The Six Pillars of Calculus

The Pillars: A Road Map
A picture is worth 1000 words

Trigonometry Review

The basic trig functions
Basic trig identities
The unit circle
Addition of angles, double and half angle formulas
The law of sines and the law of cosines
Graphs of Trig Functions

Exponential Functions

Exponentials with positive integer exponents
Fractional and negative powers
The function $f(x)=a^x$ and its graph
Exponential growth and decay

Logarithms and Inverse functions

Inverse Functions
How to find a formula for an inverse function
Logarithms as Inverse Exponentials
Inverse Trig Functions

Intro to Limits

Close is good enough
One-sided Limits
How can a limit fail to exist?
Infinite Limits and Vertical Asymptotes

Limit Laws and Computations

A summary of Limit Laws
Why do these laws work?
Two limit theorems
How to algebraically manipulate a 0/0?
Limits with fractions
Limits with Absolute Values
Limits involving Rationalization
Limits of Piece-wise Functions
The Squeeze Theorem

Continuity and the Intermediate Value Theorem

Definition of continuity
Continuity and piece-wise functions
Continuity properties
Types of discontinuities
The Intermediate Value Theorem
Examples of continuous functions

Limits at Infinity

Limits at infinity and horizontal asymptotes
Limits at infinity of rational functions
Which functions grow the fastest?
Vertical asymptotes (Redux)
Toolbox of graphs

Rates of Change

Tracking change
Average and instantaneous velocity
Instantaneous rate of change of any function
Finding tangent line equations
Definition of derivative

The Derivative Function

The derivative function
Sketching the graph of $f'$
Notation and higher-order derivatives

Basic Differentiation Rules

The Power Rule and other basic rules
The derivative of $e^x$

Product and Quotient Rules

The Product Rule
The Quotient Rule

Derivatives of Trig Functions

Two important Limits
Sine and Cosine
Tangent, Cotangent, Secant, and Cosecant

The Chain Rule

Two forms of the chain rule
Version 1
Version 2
Why does it work?
A hybrid chain rule

Implicit Differentiation

Introduction and Examples
Derivatives of Inverse Trigs via Implicit Differentiation
A Summary

Derivatives of Logs

Formulas and Examples
Logarithmic Differentiation

Derivatives in Science

In Physics
In Economics
In Biology

Related Rates

How to tackle the problems
Example (ladder)
Example (shadow)

Linear Approximation and Differentials

An example with negative $dx$

Differentiation Review

Basic Building Blocks
Advanced Building Blocks
Product and Quotient Rules
The Chain Rule
Combining Rules
Implicit Differentiation
Logarithmic Differentiation
Conclusions and Tidbits

Absolute and Local Extrema

The Extreme Value Theorem
Fermat's Theorem

The Mean Value and other Theorems

Rolle's Theorems
The Mean Value Theorem
Finding $c$

$f$ vs. $f'$

Increasing/Decreasing Test and Critical Numbers
The First Derivative Test
Concavity, Points of Inflection, and the Second Derivative Test

Indeterminate Forms and L'Hospital's Rule

What does $\frac{0}{0}$ equal?
Indeterminate Differences
Indeterminate Powers
Three Versions of L'Hospital's Rule


Another Example

Newton's Method

The Idea of Newton's Method
An Example
Solving Transcendental Equations
When NM doesn't work


Anti-derivatives and Physics
Some formulas
Anti-derivatives are not Integrals

The Area under a curve

The Area Problem and Examples
Riemann Sums Notation

Definite Integrals

What is integration good for?
More Examples

The Fundamental Theorem of Calculus

Three Different Quantities
The Whole as Sum of Partial Changes
The Indefinite Integral as Antiderivative
The FTC and the Chain Rule

Concavity, Points of Inflection, and the Second Derivative Test

Concavity and Points of Inflection


  • If the graph of $f$ lies above all of its tangent lines on an open interval, the we say it is concave up on that interval.

  • If the graph of $f$ lies below all of its tangent lines on an open interval, then we say it is concave down on that interval.

  • A point, $P$, on a continuous curve $f(x)$ is an inflection point if $f$ changes concavity there.

When a curve is concave up, it is sort of bowl-shaped, and you can think it might hold water. When it is concave down, it is sort of upside-down-bowl-like, and water would run off of it.


The intervals of concavity can be found in the same way used to determine the intervals of increase/decrease, except that we use the second derivative instead of the first. In particular, since $(f')'=f''$, the intervals of increase/decrease for the first derivative will determine the concavity of $f$:

  1. If possible, factor $f''$. If $f''$ is a quotient, factor the numerator and denominator (separately).

  2. Find all critical numbers $x=s$ of $f'$. These are the points where $(f')'=0$ or $(f')'$ doesn't exist (i.e., the points where $f''=0$ or where $f''$ doesn't exist).

  3. Draw a number line with tick marks at each critical number $s$.

  4. For each interval in which the function $f$ is defined, find the sign of the second derivative $f''$.

  5. If $f''(b) \gt 0$, then $f'$ is increasing on the interval containing $b$. This means that the slopes are increasing, so $f$ is concave up. Draw a right-side-up bowl over that interval on your number line. Similarly, if $f''(b) \lt 0$, draw an upside-down bowl.

  6. That's it! You can now see the intervals where $f$ is concave up or down.

The Second Derivative Test

  • If $f''(x)>0$ for all $x$ in an open interval, then it is concave up on that interval.

  • If $f ''(x) < 0$ for all $x$ in an open interval, then it is concave down on that interval.

Second Derivative Test:
  • If $f'(c)=0$ and $f''(c) \gt0$, then there is a local minimum at $x=c$.

  • If $f'(c)=0$ and $f''(c) \lt 0$, then there is a local maximum at $x=c$.

  • If $f'(c)=0$ and $f''(c)=0$, or if $f''(c)$ doesn't exist, then the test is inconclusive. There might be a local maximum or minimum, or there might be a point of inflection.

The reasoning behind the test is simple: if $f''(c) \gt 0$, then $f'(x)$ is increasing near $x=c$. Since $f'(c)=0$, this means that $f'(x)$ used to be negative and is about to be positive. So the curve bottoms out at $x=c$ and then heads back up. The critical number $x=c$ is the bottom of the concave-up bowl. Likewise, if $f''(c) \lt 0$ and $f'(c)=0$, then $f'(x)$ is decreasing; it used to be positive and is about to be negative. The point $x=c$ is at the top of an upside-down bowl.

Inflection Points

An inflection point is a point where concavity changes sign from plus to minus or from minus to plus.

Example: Find the concavity of $f(x) = x^3 - 3x^2$.

Solution: Since $f'(x)=3x^2-6x=3x(x-2)$,our two critical points for $f$ are at $x=0$ and $x=2$. Meanwhile, $f''(x)=6x-6$, so the only critical point for $f'$ is at $x=1$. It's easy to see that $f''$ is negative for $x \lt 1$ and positive for $x \gt 1$, so our curve is concave down for $x \lt 1$ and concave up for $x \gt 1$, and there is a point of inflection at $x=1$.

As for the critical points, $f''(0)=-6 \lt 0$, so we have a local maximum at $x=0$. $f''(2)=6 > 0$, so we have a local minimum at $x=2$. These results agree with what we got from the first derivative test.

Visual Wrap-up

$f'< 0$ $f'=0$ $f'> 0 $