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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
Definition
Onesided Limits
How can a limit fail to exist?
Infinite Limits and Vertical Asymptotes
Summary
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 Piecewise Functions
The Squeeze Theorem
Continuity and the Intermediate Value Theorem
Definition of continuity
Continuity and piecewise 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 $f'$
Differentiability
Notation and higherorder 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
Summary
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
Overview
How to tackle the problems
Example (ladder)
Example (shadow)
Linear Approximation and Differentials
Overview
Examples
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
Definitions
The Extreme Value Theorem
Fermat's Theorem
Howto
The Mean Value and other Theorems
Rolle's Theorems
The Mean Value Theorem
Finding $c$
$f$ vs. $f'$
Increasing/Decreasing Test and Critical Numbers
Howto
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
Proofs
Optimization
Strategies
Another Example
Newton's Method
The Idea of Newton's Method
An Example
Solving Transcendental Equations
When NM doesn't work
Antiderivatives
Antiderivatives and Physics
Some formulas
Antiderivatives are not Integrals
The Area under a curve
The Area Problem and Examples
Riemann Sums Notation
Summary
Definite Integrals
Definition
Properties
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


Finding $c$
Rolle's theorem and the Mean Value Theorem
say that $f'(x)$ takes on a certain value at some point $c$ between
$a$ and $b$. But how do you find such a $c$? Is there only one?
Example 1:
Suppose that $f(x) = x^3 6x^2 + 5x$. Plugging in, we see that
$f(0)=0$ and $f(1)=0$. (The third root is $x=5$, by the way).
By Rolle's theorem, there has to be a $c$ between 0 and 1 with
$f'(c)=0$. What is it?
Solution: To find $c$, we first compute
$f'(x)= 3x^2  12x + 5$. So
$$0=f'(c) = 3c^2  12 c + 5. $$
We then solve this with the quadratic formula:
$$c = \frac{12 \pm \sqrt{(12)^2  4(3)(5)}}{6} =
\frac{12 \pm \sqrt{84}}{6} = \frac{6 \pm \sqrt{21}}{3}$$
Since we are looking for something between 0 and 1, we want $c =
(6\sqrt{21})/3 \approx 0.4725$ instead of $(6+\sqrt{21})/3 \approx
3.5275$. (If we had used Rolle's theorem with $a=1$ and $b=5$, we
would have wanted the larger value of $c$.)

This answer isn't a particularly nice number. But nothing guarantees
that realworld problems will have simple answers!
Example 2:
Consider the function $f(x) = e^x$ between $x=1$ and $x=3$.
Find a point where the tangent to the curve $y=f(x)$ is parallel to
the line between $(1,e)$ and $(3,e^3)$.
Solution: This is the Mean Value Theorem
with $f(x)=e^x$, $a=1$ and $b=3$. We solve it in steps:
 We compute $\displaystyle \frac{f(b)f(a)}{ba} = \frac{e^3e}{2}$.

We look for c. Since $f'(x)=e^x$,
${\displaystyle f'(c) = e^c = \frac{e^3e}{2}}$, so
${\displaystyle c = \ln \left ( \frac{e^3e}{2} \right ) \approx 2.1614}$.

Once we have $c$, we have our point
$$ (c, e^c) = \left (\ln \left (\frac{e^3e}{2} \right ),
\frac{e^3e}{2} \right) \approx (2.1614, 8.684).$$

