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Application of Derivatives

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Vectors in Space
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Point Estimation
- Probability
- Probability Calculations
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residuals
- Sample Mean
- Sample Proportion
- Sampling
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Standard Deviation
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Survey Bias
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Venn Diagrams

You want to record a rocket launch, so you place your camera on your trusty tripod and get it all set up to record this event. Now, only one question remains: at what rate should your camera's angle with the ground change to allow it to keep the rocket in view as it makes its flight?

Because launching a rocket involves two related quantities that change over time, the answer to this question relies on an **application of derivatives** known as related rates. In this article, you will discover some of the many applications of derivatives and how they are used in calculus, engineering, and economics.

Being able to solve the **related rates** problem discussed above is just one of many applications of derivatives you learn in calculus. You will also learn how derivatives are used to:

find

**tangent****and****normal****lines**to a curve, andfind

**maximum**and**minimum**values.

You will then be able to use these techniques to solve **optimization **problems, like maximizing an area or maximizing revenue.

Additionally, you will learn how derivatives can be applied to:

solve

**complicated limits**,make

**approximations**, andto give

**accurate graphs**.

Derivatives are very useful tools for finding the equations of tangent lines and normal lines to a curve.

The tangent line to a curve is one that touches the curve at only one point and whose slope is the derivative of the curve at that point.

To find the tangent line to a curve at a given point (as in the graph above), follow these steps:

- Given a point and a curve, find the slope by taking the derivative of the given curve.
- The given point is: \[ (2, 4) \]
- The given curve is: \[ f(x) = x^{2} \]
- The derivative of the given curve is: \[ f'(x) = 2x \]

- Plug the \( x \)-coordinate of the given point into the derivative to find the slope.\[ \begin{align}f'(x) &= 2x \\f'(2) &= 2(2) \\ &= 4 \\ &= m.\end{align} \]
- Use the point-slope form of a line to write the equation.\[ \begin{align}y-y_1 &= m(x-x_1) \\y-4 &= 4(x-2) \\y &= 4(x-2)+4 \\ &= 4x - 4.\end{align} \]

For more information and examples about this subject, see our article on Tangent Lines.

The normal line to a curve is perpendicular to the tangent line. You use the tangent line to the curve to find the normal line to the curve. The slope of the normal line is:

\[ n = - \frac{1}{m} = - \frac{1}{f'(x)}. \]

To find the normal line to a curve at a given point (as in the graph above), follow these steps:

- Find the tangent line to the curve at the given point, as in the example above.
- The tangent line to the curve is: \[ y = 4(x-2)+4 \]

- Use the slope of the tangent line to find the slope of the normal line.
- The slope of the normal line to the curve is:\[ \begin{align}n &= - \frac{1}{m} \\n &= - \frac{1}{4}\end{align} \]

- Use the point-slope form of a line to write the equation.\[ \begin{align}y-y_1 &= n(x-x_1) \\y-4 &= - \frac{1}{4}(x-2) \\y &= - \frac{1}{4} (x-2)+4\end{align} \]

In many real-world scenarios, related quantities change with respect to time. If you think about the rocket launch again, you can say that the rate of change of the rocket's height, \( h \), is related to the rate of change of your camera's angle with the ground, \( \theta \). In this case, you say that \( \frac{dg}{dt} \) and \( \frac{d\theta}{dt} \) are related rates because \( h \) is related to \( \theta \).

In related rates problems, you study related quantities that are changing with respect to time and learn how to calculate one rate of change if you are given another rate of change.

- Assign symbols to all the variables in the problem and sketch the problem if it makes sense.
- In terms of the variables you just assigned, state the information that is given and the rate of change that you need to find.
- Find an equation that relates your variables.
- Using the chain rule, take the derivative of this equation with respect to the independent variable.
- The new equation relates the derivatives.

- Substitute all the known values into the derivative, and solve for the rate of change you needed to find.

It is crucial that you do not substitute the known values too soon. If you make substitute the known values before you take the derivative, then the substituted quantities will behave as constants and their derivatives will not appear in the new equation you find in step 4.

There are lots of different articles about related rates, including Rates of Change, Motion Along a Line, Population Change, and Changes in Cost and Revenue.

When it comes to functions, linear functions are one of the easier ones with which to work. Therefore, they provide you a useful tool for approximating the values of other functions. You will build on this application of derivatives later as well, when you learn how to approximate functions using higher-degree polynomials while studying sequences and series, specifically when you study power series.

The key concepts and equations of **linear approximations and differentials** are:

A differentiable function, \( y = f(x) \), can be approximated at a point, \( a \), by the

**linear approximation**function:\[ L(x) = f(a) + f'(a)(x-a). \]

Given a function, \( y = f(x) \), if, instead of replacing \( x \) with \( a \), you replace \( x \) with \( a + dx \), then the

**differential**:\[ dy = f'(x)dx \]

is an approximation for the change in \( y \).

The actual change in \( y \), however, is:

\[ \Delta y = f(a+dx) - f(a). \]

A measurement error of \( dx \) can lead to an error in the quantity of \( f(x) \). This is known as

**propagated error**, which is estimated by:\[ dy \approx f'(x)dx \]

To estimate the relative error of a quantity ( \( q \) ) you use:\[ \frac{ \Delta q}{q}. \]

For more information on this topic, see our article on the Amount of Change Formula.

One of the most common applications of derivatives is finding the extreme values, or maxima and minima, of a function. Once you learn the methods of finding extreme values (also known collectively as extrema), you can apply these methods to later applications of derivatives, like creating accurate graphs and solving optimization problems.

The key terms and concepts of **maxima and minima** are:

Terms

**Absolute extremum**If a function, \( f \), has an absolute max or absolute min at point \( c \), then you say that the function \( f \) has an

**absolute extremum**at \( c \).**Absolute max / absolute maximum**If \( f(c) \geq f(x) \) for all \( x \) in the domain of \( f \), then you say that \( f \) has an

**absolute maximum**at \( c \).**Absolute min / absolute minimum**If \( f(c) \leq f(x) \) for all \( x \) in the domain of \( f \), then you say that \( f \) has an

**absolute minimum**at \( c \).**Local extremum**If a function, \( f \), has a local max or min at point \( c \), then you say that \( f \) has a

**local extremum**at \( c \).**Local max / local maximum**If there exists an interval, \( I \), such that \( f(c) \geq f(x) \) for all \( x \) in \( I \), you say that \( f \) has a

**local max**at \( c \).**Local min / local minimum**If there exists an interval, \( I \), such that \( f(c) \leq f(x) \) for all \( x \) in \( I \), you say that \( f \) has a

**local min**at \( c \).**Critical point**Based on the definitions above, the point \( (c, f(c)) \) is a

**critical point**of the function \( f \).**Critical number**If \( f'(c) = 0 \) or \( f'(c) \) is undefined, you say that \( c \) is a

**critical number**of the function \( f \).**Extreme value theorem**If the function \( f \) is continuous over a finite, closed interval, then \( f \) has an absolute max and an absolute min.

**Fermat's theorem**If a function \( f \) has a local extremum at point \( c \), then \( c \) is a critical point of \( f \).

Concepts

It is possible for a function to have:

both an absolute max and an absolute min,

just one absolute extremum, or

have no absolute extremum.

If a function has a local extremum, the point where it occurs must be a critical point.

However, a function does not necessarily have a local extremum at a critical point.

A continuous function over a closed and bounded interval has an absolute max and an absolute min.

Each extremum occurs at either a critical point or an endpoint of the function.

For more information on maxima and minima see Maxima and Minima Problems and Absolute Maxima and Minima.

One of the most important theorems in calculus, and an application of derivatives, is the Mean Value Theorem (sometimes abbreviated as MVT). Like the previous application, the MVT is something you will use and build on later.

The key concepts of the mean value theorem are:

**The definition of the MVT**If a function, \( f \), is continuous over the closed interval \( [a, b] \) and differentiable over the open interval \( (a, b) \), then there exists a point \( c \) in the open interval \( (a, b) \) such that

\[ f'(c) = \frac{f(b)-f(a)}{b-a}. \]

The special case of the MVT known as

**Rolle's theorem**If a function, \( f \), is continuous over the closed interval \( [a, b] \), differentiable over the open interval \( (a, b) \), and if \( f(a) = f(b) \), then there exists a point \( c \) in the open interval \( (a, b) \) such that

\[ f'(c) = 0 \]

The

**corollaries of the mean value theorem****Functions with a derivative of zero**Let \( f \) be differentiable on an interval \( I \). If \( f'(x) = 0 \) for all \( x \) in \( I \), then \( f'(x) = \) constant for all \( x \) in \( I \).

**Constant difference theorem**If the functions \( f \) and \( g \) are differentiable over an interval \( I \), and \( f'(x) = g'(x) \) for all \( x \) in \( I \), then \( f(x) = g(x) + C \) for some constant \( C \).

**Increasing and decreasing functions**Let \( f \) be continuous over the closed interval \( [a, b] \) and differentiable over the open interval \( (a, b) \).

If \( f'(x) > 0 \) for all \( x \) in \( (a, b) \), then \( f \) is an increasing function over \( [a, b] \).

If \( f'(x) < 0 \) for all \( x \) in \( (a, b) \), then \( f \) is a decreasing function over \( [a, b] \).

Building on the applications of derivatives to find maxima and minima and the mean value theorem, you can now determine whether a critical point of a function corresponds to a local extreme value. But what about the shape of the function's graph? Well, this application teaches you how to use the first and second derivatives of a function to determine the shape of its graph.

Key concepts of **derivatives and the shape of a graph** are:

Say a function, \( f \), is continuous over an interval \( I \) and contains a critical point, \( c \). If \( f \) is differentiable over \( I \), except possibly at \( c \), then \( f(c) \) satisfies one of the following:

If \( f' \) changes sign from positive when \( x < c \) to negative when \( x > c \), then \( f(c) \) is a local max of \( f \).

If \( f' \) changes sign from negative when \( x < c \) to positive when \( x > c \), then \( f(c) \) is a local min of \( f \).

If \( f' \) has the same sign for \( x < c \) and \( x > c \), then \( f(c) \) is neither a local max or a local min of \( f \).

**the Candidates Test**This is a method for finding the absolute maximum and the absolute minimum of a continuous function that is defined over a closed interval. It consists of the following:

Find all the relative extrema of the function.

Evaluate the function at the extreme values of its domain.

Order the results of steps 1 and 2 from least to greatest.

The least value is the global minimum.

The greatest value is the global maximum.

**test for concavity**If \( f \) is a function that is twice differentiable over an interval \( I \), then:

If \( f''(x) > 0 \) for all \( x \) in \( I \), then \( f \) is concave up over \( I \).

If \( f''(x) < 0 \) for all \( x \) in \( I \), then \( f \) is concave down over \( I \).

Suppose \( f'(c) = 0 \), \( f'' \) is continuous over an interval that contains \( c \).

If \( f''(c) > 0 \), then \( f \) has a local min at \( c \).

If \( f''(c) < 0 \), then \( f \) has a local max at \( c \).

If \( f''(c) = 0 \), then the test is inconclusive.

You must evaluate \( f'(x) \) at a test point \( x \) to the left of \( c \) and a test point \( x \) to the right of \( c \) to determine if \( f \) has a local extremum at \( c \).

Once you understand derivatives and the shape of a graph, you can build on that knowledge to graph a function that is defined on an unbounded domain. To accomplish this, you need to know the behavior of the function as \( x \to \pm \infty \). This application of derivatives defines limits at infinity and explains how infinite limits affect the graph of a function.

The key terms and concepts of **limits at infinity and asymptotes** are:

Terms

**end behavior**The behavior of the function, \( f(x) \), as \( x\to \pm \infty \).

**horizontal asymptote**If \( \lim_{x \to \pm \infty} f(x) = L \), then \( y = L \) is a horizontal asymptote of the function \( f(x) \).

**infinite limit at infinity**The function \( f(x) \) becomes larger and larger as \( x \) also becomes larger and larger.

**limit at infinity**The limiting value, if it exists, of a function \( f(x) \) as \( x \to \pm \infty \).

**oblique asymptote**The line \( y = mx + b \), if \( f(x) \) approaches it, as \( x \to \pm \infty \) is an oblique asymptote of the function \( f(x) \).

Concepts

The limit of the function \( f(x) \) is \( L \) as \( x \to \pm \infty \) if the values of \( f(x) \) get closer and closer to \( L \) as \( x \) becomes larger and larger.

The limit of the function \( f(x) \) is \( \infty \) as \( x \to \infty \) if \( f(x) \) becomes larger and larger as \( x \) also becomes larger and larger.

The limit of the function \( f(x) \) is \( - \infty \) as \( x \to \infty \) if \( f(x) < 0 \) and \( \left| f(x) \right| \) becomes larger and larger as \( x \) also becomes larger and larger.

For the polynomial function \( P(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + \ldots + a_{1}x + a_{0} \), where \( a_{n} \neq 0 \), the end behavior is determined by the leading term: \( a_{n}x^{n} \).

If \( n \neq 0 \), then \( P(x) \) approaches \( \pm \infty \) at each end of the function.

For the rational function \( f(x) = \frac{p(x)}{q(x)} \), the end behavior is determined by the relationship between the degree of \( p(x) \) and the degree of \( q(x) \).

If the degree of \( p(x) \) is less than the degree of \( q(x) \), then the line \( y = 0 \) is a horizontal asymptote for the rational function.

If the degree of \( p(x) \) is equal to the degree of \( q(x) \), then the line \( y = \frac{a_{n}}{b_{n}} \), where \( a_{n} \) is the leading coefficient of \( p(x) \) and \( b_{n} \)

If the degree of \( p(x) \) is greater than the degree of \( q(x) \), then the function \( f(x) \) approaches either \( \infty \) or \( - \infty \) at each end.

Continuing to build on the applications of derivatives you have learned so far, **optimization problems** are one of the most common applications in calculus. These are defined as calculus problems where you want to solve for a maximum or minimum value of a function.

- Introduce all variables.
- If it makes sense, draw a figure and
**label all your variables**.

- If it makes sense, draw a figure and
- Determine which quantity (which of your variables from step 1) you need to maximize or minimize.
- Determine for what range of values of the other variables (if this can be determined at this time) you need to maximize or minimize your quantity.

- Write a formula for the quantity you need to maximize or minimize in terms of your variables.
- This formula will most likely involve more than one variable.

- Write any equations you need to relate the independent variables in the formula from step 3.
- Use these equations to write the quantity to be maximized or minimized as a function of one variable.

- Identify the domain of consideration for the function in step 4.
- Make sure you consider the
**physical**problem to be solved.

- Make sure you consider the
- Locate the maximum or minimum value of the function from step 4.
- This step usually involves looking for
**critical points**and**evaluating a function at endpoints**.

- This step usually involves looking for

A powerful tool for evaluating limits, **L’Hôpital’s Rule** is yet another application of derivatives in calculus. This application uses derivatives to calculate limits that would otherwise be impossible to find. These limits are in what is called **indeterminate forms**.

The key terms and concepts of **L’Hôpital’s Rule** are:

Terms

When evaluating a limit, the forms \[ \frac{0}{0}, \ \frac{\infty}{\infty}, \ 0 \cdot \infty, \ \infty - \infty, \ 0^{0}, \ \infty^{0}, \ \mbox{ and } 1^{\infty} \] are all considered indeterminate forms because you need to further analyze (i.e., by using L’Hôpital’s rule) whether the limit exists and, if so, what the value of the limit is.

**L’Hôpital’s Rule**If two functions, \( f(x) \) and \( g(x) \), are differentiable functions over an interval \( a \), except possibly at \( a \), and \[ \lim_{x \to a} f(x) = 0 = \lim_{x \to a} g(x) \] or \[ \lim_{x \to a} f(x) \mbox{ and } \lim_{x \to a} g(x) \mbox{ are infinite, } \] then \[ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}, \] assuming the limit involving \( f'(x) \) and \( g'(x) \) either exists or is \( \pm \infty \).

Concepts

You can use L’Hôpital’s rule to evaluate the limit of a quotient when it is in either of the indeterminate forms \( \frac{0}{0}, \ \frac{\infty}{\infty} \).

You can also use L’Hôpital’s rule on the other indeterminate forms if you can rewrite them in terms of a limit involving a quotient when it is in either of the indeterminate forms \( \frac{0}{0}, \ \frac{\infty}{\infty} \).

In many applications of math, you need to find the zeros of functions. Unfortunately, it is usually very difficult – if not impossible – to explicitly calculate the zeros of these functions. **Newton's method **saves the day in these situations because it is a technique that is efficient at approximating the zeros of functions.

The key terms and concepts of **Newton's method** are:

Terms

**iterative process**A process in which a list of numbers like \[ x_{0}, x_{1}, x_{2}, \ldots \] is generated by beginning with a number \( x_{0} \) and then defining \[ x_{n} = F \left( x_{n-1} \right) \] for \( n \neq 1 \).

**Newton's method**A method for approximating the roots of \( f(x) = 0 \). It uses an initial guess of \( x_{0} \). Each subsequent approximation is defined by the equation \[ x_{n} = x_{n-1} - \frac{f(x_{n-1})}{f'(x_{n-1})}. \]

Concepts

Newton's method approximates the roots of \( f(x) = 0 \) by starting with an initial approximation of \( x_{0} \). From there, it uses tangent lines to the graph of \( f(x) \) to create a sequence of approximations \( x_1, x_2, x_3, \ldots \).

Failures of Newton's method:

This method fails when the list of numbers \( x_1, x_2, x_3, \ldots \) does not approach a finite value, or

when it approaches a value other than the root you are looking for.

Any process in which a list of numbers \( x_1, x_2, x_3, \ldots \) is generated by defining an initial number \( x_{0} \) and defining the subsequent numbers by the equation \[ x_{n} = F \left( x_{n-1} \right) \] for \( n \neq 1 \) is an iterative process.

Newton's method is an example of an iterative process, where the function \[ F(x) = x - \left[ \frac{f(x)}{f'(x)} \right] \] for a given function of \( f(x) \).

Having gone through all the applications of derivatives above, now you might be wondering: what about turning the derivative process around? What if I have a function \( f(x) \) and I need to find a function whose derivative is \( f(x) \)? What application does this have?

To answer these questions, you must first define **antiderivatives**.

An antiderivative of a function \( f \) is a function whose derivative is \( f \).

One of many examples where you would be interested in an antiderivative of a function is the study of motion.

The key terms and concepts of **antiderivatives **are:

Terms

**antiderivative**A function \( F(x) \) such that \( F'(x) = f(x) \) for all \( x \) in the domain of \( f \) is an antiderivative of \( f \).

The most general antiderivative of a function \( f(x) \) is the indefinite integral of \( f \). The notation \[ \int f(x) dx \] denotes the indefinite integral of \( f(x) \).

**initial value problem**A problem that requires you to find a function \( y \) that satisfies the differential equation \[ \frac{dy}{dx} = f(x) \] together with the initial condition of \[ y(x_{0}) = y_{0}. \]

Concepts

If the function \( F \) is an antiderivative of another function \( f \), then every antiderivative of \( f \) is of the form \[ F(x) + C \] for some constant \( C \).

Solving the initial value problem \[ \frac{dy}{dx} = f(x), \mbox{ with the initial condition } y(x_{0}) = y_{0} \] requires you to:

first find the set of antiderivatives of \( f \) and then

look for the particular antiderivative that also satisfies the initial condition.

The applications of derivatives in engineering is really quite vast. To touch on the subject, you must first understand that there are many kinds of engineering. To name a few;

Mechanical Engineering

Civil Engineering

Industrial Engineering

Electrical Engineering

Aerospace Engineering

Chemical Engineering

Computer Engineering

\( \vdots \)

All of these engineering fields use calculus. They all use applications of derivatives in their own way, to solve their problems.

An example that is common among several engineering disciplines is the use of derivatives to study the forces acting on an object. For instance,

Mechanical Engineers could study the forces that on a machine (or even within the machine).

Civil Engineers could study the forces that act on a bridge.

Industrial Engineers could study the forces that act on a plant.

Aerospace Engineers could study the forces that act on a rocket.

And so on.

In every case, to study the forces that act on different objects, or in different situations, the engineer needs to use applications of derivatives (and much more).

Even the financial sector needs to use calculus! Applications of derivatives are used in economics to determine and optimize:

supply and demand,

profit and cost, and

revenue and loss.

__Launching a Rocket – Related Rates Example__

Your camera is set up \( 4000ft \) from a rocket launch pad. The rocket launches, and when it reaches an altitude of \( 1500ft \) its velocity is \( 500ft/s \). What rate should your camera's angle with the ground change to allow it to keep the rocket in view as it makes its flight?

**Solution**:

- Sketch the problem.
- Here, \( \theta \) is the angle between your camera lens and the ground and \( h \) is the height of the rocket above the ground.

- Clarify what exactly you are trying to find.
- The problem asks you to find the rate of change of your camera's angle to the ground when the rocket is \( 1500ft \) above the ground. Both of these variables are changing with respect to time.
- This means you need to find \( \frac{d \theta}{dt} \) when \( h = 1500ft \). You also know that the velocity of the rocket at that time is \( \frac{dh}{dt} = 500ft/s \).

- Determine what equation relates the two quantities \( h \) and \( \theta \).
- Looking back at your picture in step \( 1 \), you might think about using a trigonometric equation. What relates the opposite and adjacent sides of a right triangle? The \( \tan \) function! So, you have:\[ \tan(\theta) = \frac{h}{4000} .\]
- Rearranging to solve for \( h \) gives:\[ h = 4000\tan(\theta). \]

- Differentiate this to get:\[ \frac{dh}{dt} = 4000\sec^{2}(\theta)\frac{d\theta}{dt} .\]
- Find \( \frac{d \theta}{dt} \) when \( h = 1500ft \).
- To find \( \frac{d \theta}{dt} \), you first need to find \(\sec^{2} (\theta) \). How can you do that?
- Going back to trig, you know that \( \sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}} \).
- And, from the givens in this problem, you know that \( \text{adjacent} = 4000ft \) and \( \text{opposite} = h = 1500ft \).
- So, you can use the Pythagorean theorem to solve for \( \text{hypotenuse} \).\[ \begin{align}a^{2}+b^{2} &= c^{2} \\(4000)^{2}+(1500)^{2} &= (\text{hypotenuse})^{2} \\\text{hypotenuse} &= 500 \sqrt{73}ft.\end{align} \]
- Therefore, when \( h = 1500ft \), \( \sec^{2} ( \theta ) \) is:\[ \begin{align}\sec^{2}(\theta) &= \left( \frac{\text{hypotenuse}}{\text{adjacent}} \right)^{2} \\&= \left( \frac{500 \sqrt{73}}{4000} \right)^{2} \\&= \frac{73}{64}.\end{align} \]

- Plug in the values for \( \sec^{2}(\theta) \) and \( \frac{dh}{dt} \) into the function you found in step 4 and solve for \( \frac{d \theta}{dt} \).\[ \begin{align}\frac{dh}{dt} &= 4000\sec^{2}(\theta)\frac{d\theta}{dt} \\500 &= 4000 \left( \frac{73}{64} \right) \frac{d\theta}{dt} \\\frac{d\theta}{dt} &= \frac{8}{73}.\end{align} \]

- To find \( \frac{d \theta}{dt} \), you first need to find \(\sec^{2} (\theta) \). How can you do that?
**Therefore, the rate that your camera's angle with the ground should change to allow it to keep the rocket in view as it makes its flight is:\[ \frac{d\theta}{dt} = \frac{8}{73} rad/s. \]**

__Engineering Application – Optimization Example__

You are an agricultural engineer, and you need to fence a rectangular area of some farmland. One side of the space is blocked by a rock wall, so you only need fencing for three sides. Given that you only have \( 1000ft \) of fencing, what are the dimensions that would allow you to fence the maximum area? What is the maximum area?

**Solution**:

- Let \( x \) be the length of the sides of the farmland that run perpendicular to the rock wall, and let \( y \) be the length of the side of the farmland that runs parallel to the rock wall. Then the area of the farmland is given by the equation for the area of a rectangle:\[ A = x \cdot y. \]
- Since you want to find the maximum possible area given the constraint of \( 1000ft \) of fencing to go around the perimeter of the farmland, you need an equation for the perimeter of the rectangular space.
- Don't forget to consider that the fence only needs to go around \( 3 \) of the \( 4 \) sides! So, your constraint equation is:\[ 2x + y = 1000. \]
- Now, you want to solve this equation for \( y \) so that you can rewrite the area equation in terms of \( x \) only:\[ y = 1000 - 2x. \]
- Rewriting the area equation, you get:\[ \begin{align}A &= x \cdot y \\A &= x \cdot (1000 - 2x) \\A &= 1000x - 2x^{2}.\end{align} \]

- Before jumping right into maximizing the area, you need to determine what your domain is.
- First, you know that the lengths of the sides of your farmland must be positive, i.e., \( x \) and \( y \) can't be negative numbers.
- Since \( y = 1000 - 2x \), and you need \( x > 0 \) and \( y > 0 \), then when you solve for \( x \), you get:\[ x = \frac{1000 - y}{2}. \]
- Minimizing \( y \), i.e., if \( y = 1 \), you know that:\[ x < 500. \]
- So, you need to determine the maximum value of \( A(x) \) for \( x \) on the open interval of \( (0, 500) \).
- However, you don't know that a function necessarily has a maximum value on an open interval, but you do know that a function does have a max (and min) value on a closed interval. Therefore, you need to consider the area function \( A(x) = 1000x - 2x^{2} \) over the closed interval of \( [0, 500] \).

- First, you know that the lengths of the sides of your farmland must be positive, i.e., \( x \) and \( y \) can't be negative numbers.
- Find the max possible area of the farmland by maximizing \( A(x) = 1000x - 2x^{2} \) over the closed interval of \( [0, 500] \).
- Since \( A(x) \) is a continuous function on a closed, bounded interval, you know that, by the extreme value theorem, it will have maximum and minimum values. These extreme values occur at the endpoints and any critical points.
- At the endpoints, you know that \( A(x) = 0 \).
- Since the area must be positive for all values of \( x \) in the open interval of \( (0, 500) \), the max must occur at a critical point. To find critical points, you need to take the first derivative of \( A(x) \), set it equal to zero, and solve for \( x \).\[ \begin{align}A(x) &= 1000x - 2x^{2} \\A'(x) &= 1000 - 4x \\0 &= 1000 - 4x \\x &= 250.\end{align} \]
- The only critical point is \( x = 250 \). Therefore, the maximum area must be when \( x = 250 \).
- Plugging this value into your perimeter equation, you get the \( y \)-value of this critical point:\[ \begin{align}y &= 1000 - 2x \\y &= 1000 - 2(250) \\y &= 500.\end{align} \]
**Therefore, to maximize the area of the farmland, \( x = 250ft \) and \( y = 500ft \). The area is \( 125000ft^{2} \).**The graph below visualizes this.

- Since \( A(x) \) is a continuous function on a closed, bounded interval, you know that, by the extreme value theorem, it will have maximum and minimum values. These extreme values occur at the endpoints and any critical points.

__Economic Application – Optimization Example__

You are the Chief Financial Officer of a rental car company. You found that if you charge your customers \( p \) dollars per day to rent a car, where \( 20 < p < 100 \), the number of cars \( n \) that your company rent per day can be modeled using the linear function

\[ n(p) = 600 - 6p. \]

If the company charges \( $20 \) or less per day, they will rent all of their cars. If the company charges \( $100 \) per day or more, they won't rent any cars.

How much should you tell the owners of the company to rent the cars to maximize revenue?

**Solution**:

- Let \( p \) be the price charged per rental car per day. Let \( n \) be the number of cars your company rents per day. Let \( R \) be the revenue earned per day.
- Find an equation that relates all three of these variables.
- Revenue earned per day is the number of cars rented per day times the price charged per rental car per day:\[ R = n \cdot p. \]

- Substitute the value for \( n \) as given in the original problem.\[ \begin{align}R &= n \cdot p \\R &= (600 - 6p)p \\R &= -6p^{2} + 600p.\end{align} \]
- Determine what your domain is.
- Since you intend to tell the owners to charge between \( $20 \) and \( $100 \) per car per day, you need to find the maximum revenue for \( p \) on the closed interval of \( [20, 100] \).

- Find the maximum possible revenue by maximizing \( R(p) = -6p^{2} + 600p \) over the closed interval of \( [20, 100] \).
- Since \( R(p) \) is a continuous function over a closed, bounded interval, you know that, by the extreme value theorem, it will have maximum and minimum values. These extreme values occur at the endpoints and any critical points.
- Find the critical points by taking the first derivative, setting it equal to zero, and solving for \( p \).\[ \begin{align}R(p) &= -6p^{2} + 600p \\R'(p) &= -12p + 600 \\0 &= -12p + 600 \\p = 50.\end{align} \]
- The only critical point is \( p = 50 \). Therefore, the maximum revenue must be when \( p = 50 \).
- Plugging this value into your revenue equation, you get the \( R(p) \)-value of this critical point:\[ \begin{align}R(p) &= -6p^{2} + 600p \\R(50) &= -6(50)^{2} + 600(50) \\R(50) &= 15000.\end{align} \]
**Therefore, to maximize revenue, you should tell the owners to charge \( $50 \) per car per day.**The graph below visualizes this.

- Since \( R(p) \) is a continuous function over a closed, bounded interval, you know that, by the extreme value theorem, it will have maximum and minimum values. These extreme values occur at the endpoints and any critical points.

- In calculus, there are many applications of derivatives, including:
- Tangent Lines and Normal Lines to Curves
- Related Rates
- Linear Approximations and Differentials
- Maxima and Minima
- The Mean Value Theorem
- Derivatives and the Shape of a Graph
- Limits at Infinity and Asymptotes
- Applied Optimization Problems
- L’Hôpital’s Rule
- Newton’s Method
- Antiderivatives

- Derivatives are useful beyond the realm of math, in fields like:
- Engineering
- Physics
- Economics
- Business
- Health

You find the application of the second derivative by first finding the first derivative, then the second derivative of a function. The applications of the second derivative are:

- determining concavity/convexity
- finding inflection points
- finding local extrema

You can use second derivative tests on the second derivative to find these applications.

The practical applications of derivatives are:

- Related Rates
- Linear Approximations and Differentials
- Maxima and Minima
- The Mean Value Theorem
- Derivatives and the Shape of a Graph
- Limits at Infinity and Asymptotes
- Applied Optimization Problems
- L’Hôpital’s Rule
- Newton’s Method
- Antiderivatives

Essentially, calculus, and its applications of derivatives, are the heart of engineering.

More about Application of Derivatives

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