Logarithm

Definition. If $x>0$, define
MATH

It follows from the Fundamental Theorem of Calculus that $\ln x$ is differentiable and
MATH
For each fixed $y>0$, define $f(x)=\ln (xy)$. Then MATH. Therefore there is a constant $c$ so that
MATH
Setting $x=1$, we see that
MATH
Therefore
MATH

Corollary. If $n$ is a natural number and $x>0$, then
MATH

Corollary. If $x,y>0$, then
MATH

Exponential Function

Since MATH for all $x$, $\ln $ is an increasing function on $(0,+\infty )$. It is unbounded from above:
MATH
and it is unbounded from below:
MATH
Therefore the inverse function $\ln ^{-1}$ of $\ln $ exists. The exponential function $\exp $ is defined as $\ln ^{-1}$. Thus $\exp (x)$ is the unique positive number such that
MATH
holds for all real number $y.$

Derivative of Exp

Since MATH holds for all $x>0$, it follows that, writing $y=\ln \ x$ and $y_0=\ln \ x_0,$
MATH
Therefore $\exp $ is differentiable at any $y_0$ with MATH
MATH

Write MATH, MATHThen MATH, MATH. Then
MATH
so
MATH

Definition of $e$

Definition: $e=\exp (1).$

Thus $e$ is the unique number satisfying
MATH

As
MATH
and
MATH
it follows that
MATH

Since MATH, it follows that
MATH
holds for all integers $n$ and
MATH
holds for all integers $m,n$ with $m\neq 0$, i.e.,
MATH
holds for all rational numbers.

Definition. For any number $x$, $e^x=\exp (x).$

Note that if $x$ is rational, then
MATH

Definition. If $a>0$, then for any real number $x,$
MATH

In case $e=a$, this definition is consistent with the earlier definition.

Properties of $a^x$

(1) If $a>0$, then $(a^b)^c=a^{(bc)}$ for all $b,c.$

(2) If $a>0$, then $a^1=a$, MATH for all $x,y.$

Uniqueness of the solution to MATH

Theorem. If $f$ is differentiable and
MATH
then there is a number $c$ such that
MATH

Proof. Consider
MATH
Then
MATH
Therefore $g$ must be constant $c.$

Properties of Integrals

Corollary: If $f$ is integrable on $[a,b]$, then $|f|$ is integrable on $[a,b]$ and
MATH

Proof: MATH

unless there exists a constant $c$ such that $g=cf.$

Proof. Write
MATH

The algebraic properties of integrals show that

(i) $<f,f>>0$ unless $f\equiv 0.$

(ii) $<f,g>=<g,f>$

(iii) MATH

(iii') MATH

We need to show that
MATH

Case (1) $g\equiv 0$. The result holds since both sides are zero.

Case (2)$<g,g>>0$. Then
MATH

MATH

MATH
Therefore
MATH

If the equality holds then
MATH
consequently
MATH

If $f$ and $g$ are integrable on $[a,b]$, then
MATH

Proof.
MATH

MATH

MATH

Continuity of Indefinite Integrals

Recall that the first fundamental theorem of calculus asserts that if $f$ is integrable on $[a,b]$ and if $f$ is continuous at $c$ then the function defined by $F(x)=\int_a^xf$ is differentiable at $c$ and MATH. Note that the function $f$ is not assumed to be continuous throughout $[a,b]$. The following result, however, does hold true.

Theorem. If $f$ is integrable on $[a,b]$ and $F$ is defined by
MATH
then $F$ is continuous on $[a,b].$

Proof. Fix $c\in [a,b]$. Since $f$ is integrable on $[a,b]$ it is, by definition, bounded on $[a,b]$. Choose $M>0$ so that $|f(x)|\leq M$ for all $x\in [a,b]$. Let $\epsilon >0$ be given. Set MATH. We now show that MATH for $|h|<\delta .$

Case (1). $0<h<\delta $. Then
MATH

Since
MATH
it follows that
MATH
Therefore,
MATH

Case (2). $-\delta <h<0$. Then
MATH
Since
MATH
it follows that
MATH
Therefore,
MATH
Thus MATH in both cases.

Mean Value Theorem for Integrals

Theorem. Suppose that $f$ is continuous on $[a,b]$ and that $g$ is integrable and nonnegative on $[a,b]$. Then there exists $\xi $ in $[a,b]$ such that
MATH

Corollary. Suppose that $f$ is continuous on $[a,b]$. Then there exists $\xi $ in $[a,b]$ such that
MATH

Proof. Since $f$ is continuous on $[a,b]$, there exists $\alpha $ in $[a,b]$ such that
MATH
and there exists $\beta $ in $[a,b]$ such that
MATH
It follows from the inequality
MATH
that
MATH

If $g\equiv 0$ then the result clearly holds. If $g$ is not identically $0$ then $\int_a^bg(x)dx>0$ and so
MATH
Now the function $h$ given by
MATH
is positive for $t=\beta $ and negative for $t=\alpha $. It follows from the Intermediate Value Theorem that there exists $\xi $ (between $\alpha $ and $\beta )$ with $h(\xi )=0.$

Riemann Sums

Let $a<b$. Let $P$ be a partition of the interval $[a,b]$ consisting of
MATH

For a bounded function $f$ defined on $[a,b]$ and for each $i$ fix a point $t_i$ in the interval $[x_{i-1},x_i]$. Then
MATH
Any sum of the form
MATH
is called a Riemann sum of $f$ for $P.$

Theorem. Suppose that $f$ is continuous on $[a,b]$. Then for every $\epsilon >0$ there is some $\delta >0$ such that, if $P$ be a partition of the interval $[a,b]$ consisting of
MATH
with MATH for all $i$, then
MATH
for any Riemann sum formed by choosing $t_i$ in $[x_{i-1},x_i].$

Proof. Given $\epsilon >0$, choose $\delta >0$ so that for all $x,y$ in $[a,b]$
MATH
This is possible since $f$ is uniformly continuous on $[a,b]$. Now for any partition $P$ satisfies the stated condition we have
MATH
But we have also
MATH
and
MATH
The result follows from the above three inequalities.

Length of a Parametric Curve

Let a curve be given parametrically by
MATH
For any partition $P$
MATH
of $[a,b]$ the total length of the polygonal line segments connecting
MATH
where $x_i=x(t_i)$ and $y_i=y(t_i)$ for all $i$, is given by
MATH

MATH
We now show that for each given $\epsilon >0$ there exists a $\delta >0$ such that if all $t_i-t_{i-1}$ in the partition $P\ $are $<\delta $, then
MATH

(1) For the given $\epsilon >0$, there exists $\delta _1>0$ so that
MATH
whenever $P$ is a partition
MATH
with MATH for all $i$.

(2) Since the function MATH is uniformly continuous for all $s > 0$, it follows that there exists $\epsilon _1$ such that


MATH

(3) Since the function $x^{\prime }$ is continuous on $[a,b]$, it is uniformly continuous there. Thus there exists $\delta _2>0$ such that
MATH
Let MATH. Consider a partition $P$ consisting of
MATH
with MATH for all $i$. The Mean Value Theorem shows that there exist $\xi _i$ and $\eta _i$ in $(t_{i-1},t_i)$ such that
MATH
Since MATH, it follows that
MATH
and so
MATH

MATH

The desired result follows since
MATH

Length of a Parametric Curve

If $x^{\prime }$ and $y^{\prime }$ are continuous on $[a,b]$, the length of the parametric curve
MATH
is defined to be the definite integral
MATH

Length of the Graph of a Continuously Differentiable Function

If $f^{\prime }$ is continuous on $[a,b]$, then the length of its graph is the same as the length of the parametric curve
MATH
Therefore it is given by
MATH

Length of a Polar Curve

If the curve is given by the polar equation
MATH
It is just a particular instance of the parametric equations
MATH
Since
MATH
and
MATH
it follows that
MATH
so the length of its graph is given by
MATH

Indefinite Integral of a Quadratic Rational Function

MATH

Example. Find MATH

Set MATH. Then MATH and MATH. Thus
MATH
Since MATH, MATH, we have
MATH

MATH

Method of Completing the Square


MATH

Example. MATH

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