Unit 1: The Intergral
Preface
This is the note of the MIT Course 18.01.2 Calculus. For more details, refer to MIT 18.x Catalog, and it can also be found at MIT Open Learning Library.
Mean Value Theorem
The Mean Value Theorem (MVT)
If $x(t)$ is continuous on $a\le t\le b$, and differentiable on $a<t<b$, that is, $x^{\prime}(t)$ is defined for all $t$, $a<t<b$, then $$ \frac{x(b)-x(a)}{b-a}=x^{\prime}(c)\qquad \text{for some }c\text{, with }a<c<b $$ Equivalently, in geometric terms, there is at least one point $c$, with $a<c<b$, at which the tangent line is parallel to the secant line through $(a,x(a))$ and $(b,x(b))$:
Upper and Lower Bounds
We have introduced the notion of upper and lower bounds.
A number $M$ is an uppper bound of a function $f(x)$ if
$$
f(x)\le M\quad \text{for all }x
$$
and a number $m$ is a lower bound of a function $f(x)$ if
$$
m\le f(x)\quad \text{for all }x
$$
We can consider upper and lower bounds on the entire real number line, or on an interval.
$$
m\le f(x)\le M
$$
In other words, an upper bound of a function is a number that is larger than or euqual to all values of the function. A lower bound of a function is a number which is smaller than or equal to all values of the function.
Old News
We have been relying on the following fundamental facts whenever we try to understand a function using its derivative. But in fact, these facts are consequences of the MVT.
- If $x^{\prime}(t)\ge 0$ for all $t$ in $(A,B)$, then $x(t)$ is
increasing or staying the sameover $[A,B]$. - If $x^{\prime}(t)>0$ for all $t$ in $(A,B)$, then $x(t)$ is
strictly increasingover $[A,B]$. - If $x^{\prime}(t)\le 0$ for all $t$ in $(A,B)$, then $x(t)$ is
decreasing or staying the sameover $[A,B]$. - If $x^{\prime}(t)< 0$ for all $t$ in $(A,B)$, then $x(t)$ is
strictly increasingover $[A,B]$. - IF $x^{\prime}(t)=0$ for all $t$ in $(A,B)$, the $x(t)$ is
constantover $[A,B]$.
These facts need proofs and their proofs are based on the MVT. The subtlety is that the MVT relates the infinitesimal behviour of the function, the derivative, which is defined at a point, to the macroscopic behviour of the function, the total change over an interval.
Bounding The Average Rate of Change
The equality in the MVT can be used to restrict the range of possible values of the average rate of change and the total change.
More precisely, if there are numbers $m$ and $M$ such that $$ m\le x^{\prime}(c)\le M\qquad\text{for all }c\text{ with }a<c<b $$ that is, $m$ is a lower bound and $M$ is an upper bound on $x^{\prime}(c)$ over $(a,b)$, then the MVT implies the following: $$ \begin{aligned} m\ \ &\le\ \frac{x(b)-x(a)}{b-a}\ \le\ M\qquad \text{(Bounds on the average rate of change)}\\ m\cdot (b-a)\ &\le\ x(b)-x(a)\ \le\ M\cdot (b-a)\qquad \text{(Bounds on the total change)} \end{aligned} $$
In other words, a lower bound on the derivative is also a lower bound on the average rate of change, and an upper bound on the derivative is also an upper bound on the average rate of change. Also, the product of a lower bound on the derivative with the length of an interval, is a lower bound on the total change of the function over that interval. Similarly, the product of an upper bound on the derivative with the length of an interval, is an upper bound on the total change of the function over that interval.
When we know the maximum and minimum values of $x^{\prime}(t)$, we can use them as bounds on $x^{\prime}(t)$ and obtain the following. $$ \begin{alignat*}{2} \min_{a\le t\le b}x^{\prime}(t) &\le \frac{x(b)-x(a)}{b-a} &&\le \max_{a\le t\le b}x^{\prime}(t) \quad\text{(Bounds on the average rate of change)} \\ \min_{a\le t\le b}x^{\prime}(t)\cdot (b-a) &\le x(b)-x(a) &&\le \max_{a\le t\le b}x^{\prime}(t)\cdot (b-a) \quad\text{(Bounds on the total change)} \end{alignat*} $$ In other words, the average rate of change must be in between the maximum and the minimum of the derivative, and the total change must be in between the maximum and minimum of the derivative multiplied by the length of the interval.
Differentials & Antiderivatives
Differential Notation
Let $y=F(x)$, the differential of y is defined as
$$
dy=f^{\prime}(x)\cdot dx
$$
This is also called the differrential of F and denoted dF.
Rearranging this equation, we get the Leibniz notation for the derivative, which says the derivative is the ratio of the two differentials $dy$ and $dx$. $$ F^{\prime}(x)=\frac{dy}{dx}\qquad \left(\text{or}\ \frac{dF}{dx}\right) $$ We may think of the differential of $x$, $dx$, as a “little bit” of $x$, and the differential of $y$, $dy$, as a “little bit” of $y$. Here what we mean by a “little bit” is really an infinitely small bit, we call these infinitely small quantities “infinitesimals”. The point is that even though both $dy$ and $dx$ are infinitely small, their ration is NOT. Their ratio is the derivative $F^{\prime}(x)$. In other words, the differential notation says that $dy$ is proportional to $dx$ with constant of proportionaliyu $F^{\prime}(x)$ even though both $dy$ and $dx$ are infinitely small. We use the differential notation as a tool to keep track of how much $y$ changes when $x$ changes by a tiny tiny … tiny bit.
The geometric picture for the differential is the same as that for linear approximation.
Let us compare the differential notation with the formula for linear approximations.
$$ \begin{alignat*}{2} \text{Linear Approximation at }x\quad \Delta F\quad &\approx\quad F^{\prime}(x)\cdot \Delta x,\quad &&\Delta x\text{ is a finite change in }x\\ \text{Differential Notation}\quad dF\quad &=\quad F^{\prime}(x)\cdot dx,\quad && dx\text{ is a tiny tiny … tiny bit of }x \end{alignat*} $$
Antiderivatives
An antiderivative of $f(x)$ is any function $F(X)$ such that
$$
F^{\prime}(x)=f(x)
$$
Indefinite Integral
Given a function $f(x)$, the indefinite integral or the antiderivative of $f(x)$ is denoted $\displaystyle\int f(x) \ dx$. It is the family of functions
$$
\int f(x)\ dx\ =\ F(x)+C
$$
where $F(X)$ is any antiderivative of $f(x)$, and $C$ is any constant.
We call $\int$ the integral sign, $f(x)$ the integral, and $C$ the constant of integration.
The constant of integration is present in this definition because the derivative of a function determines only the shape of the function, but the derivative does not change if the function is shifted up or down by the same constant everywhere.
Uniqueness of the Indefinite Integral
The indefinite integral $$ \int f(x)\ dx\ =\ F(x)+C $$ is termed “indefinite” since it contains an undetermined constant $C$ and is not just one function but a family of infinitely many functions, parameterized by $C$.
On the other hand, the constant is the only ambiguity of the indefinite integral due to the MVT, which guarantees that any two antiderivatives of the same function can differ only by a constant.
Integrals of Powers
$$ \int x^P\ dx\ =\quad \begin{cases} \dfrac{x^{P+1}}{p+1}, & \text {if $p\ne -1$} \\ \ln{(\vert x\vert)}, &\text{if $p=-1$} \end{cases} $$
First Rules of Integration
$$ \begin{array}{lcc} & \text{Integration Rules} & \text{Differentiation Rules} \\ \text{Constant Multiple: } & \displaystyle\int k f(x)\ dx=k\int f(x)\ dx & d(k F)=k\ dF \\ \text{Sum: } & \displaystyle\int{(f(x)+g(x))\ dx}=\int{f(x)\ dx}+\int{g(x)\ dx} & d(F+G)=dF+dG \end{array} $$ On the other hand, the following naive product and quotient rules do NOT work.
$$ \int f\cdot g\ dx\quad\textbf{DOES NOT EQUAL}\quad(\int f\ dx)\cdot(\int g\ dx)\\ \int \frac f g\ dx\quad\textbf{DOES NOT EQUAL}\quad\frac{\int f\ dx}{\int g\ dx} $$
Method of Substituion
The method of substitution is the integration analogue of the chain rule.
If $$ g(x)\ dx\quad =\quad f(u(x))u^{\prime}(x)\ dx $$ that is, the differential $g(x)\ dx$ is the result of a chain rule, then $$ \begin{aligned} \int g(x)\ dx\quad =&\quad\int f(u(x))u^{\prime}(x)\ dx\\ =&\quad\int f(u)\ du\\ =&\quad F(u(x))+C \end{aligned} $$ where $F(u)$ is an antiderivative of $f(u)$.
Differentialn Equations
Separation of Variables
A differential euqation is separable if it can be written in the form $\frac{dy}{dx}=f(x)\cdot g(y)$.
To solve a separable differential equation, separate all terms involving $x$ from terms involving $y$, and anti-differentiate both sides.
$$ \begin{aligned} \frac{dy}{dx}=&f(x)\cdot g(y)\\ \frac{dy}{g(y)}=&f(x)\ dx\\ \int\frac{dy}{g(y)}=&\int f(x)\ dx \end{aligned} $$
Theorem
Given a differential equation $\frac{dy}{dx}=f(x)\cdot g(y)$ and an initial condition $y(a)=b$, if $f$, $g$ and $g’$ are continuous near $(a,b)$, then there is a unique function $y$ whose derivative is given by $f(x)\cdot g(y)$ and that passes through the point $(a,b)$.
Slope Fields
The slope field is a diagram that helps us to visualize the information in a first order differential equation. The slope field is obatined as follows. At each point $(x,y)$, you draw a short segment whose slope is the value of $y^{\prime}$ at the point $(x,y)$. The solution curves must be tangent to the slope filed at all points.
Euler’s Method
Given the differential equation $\frac{dy}{dx}=x+y$:
- Choose a step size $h$ (The smaller the step size, the more accurate the approximation.)
- let $(x_0,y_0)$ be the initial condition.
- We can use the differential equation and step size to determine the value of the function at $x_1=x+h$: $$ \begin{aligned} x_1=&x_0+h\\ y_1=&y_0+(x_0+y_0)\cdot h\quad\text{Linear Approximation} \end{aligned} $$
- Iterate this process: $$ \begin{aligned} x_{k+1}=&x_k+h\\ y_{k+1}=&y_k+(x_k+y_k)\cdot h\quad\text{Linear Approximation} \end{aligned} $$
Modeling a Zipline
Hyperbolic Sine and Cosine
Recall that the hyperbolic cosine and sine are defined by the relationships $$ \begin{aligned} \cosh(t)=& \frac 1 2 (e^t+e^{-t})\\ \sinh(t)=& \frac 1 2 (e^t-e^{-t}) \end{aligned} $$ The basic trignonometric functions are related to the geometry of a circle. These hyperbolic trig functions are related to the geometry of hyperbola.
Geometric Description of Trig Functions
The poinr labeled in the image, $(\cos(\theta),\sin(\theta))$, is defined to be the point on the cricle $x^2+y^2=1$ such that the shaded area is $\theta$.
A circle of radius 1 centered at the origin is plotted in the x y plane. A point is indicated on the circle in the first quadrant and makes an angle of theta with the positive x axis. A portion of the circle is shaded that indicates the area swept over the range of angles from minus theta to theta.
Geometric Description of Hyperbolic Trig Functions
The point labeled in the image, $(\cosh(t),\sinh(t))$, is defined to be the point on the hyperbolan $x^2-y^2=1$ such that the area of the shaded region is $t$.