Applications of the Derivative

Applications of the Derivative

$Learning Objectives$

Use derivatives to find tangent lines, normal lines, and angles of intersection between curves.
Apply related rates to solve problems involving multiple variables changing with respect to time.
Analyze rectilinear motion and compute velocity, speed, and acceleration.
Classify interior critical points as local maxima, local minima, or neither using the first and second derivative tests; distinguish these local results from absolute extrema.
Apply marginal analysis to economic functions and use L'Hopital's Rule to evaluate indeterminate limits.

Calculus is not just about computing derivatives; it's about using them to understand how quantities change. We use derivatives to optimize systems (finding max profit, min cost), analyze motion (velocity, acceleration), calculate marginal analysis in economics, and solve related rates problems.

Tangent and Normal Lines

$Tangent Line$

The tangent line to a curve $y = f(x)$ at $x = c$ is the best linear approximation of the curve at that point. Its slope is $m = f'(c)$ . Equation: $y - f(c) = f'(c)(x - c)$

$Normal Line$

The normal line is perpendicular to the tangent line at the point of tangency. Its slope is the negative reciprocal: $-1/f'(c)$ . Equation: $y - f(c) = -\frac{1}{f'(c)}(x - c)$

Angle of Intersection Between Curves

When two curves intersect, the angle

\theta

between them is defined as the angle between their respective tangent lines at the point of intersection.

$How to Find the Angle of Intersection$

STEP-BY-STEP

Find the intersection point(s) of the two curves.
Calculate the slopes $m_1 = f'(x)$ and $m_2 = g'(x)$ at the point of intersection.
Use the tangent addition formula to find the acute angle $\theta$ .

Angle of Intersection Formula

Angle between two intersecting curves.

\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|

Variables

Symbol	Description	Unit
$\theta$	Acute angle of intersection	-
$m_1, m_2$	Slopes of the tangent lines	-

Rates of Change and Rectilinear Motion

$Rectilinear Motion$

Motion that occurs along a straight line.

s(t)

represents the position of an object moving along a straight line at time

t

Kinematic Equations

The fundamental equations relating position, velocity, speed, and acceleration in rectilinear motion.

v(t) = s'(t)

a(t) = v'(t) = s''(t)

\text{Speed} = |v(t)|

Variables

Symbol	Description	Unit
$s(t)$	Position function with respect to time	-
$v(t)$	Velocity function (Instantaneous rate of change of position)	-
$a(t)$	Acceleration function (Rate of change of velocity)	-
$\|v(t)\|$	Speed (Magnitude of velocity)	-

Marginal Analysis in Economics

In economics, the term "marginal" refers to a derivative. It represents the approximate change in a quantity resulting from a one-unit increase in production or sales.

$C(x)$

The total cost of producing $x$ units.

$C'(x)$

The instantaneous rate of change of cost, roughly approximating the cost of producing the next unit.

$R(x)$

The total revenue from selling $x$ units.

$R'(x)$

The instantaneous rate of change of revenue.

Profit Optimization

Maximum profit occurs when Marginal Revenue equals Marginal Cost.

P(x) = R(x) - C(x)

R'(x) = C'(x)

Variables

Symbol	Description	Unit
$P(x)$	Profit function	-
$R(x)$	Revenue function	-
$C(x)$	Cost function	-
$R'(x)$	Marginal revenue	-
$C'(x)$	Marginal cost	-

Related Rates

In related rates problems, we relate the rates of change of two or more variables that are changing with respect to time. The key is to differentiate implicitly with respect to time

t

$Common Strategy for Related Rates$

STEP-BY-STEP

Draw a picture and label variables.
Write an equation relating the variables.
Differentiate implicitly with respect to time ( $t$ ).
Substitute known values after differentiating.
Solve for the unknown rate.

$Interactive Simulation$

Interact with the simulation below to explore related rates using a sliding ladder context.

Related Rates: Sliding Ladder

Distance from Wall (x)

Length (L)10 m

x6.00 m

y8.00 m

Assume dx/dt = 2 m/s

dy/dt-1.50 m/s

Notice how dy/dt (speed of top of ladder) increases dramatically as x approaches L (bottom pulls away).

$Interactive Simulation$

Interact with the simulation below to explore related rates using a conical hydraulic reservoir context.

Hydraulic Related Rates: Conical Reservoir

Depth 4.0 mRadius 2.00 m

dh/dt

= 0.1194 m/min

Controls

Tank height

H

10 m

Top radius

R

5.0 m

Inflow

Q=dV/dt

1.5 m³/min

Water depth

h

4.0 m

Tank visual

Physical cross-section · uniform scale

Current r

2.00 m

Water V

16.8 m³

dh/dt

0.1194 m/min

Rate graph

Depth $h$ vs. rise rate $dh/dt$

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Optimization: Maxima and Minima

Finding the best (optimal) value is a core engineering task, such as finding the maximum volume or minimum cost. Derivative tests classify local behavior, while absolute optimization requires comparing values over the full feasible domain.

$Critical Point$

A point $c$ in the domain of $f$ is a critical point if $f'(c)=0$ or $f'(c)$ does not exist.

$Absolute vs. Relative Extrema$

Absolute (Global) Maximum: The absolute highest value of $f(x)$ over its entire domain.
Absolute (Global) Minimum: The absolute lowest value of $f(x)$ over its entire domain.
Relative (Local) Maximum: A peak in the graph. The value $f(c)$ is higher than all nearby values in an open interval containing $c$ .
Relative (Local) Minimum: A valley in the graph. The value $f(c)$ is lower than all nearby values in an open interval containing $c$ .

$Classifying Interior Local Extrema$

STEP-BY-STEP

Consider only a point $c$ in the interior of the domain. A critical point must belong to the domain and satisfy $f'(c)=0$ or have $f'(c)$ undefined; it is only a candidate for a local extremum.
Apply the first derivative test when $f$ $f$ is continuous on an open interval containing $c$ $c$ and differentiable there except possibly at $c$ $c$ . Compare the sign of $f'$ $f^{'}$ immediately to the left and right:
- Positive to negative gives a local maximum.
- Negative to positive gives a local minimum.
- The same strict nonzero sign on both sides gives neither a local maximum nor a local minimum.
- If the hypotheses or a definite one-sided sign pattern are unavailable, the test does not classify the point.
At a stationary point where $f'(c)=0$ $f^{'} (c) = 0$ , use the second derivative test only when $f''$ $f^{''}$ is continuous on an open interval containing $c$ $c$ :
- $f''(c)>0$ gives a local minimum.
- $f''(c)<0$ gives a local maximum.
- $f''(c)=0$ is inconclusive; use the first derivative test or the definition of a local extremum.

Classifying Interior Local Extrema

Use this decision path to classify interior local behavior at an in-domain point c of a real-valued function on an interval.

A critical point is an in-domain point c where f'(c)=0 or f'(c) is undefined. This chart classifies only interior local extrema. The first-derivative test requires f to be continuous on an open interval containing c and differentiable there except possibly at c; compare the signs of f' on intervals immediately to each side. Positive-to-negative gives a local maximum, negative-to-positive gives a local minimum, and the same strict nonzero sign of f′ on both sides gives neither. The second-derivative test applies only when f'(c)=0 and f'' is continuous on an open interval containing c: f''(c)>0 gives a local minimum, f''(c)<0 a local maximum, and f''(c)=0 is inconclusive. Boundary points require one-sided local analysis. Neither derivative test proves an absolute/global extremum; that requires comparing values over the full domain, including endpoints when they belong to it.

Choose a point c in the domain of f → Is c an interior point?; Is c an interior point? — Yes → Is f'(c)=0 or undefined?; Is c an interior point? — No → Boundary point: use one-sided local analysis; check separately for absolute extrema; Is f'(c)=0 or undefined? — Yes → Is f'(c)=0 and f'' continuous near c?; Is f'(c)=0 or undefined? — No → No interior extremum: f'(c) exists and is nonzero; Is f'(c)=0 and f'' continuous near c? — Yes → Is f''(c)>0?; Is f'(c)=0 and f'' continuous near c? — No → Are first-derivative test hypotheses met?; Is f''(c)>0? — Yes → Interior local minimum by second-derivative test; Is f''(c)>0? — No → Is f''(c)<0?; Is f''(c)<0? — Yes → Interior local maximum by second-derivative test; Is f''(c)<0? — No → f''(c)=0: test inconclusive; use first-derivative test or definition; Are first-derivative test hypotheses met? — Yes → Is f' positive left and negative right?; Are first-derivative test hypotheses met? — No → First-derivative test unavailable: use another valid method; Is f' positive left and negative right? — Yes → Interior local maximum by first-derivative test; Is f' positive left and negative right? — No → Is f' negative left and positive right?; Is f' negative left and positive right? — Yes → Interior local minimum by first-derivative test; Is f' negative left and positive right? — No → Is f′ the same strict nonzero sign on both sides?; Is f′ the same strict nonzero sign on both sides? — Yes → Neither local maximum nor local minimum; Is f′ the same strict nonzero sign on both sides? — No → No listed sign pattern: use the local definition or another method

Choose a point c in the domain of f: terminator
Is c an interior point?: decision
Is f'(c)=0 or undefined?: decision
Boundary point: use one-sided local analysis; check separately for absolute extrema: terminator
No interior extremum: f'(c) exists and is nonzero: terminator
Is f'(c)=0 and f'' continuous near c?: decision
Are first-derivative test hypotheses met?: decision
Is f''(c)>0?: decision
Is f' positive left and negative right?: decision
First-derivative test unavailable: use another valid method: terminator
Is f''(c)<0?: decision
Interior local minimum by second-derivative test: terminator
Is f' negative left and positive right?: decision
Interior local maximum by first-derivative test: terminator
Interior local maximum by second-derivative test: terminator
f''(c)=0: test inconclusive; use first-derivative test or definition: terminator
Is f′ the same strict nonzero sign on both sides?: decision
Interior local minimum by first-derivative test: terminator
Neither local maximum nor local minimum: terminator
No listed sign pattern: use the local definition or another method: terminator

$Interactive Simulation$

Interact with the simulation below to explore a classic optimization problem: constructing an open-top box with maximum volume from a flat sheet.

Box Optimization Problem

Cut Size (x): 2.00

Length16.00

Width16.00

Height (x)2.00

Volume512.0

Max Volume occurs at x ≈ 3.33

Current % of Max: 86.4%

Curve Sketching

Derivatives provide powerful tools for understanding the shape of a graph without plotting points manually.

$First and Second Derivative Tests$

Increasing/Decreasing: If $f'(x) > 0$ , the curve is increasing. If $f'(x) < 0$ , it is decreasing.
Concavity: The second derivative $f''(x)$ $f^{''} (x)$ tells us about concavity.
- $f''(x) > 0$ : Concave Up (looks like a cup U).
- $f''(x) < 0$ : Concave Down (looks like a frown).
Point of Inflection: A point where the concavity changes (e.g., from concave up to concave down). This occurs where $f''(x) = 0$ or is undefined, and the sign of $f''(x)$ actually changes across the point.

L'Hopital's Rule

Used to evaluate limits of indeterminate forms

0/0

\infty/\infty

. It states that the limit of the ratio of functions is the limit of the ratio of their derivatives.

L'Hopital's Rule

Limit of a ratio of functions.

\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}

Variables

Symbol	Description	Unit
$f(x), g(x)$	Differentiable functions producing indeterminate forms	-
$c$	The limit point	-

Civil Engineering Optimization: Pipeline and Canal Routing

In infrastructure design, civil engineers utilize optimization to find the most cost-effective path for utility pipelines, roads, or canals crossing multiple regions (such as dry land versus riverbeds) where excavation and construction costs differ.

$Interactive Simulation$

Interact with the simulation below to find the mathematically optimal path that minimizes total project cost based on cost-per-meter ratios.

Canal Optimization

Adjust the depth $y$ of the trapezoidal canal. The total area is held constant at 50 m². Notice how the wetted perimeter $P$ changes, reaching a minimum when the cross-section approaches a half-hexagon.

Depth

y

: 5.37 m

Cross-Section Properties:

Area A: 50.0 m² (Constant)

Depth y: 5.37 m

Bottom Width b: 6.21 m

Wetted Perimeter P: 18.61 m

Optimal Design (Minimum P)

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Key Takeaways

Tangent lines approximate curves locally. The angle of intersection between two curves uses the slopes of their tangent lines.
Velocity and acceleration are the first and second derivatives of position in rectilinear motion.
Marginal Analysis uses the derivative to find the incremental cost or revenue, maximizing profit where $R'(x) = C'(x)$ .
Related Rates problems require implicit differentiation with respect to time. Always substitute after differentiating.
Optimization uses derivative tests to classify interior critical points as local maxima, local minima, or neither; absolute extrema require comparison over the full feasible domain, including endpoints when present.
L'Hopital's Rule simplifies finding limits of indeterminate forms using derivatives.

$References$

OpenStax, Calculus Volume 1, §4.5, “Derivatives and the Shape of a Graph”

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Learning Objectives

Tangent and Normal Lines

Tangent Line

Normal Line

Angle of Intersection Between Curves

How to Find the Angle of Intersection

Angle of Intersection Formula

Rates of Change and Rectilinear Motion

Rectilinear Motion

Kinematic Equations

Marginal Analysis in Economics

Cost Function C(x)C(x)C(x)

Marginal Cost C′(x)C'(x)C′(x)

Revenue Function R(x)R(x)R(x)

Marginal Revenue R′(x)R'(x)R′(x)

Profit Optimization

Related Rates

Common Strategy for Related Rates

Interactive Simulation

Related Rates: Sliding Ladder

Interactive Simulation

Hydraulic Related Rates: Conical Reservoir

Optimization: Maxima and Minima

Critical Point

Absolute vs. Relative Extrema

Classifying Interior Local Extrema

Interactive Simulation

Box Optimization Problem

Curve Sketching

First and Second Derivative Tests

L'Hopital's Rule

L'Hopital's Rule

Civil Engineering Optimization: Pipeline and Canal Routing

Interactive Simulation

Canal Optimization

References

$Learning Objectives$

$Tangent Line$

$Normal Line$

$How to Find the Angle of Intersection$

$Rectilinear Motion$

$C(x)$

$C'(x)$

$R(x)$

$R'(x)$

$Common Strategy for Related Rates$

$Interactive Simulation$

$Interactive Simulation$

$Critical Point$

$Absolute vs. Relative Extrema$

$Classifying Interior Local Extrema$

$Interactive Simulation$

$First and Second Derivative Tests$

$Interactive Simulation$

$References$