Limits Derivatives Integrals Infinite Series Parametrics Polar Coordinates Conics
Limits
Epsilon-Delta Definition
Finite Limits
One-Sided Limits
Infinite Limits
Trig Limits
Pinching Theorem
Indeterminate Forms
L'Hopitals Rule
Limits That Do Not Exist
Continuity & Discontinuities
Intermediate Value Theorem
Derivatives
Power Rule
Product Rule
Quotient Rule
Chain Rule
Trig and Inverse Trig
Implicit Differentiation
Exponentials & Logarithms
Logarithmic Differentiation
Hyperbolic Functions
Higher Order Derivatives
Differentials
Slope, Tangent, Normal...
Linear Motion
Mean Value Theorem
Graphing
1st Deriv, Critical Points
2nd Deriv, Inflection Points
Related Rates Basics
Related Rates Areas
Related Rates Distances
Related Rates Volumes
Optimization
Integrals
Definite Integrals
Integration by Substitution
Integration By Parts
Partial Fractions
Improper Integrals
Basic Trig Integration
Sine/Cosine Integration
Secant/Tangent Integration
Trig Integration Practice
Trig Substitution
Linear Motion
Area Under/Between Curves
Volume of Revolution
Arc Length
Surface Area
Work
Moments, Center of Mass
Exponential Growth/Decay
Laplace Transforms
Describing Plane Regions
Infinite Series
Divergence (nth-Term) Test
p-Series
Geometric Series
Alternating Series
Telescoping Series
Ratio Test
Limit Comparison Test
Direct Comparison Test
Integral Test
Root Test
Absolute Convergence
Conditional Convergence
Power Series
Taylor/Maclaurin Series
Interval of Convergence
Remainder & Error Bounds
Fourier Series
Study Techniques
Choosing A Test
Sequences
Infinite Series Table
Practice Problems
Exam Preparation
Exam List
Parametrics
Parametric Curves
Parametric Surfaces
Slope & Tangent Lines
Area
Arc Length
Surface Area
Volume
Polar Coordinates
Converting
Slope & Tangent Lines
Area
Arc Length
Surface Area
Conics
Parabolas
Ellipses
Hyperbolas
Conics in Polar Form
Vectors Vector Functions Partial Derivatives/Integrals Vector Fields Laplace Transforms Tools
Vectors
Unit Vectors
Dot Product
Cross Product
Lines In 3-Space
Planes In 3-Space
Lines & Planes Applications
Angle Between Vectors
Direction Cosines/Angles
Vector Projections
Work
Triple Scalar Product
Triple Vector Product
Vector Functions
Projectile Motion
Unit Tangent Vector
Principal Unit Normal Vector
Acceleration Vector
Arc Length
Arc Length Parameter
Curvature
Vector Functions Equations
MVC Practice Exam A1
Partial Derivatives
Directional Derivatives
Lagrange Multipliers
Tangent Plane
MVC Practice Exam A2
Partial Integrals
Describing Plane Regions
Double Integrals-Rectangular
Double Integrals-Applications
Double Integrals-Polar
Triple Integrals-Rectangular
Triple Integrals-Cylindrical
Triple Integrals-Spherical
MVC Practice Exam A3
Vector Fields
Curl
Divergence
Conservative Vector Fields
Potential Functions
Parametric Curves
Line Integrals
Green's Theorem
Parametric Surfaces
Surface Integrals
Stokes' Theorem
Divergence Theorem
MVC Practice Exam A4
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Prepare For Calculus 1
Trig Formulas
Describing Plane Regions
Parametric Curves
Linear Algebra Review
Word Problems
Mathematical Logic
Calculus Notation
Simplifying
Practice Exams
More Math Help
Tutoring
Tools and Resources
Learning/Study Techniques
Math/Science Learning
Memorize To Learn
Music and Learning
Note-Taking
Motivation
Instructor or Coach?
Books
Math Books

You CAN Ace Calculus

17calculus > derivatives > graphing

### Calculus Main Topics

Derivatives

Derivative Applications

Optimization

Single Variable Calculus

Multi-Variable Calculus

### Tools

math tools

general learning tools

Using Derivatives To Graph Functions

We can use derivatives to help when graphing a function. But wait, don't we have calculators and computers for that? Yes, we do. But calculators are very imprecise when graphing and really just give us a feel for what the function looks like. And, although computer graphing is more precise than calculators, they still don't usually give us exact values and are really just used to show generally what is going on. Once you learn how to use derivatives to describe a graph and how to interpret the results, you will have a much better idea of what is going on and how to use the graph for your application.

The first derivative and the second derivative give you information about the shape of the graph. Each derivative tells you different things but they do parallel one another, i.e. the pattern with the first derivative is repeated in the second derivative, as shown in this table. These may not make sense initially but, after you have studied each, they will.

First Derivative Second Derivative 1. increasing/decreasing intervals 4. concavity 2. critical points    $$f'(x)=0$$ 5. inflection points    $$f''(x)=0$$ 3. first derivative test 6. second derivative test

First Derivative

1. increasing/decreasing intervals

2. critical points    $$f'(x)=0$$

3. first derivative test

Second Derivative

4. concavity

5. inflection points    $$f''(x)=0$$

6. second derivative test

For the discussion of a topic, select a link above, starting with the the first derivative, increasing and decreasing intervals. The topics need to be discussed in order, since they build on one another. Once you have studied each of these topics, come back to this page and we will help you put all your newly-found knowledge together.

Asymptotes

We touched on asymptotes on the continuity page. After you are comfortable with that material, come back here and watch this next video. It has lots of good examples of specific types of vertical asymptotes.

 Krista King Math - How To Find Vertical Asymptotes

Putting It All Together

Okay, at this point you should have read all six topics listed in the table above and worked those practice problems. These practice problems put all those concepts together to build a picture of what a graph should look like and how it behaves. In addition, you will need to have covered specifics about domain and range.

### Search 17Calculus

Practice Problems

Instructions - - Unless otherwise instructed, use the first and second derivative as well as other concepts linked to on this page to sketch the following functions. Show your work by building tables and label your graph carefully and completely.

 Level A - Basic

Practice A01

$$\displaystyle{y=\frac{x-1}{x^2}}$$

solution

Practice A02

$$\displaystyle{f(x)=\frac{x}{x+4}}$$

solution

Practice A03

$$f(x)=3x^2-6x+5$$

solution

 Level B - Intermediate

Practice B01

$$\displaystyle{f(x)=\frac{x}{\sqrt{x^2+1}}}$$

solution

Practice B02

Use the first and second derivatives and other topics on this page to determine if $$\displaystyle{y=5x^{7/5}-x^{3/5}}$$ has a cusp at $$x=0$$.

solution