# Chapter 4: Calculus – Interpretation and Methods for Integration and Differentiation

This chapter discusses the interpretation of derivatives that helps to understand the origin of formulas for estimating values for roadway elevation along the vertical curve. Understanding derivatives as a rate of change can also help with traffic operations (e.g., capacity as a maximum flow point) and transportation economics (e.g., marginal costs and congestion pricing). Integrals are used to estimate the area under a given curve. This property of integrals is used in estimating earthwork volumes and quantifying aggregate delays at intersections or roadway segments with incidents.

Learning Objectives

At the end of the chapter, the reader should be able to do the following:

- Interpret derivatives as the rate of change.
- Differentiate various functions with single variables.
- Familiarize oneself with the rules and processes (e.g., chain rule, multiplication rule) of derivation.
- Use the integration method for estimating the area under a curve.
- Describe the process of integration and differentiation in correspondence with each other.
- Identify topics in the introductory transportation engineering courses that build on the concepts discussed in this chapter.

## Interpret Derivatives as the Rate of Change

In this section, you will learn about derivatives, tangent slopes, tangent lines, quotient rule, chain rule and power rule by watching the videos. Also, short problems to check your understanding are included.

### Interpreting Derivatives

### Derivative as Slope of Curve

Check Your Understanding: Interpreting Derivatives & Derivative as Slope of Curve

### The Derivative as Slope of Tangent Line

Check Your Understanding: The Derivative as Slope of Tangent Line

### Tangent Slope as Instantaneous Rate of Change

Check Your Understanding: Tangent Slope as Instantaneous Rate of Change

### Approximating Instantaneous Rate of Change with Average Rate of Change

## Familiarize Oneself with the Rules and Processes (e.g., chain rule, multiplication rule) of Derivation

### Basic Derivative Rules

### Basic Derivative Rules (Part 2)

Check Your Understanding: Basic Derivative Rules

### Product Rule

Check Your Understanding: Product Rule

### Quotient Rule

Check Your Understanding: Quotient Rule

### Chain Rule

Check Your Understanding: Chain Rule

### The Power Rule

Check Your Understanding: The Power Rule

### Derivatives of Trigonometric Functions

Check Your Understanding: Derivatives of Trigonometric Functions

## Differentiate Various Functions with Single Variables

In this section, you will learn how to differentiate various functions, products, quotients, rational functions, trigonomic functions, and logarithmic by watching the videos. Also, short problems to check your understanding are included.

### Differentiability at a Point: Algebraic

Check Your Understanding: Differentiability at a Point; Algebraic

### Differentiating Polynomials

Check Your Understanding: Differentiating Polynomials

### Fractional Powers Differentiation

Check Your Understanding: Fractional Powers Differentiation

### Radical Functions Differentiation Introduction

### Worked Example

Check Your Understanding: Radical Functions Differentiation

### Differentiating Products

Check Your Understanding: Differentiating Products

### Differentiate Quotients

Please read this link on differentiate quotients.

Check Your Understanding: Differentiate Quotients

### Differentiating Rational Functions

Check Your Understanding: Differentiating Rational Functions

### Differentiate Trigonometric Functions

Check Your Understanding: Differentiate Trigonometric Functions

### Differentiate Exponential Functions

Check Your Understanding: Differentiate Exponential Functions

### Differentiate Logarithmic Functions

Check Your Understanding: Differentiate Logarithmic Functions

### Differentiating Using Multiple Rules

#### Common Derivatives Review

Please read this link on common derivatives.

#### Derivative Rules Review

Please read this link on derivative rules.

Check Your Understanding: Differentiating Using Multiple Rules

## Integrals of Common Functions

In this section, you will learn about indefinite integrals of sin(x), cos(x), and e^{x }and definite integrals functions, and U-substitution exponential function by watching the videos. Also, short problems to check your understanding are included.

### Indefinite Integrals of Common Functions

#### Indefinite Integral of 1/x

Check Your Understanding: Indefinite Integral of 1/x

#### Indefinite Integrals of sin(x), cos(x), and e^{x}

Check Your Understanding: Indefinite Integrals of sin(x), cos(x), and e^{x}

### Definite Integrals of Common Functions

#### Reverse Power Rule

Check Your Understanding: Reverse Power Rule

#### Rational Functions

Check Your Understanding: Rational Functions

#### Radical Functions

Check Your Understanding: Radical Functions

#### Trig Functions

Check Your Understanding: Trig Functions

#### Natural Logs

Check Your Understanding: Natural Logs

#### Absolute Value Functions

Check Your Understanding: Absolute Value Functions

#### Piecewise Functions

Check Your Understanding: Piecewise Functions

#### Integrating with U-Substitution

#### U-Substitution: Definite Integral of Exponential Function

Check Your Understanding: Integrating with U-Substitution

## Use the Integration Method for Estimating the Area Under a Curve

In this section, you will learn about integral calculus, definite integrals, and the Riemann calculus technique by watching the videos. Also, short problems to check your understanding are included.

### Introduction to Integral Calculus

### Introduction to Definite Integrals

### Introduction to Riemann Approximation

#### Riemann Sums Review

Go to this link to read information on Riemann Sums.

Check Your Understanding: Riemann Approximation

### Definite Integral as the Limit of a Riemann Sum

Check Your Understanding: Definite Integral as the Limit of a Riemann Sum

### Finding Area Under a Curve Using Integration

Check Your Understanding: Finding Area Under a Curve Using Integration

## Describe the Process of Integration and Differentiation in Correspondence with Each Other

### The Fundamental Theorem of Calculus

Check Your Understanding: The Fundamental Theorem of Calculus

## Relevance to Transportation Engineering Coursework

This section explains the relevance of designing vertical curves, estimating congestion pricing schedules, traffic operations and roadway capacity, and estimating aggregate delays in transportation engineering coursework.

### Design of Vertical Curves

In roadway design, one grade or slope is connected with the next grade (For example, an upgrade of 3% and a downgrade of 2%) using a parabolic curve defined by a quadratic equation. First-order derivatives of these equations are used to define the highest or lowest point on these curves. If the road is being transitioned from an uphill to a downhill (or vice versa), then the highest point (or lowest point) also represents the point at which the slope is zero. Furthermore, the second derivative of the quadratic equation is a constant, i.e., independent of one’s location on the curve. It provides for a smooth transition at a constant rate between the two grades. In this chapter, the sections titled “Interpret Derivatives as the Rate of Change” and “Differentiate Various Functions with Single Variables” provided the details on materials relevant to differentiating polynomials.

### Estimating Congestion Pricing Schedules

The idea of congestion pricing is to direct roadway users to use transportation network elements (i.e., routes) that would minimize costs (in the form of travel times) for all road users. Developing a fundamental understanding of this idea requires the first-order differential of the function for the costs incurred by all road users to be set to zero. In this chapter, the sections titled “Interpret Derivatives as the Rate of Change” and “Differentiate Various Functions with Single Variables” provide the details on the relevant materials.

### Traffic Operations and Roadway Capacity

Roadway capacity is defined as the maximum flow rate across a point on the roadway segment. It is defined based on the maxima of speed-flow or speed-density relationships. The maxima is found by setting the first derivative of a speed-density (or speed-flow) relationship to zero. In this chapter, the sections titled “Interpret Derivatives as the Rate of Change” and “Differentiate Various Functions with Single Variables” provide the relevant background for this application.

### Earthwork Volumes

Roadway construction often requires earthwork, i.e., the addition or removal of material from the sites. The design process yields the roadway elevation and cross-section along a segment. These point locations are then used to estimate the aggregate earthwork volume using the principles described in the section titled “Use the Integration Method for Estimating the Area Under a Curve” of this chapter.

### Estimating Aggregate Delays

The estimation of aggregate delays requires the estimation of the area under the curve for the input-output diagrams. These diagrams have time on the x-axis and the number of road users entering/exiting a roadway location on the y-axis. The time of entrance for a road user defies the input curve, while the time of exit defines the output curve. The area between the two curves defines the aggregate delay for the roadway entity under consideration (i.e., an intersection or roadway segment). The above section titled “Use the Integration Method for Estimating the Area Under a Curve” provides the details needed for this chapter.

## Key Takeaways

Key Takeaways

- The interpretation of first-order derivative as a rate of change and definition of maxima or minima of function as points where this derivative is equal to zero is critical to several transport applications. It is used in roadway design for a smooth transition between grades, minimization of total user costs via congestion pricing, and estimating roadway capacity based on speed-flow or flow-density relationships.
- Integration of a function over a defined interval may be interpreted as the area under the curve. This interpretation is used to estimate the volume of earthwork required, which is a common aspect of roadway construction. This information is important for anticipating the resources and equipment needed for planning and budgeting purposes. The same interpretation of integral as area under the curve is used to estimate the area under the input-output diagrams in traffic operations to estimate aggregate delay on intersections and roadway locations as well

## Glossary: Key Terms

**Derivative[1]** – the limit of the ratio of the change in a function to the corresponding change in its independent variable as the latter change approaches zero

**Rate of Change[1]** – a value that results from dividing the change in a function of a variable by the change in the variable

**Slope[1]** – the slope of the line tangent to a plane curve at a point

**Tangent[1]** – meeting a curve or surface in a single point if a sufficiently small interval is considered

**Rational Function[1]** – a function that is the quotient of two polynomials

**Integration[1]** – the operation of finding whose differential is known\

**Indefinite[1]** – having no exact limits

**Definite[1]** – having distinct or certain limits

**Natural Logarithm[1]** – a logarithm with e as a base

**Absolute Value[1]** – a nonnegative number equal in numerical value to a given real number

**Piecewise[1]** – with respect to a number of discrete intervals, sets, or pieces

**Riemann Integral[1]** – a definite integral defined as the limit of sums found by partitioning the interval comprising the domain of definition into subintervals, by finding the sum of products each of which consists of the width of a subinterval multiplied by the value of the function at some point in it, and by letting the maximum width of the subintervals approach zero

**Fundamental Theorem of Calculus[2]** – the theorem, central to the entire development of calculus, that establishes the relationship between differentiation and integration

**Fundamental Theorem of Calculus, Part 1[2]** – uses a definite integral to define an antiderivative of a function

**Fundamental Theorem of Calculus, Part 2[2]** – (also, evaluation theorem) we can evaluate a definite integral by evaluating the antiderivative of the integrand at the endpoints of the interval and subtracting

[2] “Calculus Volume 1” by Gilbert Strang, Edwin “Jed” Herman on OpenStax, Chapter 5.3: The Fundamental Theorem of Calculus: https://openstax.org/books/calculus-volume-1/pages/5-3-the-fundamental-theorem-of-calculus

## Media Attributions

Note: All Khan Academy content is available for free at (www.khanacademy.org).

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- Video 1: Interpreting Derivatives by Linda Green is licensed by Creative Commons Attribution 3.0 Unported (CC BY 3.0)
- Video 2: Derivative as Slope of Curve by Khan Academy is licensed by Creative Commons NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 3: The Derivative as Slope of Tangent Line by Khan Academy is licensed by Creative Commons NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 4: Tangent Slope as Instantaneous Rate of Change by Khan Academy is licensed by Creative Commons NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 5: Approximating Instantaneous Rate of Change with Average Rate of Change by Khan Academy is licensed by Creative Commons NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 6: Basic Derivative Rules by Khan Academy is licensed by Creative Commons NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
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- Video 27: Indefinite Integral of 1/x by Khan Academy is licensed by Creative Commons NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
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the limit of the ratio of the change in a function to the corresponding change in its independent variable as the latter change approaches zero

a value that results from dividing the change in a function of a variable by the change in the variable

meeting a curve or surface in a single point if a sufficiently small interval is considered

the slope of the line tangent to a plane curve at a point

a function that is the quotient of two polynomials

having no exact limits

having distinct or certain limits

a logarithm with e as a base

a nonnegative number equal in numerical value to a given real number

with respect to a number of discrete intervals, sets, or pieces

the operation of finding whose differential is known

a definite integral defined as the limit of sums found by partitioning the interval comprising the domain of definition into subintervals, by finding the sum of products each of which consists of the width of a subinterval multiplied by the value of the function at some point in it, and by letting the maximum width of the subintervals approach zero

the theorem, central to the entire development of calculus, that establishes the relationship between differentiation and integration