx f(a) + f'(a)(x - a).
$$

This gives a simple way to estimate values of $f(x)$ without computing the exact value, especially when $x$ is close to $a$. This idea underlies more advanced approximation techniques and also informs how derivatives capture the “best linear model” of a function at a point.

Sketching the Overall Behavior of a Function

The separate chapters on “Optimization,” “Related rates,” and “Curve sketching” will each focus on specific uses. Here, we summarize the general derivative-based strategy for understanding the shape of a graph:

1. Domain:
• Identify where $f$ is defined (this comes from earlier material on domain).
2. First derivative:
• Compute $f'(x)$.
• Find critical numbers (solutions to $f'(x)=0$ and points where $f'$ doesn’t exist but $f$ does).
• Use intervals and test points to determine where $f$ is increasing/decreasing.
3. Second derivative:
• Compute $f''(x)$ if possible.
• Find where $f''(x) = 0$ or undefined.
• Test intervals for concavity (up or down) and identify possible inflection points.
4. Classify critical points:
• Use the first derivative test or second derivative test to decide which critical points correspond to local maxima or minima.
5. Endpoints and extreme values:
• If the problem involves a closed interval, check endpoints as well to locate absolute maxima and minima.
6. Tangent lines and special behavior:
• Identify where the tangent is horizontal ($f'(x) = 0$) or vertical (derivative undefined).
• Note asymptotic or other special behaviors if they arise from the domain or form of $f$.

By combining all this information, you can build a coherent picture of the graph even before plotting anything: where it rises or falls, where it peaks or dips, and how it bends. Subsequent chapters will take these tools and show how to apply them systematically to real problems.