Calculus 1

$\text{Chapter 1: Functions and limits}$

1.1: Functions and limits

• Function
  • independent vs dependent
• Qualitative & quantitative data (continuous, discrete)
• Function presentation:
  • cartesian graph
  • formula
  • descriptive
  • table
• Vertical line test: no value of domain has 2 values in range
• Piecewise defined function: different formulae for different parts of the domain
• Symmetric property: put $-x$ as input to check
  • even: $f(-x)=f(x)$
  • odd: $f(-x)=-f(x)$
• Increasing vs decreasing function

1.2: A catalog of essential functions

• Mathematical model is a mathematical description of a real-world phenomenon

  • linear function: $y=mx+c$
  • polynomial function: $y=a{x}^3+b{x}^2+cx+d$
    • the one-term function’s degree of the polynomial dictates if the graph is odd or even
  • power function:
    • type 1: $y=x^a$
    • type 2: $y=x^{1/n}$, $n^{th}$ roots of $x$
    • reciprocal: $y=x^{-1}$
  • rational function: ratio of 2 polynomial, $f(x)=P(x)/Q(x)$

• Transformation of functions:

  • Vertical and horizontal shifts
  • Vertical and horizontal stretching, shrinking and reflecting

• Combinations of functions: plus minus divide multiply, $(f+g)(x)=f(x)\pm g(x)$

  • Domains of the new function is $f\wedge g$, because both functions have to be defined

• Composite function : $(f\circ g)(x)=f(g(x))$

1.3: The limit of a function:

• $\underset{x\to a}{\lim }f(x)=L$, limit of $f(x)$ as $x$ approaches $a$ equals to $L$, but not a
  • if $x=a$ is not defined, draw a table to check the closest value $x=a\pm 0.01$ thus assuming the limit of a function for an undefined constant
• One-sided limit:
  • $\underset{x\to a^{+}}{\lim }f(x)=L$ means $x$ approaches $a$ from the right side, $a^{-}$ means from the left
  • if $\underset{x\to a^{-}}{\lim }f(x)\ne \underset{x\to a^{+}}{\lim }f(x)$ then the curve has discontinuity and limit doesn’t exist

1.4: Calculating limits:

• Limit laws: only if both exists and finite

• Special trig limits:
  • $\underset{\theta \to 0}{\lim }\frac{\cos \theta -1}{\theta }=0$
  • $\underset{\theta \to 0}{\lim }\frac{\sin \theta }{\theta }=1$, convert to this

• Split big term into a lot of smaller terms with the same limit approaching same x

1.5: Continuity:

• A function $f(x)$ is continuous if and only if:
  • $f(a)$ is defined
  • $\underset{x\to a}{\lim }f(x)$ exist, both from the left and right
  • $\underset{x\to a}{\lim }f(x)=f(a)$
• Removable discontinuity: only one value of the graph can be written in piecewise
• Infinite discontinuity: limit tends to infinity. Jump discontinuity: stair form of graph
• Polynomials, rational, functions, trigonometric functions are always continuous
• The intermediate value theorem: in a closed interval of $a<x<b$, the line cant jump thru $f(a)$ and $f(b)$ so $f(c)=0$f if $a<c<b$
• Squeeze theorem:
  • If $f(x)\le gx\le h(x)$ when $x$ is near a and $\underset{x\to a}{\lim }f(x)=\underset{x\to a}{\lim }h(x)=L$
  • Then $\underset{x\to a}{\lim }g(x)=L$

1.6: Limits involving infinity:

• Infinite limit at $a$ makes a vertical asymptote at $x=a$
• Horizontal asymptote: when a close enough constant will tends the increase of $f(x)$ to another constant
• if $P(x)$ and $Q(x)$ are functions of $x^n$:
  • $\underset{x\to \infty }{\lim }\frac{P(x)}{Q(x)}=\lim \frac{\text{leading term of P}}{\text{leading term of Q}}$ bcz the biggest term would dominate

$\text{Chapter 2: Derivative}$

2.1: Derivative and rate of change:

• Tangent problem, limit definition

• Instantaneous rate of change, $f^{\prime }(a)=m=\underset{x\to a}{\lim }\frac{f(x)-f(a)}{x-a}=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}=\underset{\Delta x\to 0}{\lim }\frac{\Delta y}{\Delta x}$

  • first principle, second principle

  

2.2: The derivative as a function:

• If $f(x)$ is differentiable at $a$, then $f(x)$ is continuous at $a$

  • Not necessarily differentiable if $f(x)$ is continuous

• The derivative of a function might not exist if at $a$, it’s:

2.3: Basic differentiation formulas:

• $\frac{d}{dx}(x^n)=n{x}^{n-1}$ for $n$ of any real number
• Derivative rules: Product rule, Sum & difference rule
• $Sinx\underset{dx}{\overset{1}{\to }}Cosx\underset{dx}{\overset{1}{\to }}-Sinx\underset{dx}{\overset{1}{\to }}-Cosx\underset{dx}{\overset{1}{\to }}Sinx$

2.4: The product and quotient rule:

• Product rule, $(fg{)}^{\prime }=f{g}^{\prime }+g{f}^{\prime }$
• Quotient rule, $(\frac{f}{g}{)}^{\prime }=\frac{g{f}^{\prime }-f{g}^{\prime }}{g^2}$
• Trigonometry functions:
  • Find $\frac{d}{dx}Tan(x),\frac{d}{dx}Csc(x),\frac{d}{dx}Sec(x),\frac{d}{dx}Cot(x)$ by quotient rule

2.5: The chain rule, power rule combined with the chain rule:

2.6: Implicit differentiation:

• Differentiate both $x$ and $y$ at the same time, $\frac{d}{dx}y=\frac{dy}{dx}$

2.7: Related rate of change:

• Connected rate of change

2.8: Linear approximations and differentials:

• $L(x)\approx f(x)=f(a)+f^{\prime }(a).(x-a)$ bcz $=f(a)-f^{\prime }(a).a+f^{\prime }(a).x$
• $dy=f^{\prime }(x).dx$ and $dx$ can be an independent variable and take a constant

Extra:

• Marginal cost: cost of making a new product given x product is already produced
  • Total cost of x products: $f(x)$ then marginal cost
• Logistic function: infinitively increasing or decreasing
  • Derivative of logistic function is called surge function, rise quickly then level down

$\text{Chapter 3: Inverse Functions}$

3.1: Exponential functions:

• Laws of exponential
• Natural exponential, $e=\underset{x\to 0}{\lim }(1+x{)}^{1/x}$

3.2: Inverse functions and logarithms:

• One-to-one function, where 1 value of x has 1 value of y and vice versa

  • No horizontal line intersect the graph more than once, thus invertible
  • If vertical line intersects the graph more than once then it’s not a function

• The domain of f(x) becomes range of its inverse and vice versa

• The graph of $f^{-1}x$ is reflection of $fx$ over the line $y=x$

• For inverted function:

  $f(y)=x⟹f^{\prime }y.\frac{dy}{dx}=1⟹f^{\prime }y=\frac{1}{dy/dx}$

  • Gradient at point $(a,f^{-1}a)$ on $f^{-1}x$ is equal to reciprocal of $f^{\prime }x$ at $(f^{-1}a,a)$
  • $(f^{-1}{)}^{\prime }a=\frac{1}{f^{\prime }(f^{-1}a)}$

• Laws of logarithms
• ${\log }_{a}x=\frac{\ln x}{\ln a}$

3.3: Derivatives of logarithmic and exponential functions:

• $f(x)={\log }_{a}x=\frac{\ln x}{\ln a}⟹f^{\prime }(x)=\frac{1}{x}{\log }_{a}e=\frac{1}{x.\ln a}$
• $\frac{d}{dx}(\ln x)=\frac{1}{x}$
• Logarithmic differentiation:
  • Take natural log both sides and use laws of log to simplify
  • Differentiate implicitly
  • Make dy/dx the subject
• $f(x)=a^x⟹\frac{d}{dx}(a^x)=a^x\ln a$
• Using logarithmic differentiation: take log before differentiating

3.4: Exponential growth and decay:

• Law of natural growth and decay, $\frac{dy}{dt}=ky$, rate of change is proportional to its size
  • Solution is in the form: $y=C.e^{kt}$
• Radioactive decay, $\frac{dm}{dt}=-km$, rate of decay is proportional to its mass
  • Solution is in the form, $m=m_o.e^{-kt}$
• Decreasing, losing, leaking, cooling down,.. means the rate of change is negative
• First order differential equation
• Newton’s law of cooling: rate of cooling an is proportional to the temperature difference between the object and its surroundings
  • $\frac{dT}{dt}=k(T-T_0)$ then solution is $f(t)=y_0{e}^{kt}$

3.5: Inverse trigonometric functions:

• Differentiate implicitly for inverse trig

• ${\cos }^{2}x+{\sin }^{2}x=1$

• ${\sec }^{2}x=1+{\tan }^{2}x$

• ${\csc }^{2}x={\cot }^{2}x+1$

• $\frac{d}{dx}({\sin }^{-1}(x))=\frac{1}{\sqrt{1-x^2}}$

  • $-\frac{\pi }{2}\le x\le \frac{\pi }{2}$ and $-1\le f(x)\le 1$

• $\frac{d}{dx}({\cos }^{-1}(x))=-\frac{1}{\sqrt{1-x^2}}$

  • $0\le x\le \pi$ and $-1\le f(x)\le 1$

• $\frac{d}{dx}({\tan }^{-1}(x))=-\frac{1}{1+x^2}$

  • $-\pi /2\le x\le \pi /2$ and $-\infty \le f(x)\le \infty$

3.6: Hyperbolic functions:

• $\sinh (x)=\frac{e^x-e^{-x}}{2}$
• $\cosh (x)=\frac{e^x+e^{-x}}{2}$
• Inverse hyperbolic functions

3.7: Indeterminate forms and l’hospital’s rule:

• Form of type $\frac{0}{0}$ and $\frac{\infty }{\infty }$ where quotient of 2 functions have same limit makes it indeterminable

• Form of type $0\times \pm \infty$ where $\underset{x\to a}{\lim }f(x).g(x)$ as $\underset{x\to a}{\lim }f(x)=0$ and $\underset{x\to a}{\lim }g(x)=\pm \infty$

  • Convert the multiplication to $\underset{x\to a}{\lim }\frac{f(x)}{g(x)}$ to solve by l’hospital’s rule

• Form of type $\infty -\infty$ where $\underset{x\to a}{\lim }[f(x)-g(x)]$ as $\underset{x\to a}{\lim }f(x)=\infty$ and $\underset{x\to a}{\lim }f(x)=\infty$

  • Convert the difference to one fraction with common denominator

• Forms of powers types: ($\underset{x\to a}{\lim }[f(x){]}^{g(x)})$

  • $0^0$ where $\underset{x\to a}{\lim }f(x)=0$ and $\underset{x\to a}{\lim }g(x)=0$
  • ${\infty }^0$ where $\underset{x\to a}{\lim }f(x)=\infty$ and $\underset{x\to a}{\lim }g(x)=0$
  • $1^{\infty }$ where $\underset{x\to a}{\lim }f(x)=1$ and $\underset{x\to a}{\lim }g(x)=\pm \infty$

  $⟹$ $\ln y=\ln f(x{)}^{g(x)}=g(x).\ln |f(x)|$ or write $f(x)=e^{\ln (f(x)}$

  $⟹$ then find $\underset{x\to a}{\lim }\ln y$ then find $y=e^{\ln y}$

• l’Hospital’s rule, a systematic method, for evaluating only indeterminate forms

  • $\underset{x\to a}{\lim }\frac{f(x)}{g(x)}=\underset{x\to a}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}$
  • Also true for $x\to a^{+},a^{-},\infty ,-\infty$ and one-sided limits

• Use l’hospital’s rule 2 times if the result of the first time applied is still indeterminable

$\text{Chapter 4: Applications of differentiation}$

4.1: Max and min values:

• Extreme values are absolute max and min values or endpoints(if included)
  • They could be one of the critical number or one of the end points
• Local maximum and local minimum value are bigger than others value near it
  • $a$ is a local max or min if $f^{\prime }(a)=0$, by Fermat’s theorem
• Extreme value theorem: If $f$is continuous on a closed interval $[a,b]$ then $c$ and $d$ are min and max somewhere in the middle of $[a,b]$
• Fermat’s theorem: if $f$ has local min or max at $c$ and if $f^{\prime }(c)$ exist then $f^{\prime }(c)=0$
• Critical number of function $f(a)$ are numbers that $f^{\prime }(a)=0$ or $f^{\prime }(a)$ doesn’t exist
  • If $f$ has local min or max at $c$, then $c$ is a critical point of $f$
  • End points has has no $f^{\prime }(a)$, thus they are critical numbers

4.2: The mean value theorem:

• By Rolle’s theorem, if:

  • $f$ is continuous on the closed interval $[a,b]$
  • $f$ is differentiable on on interval $(a,b)$
  • $f(a)=f(b)$

  $⟹$ then there is a number in the interval $(a,b)$ such $f^{\prime }(c)=0$

• Prove that the equation has only 1 root:

  • Given the equation is continuous, find $a$ and $b$ where $f(a)<0$ and $f(b)>0$
    • Thus, $a<\text{root}<b$
  • Show that $f^{\prime }(x)\ne 0$ or $f^{\prime }(x)<>0$ for all $x$ so the curve never head back down again

• By the mean value theorem, if:

  • $f$ is continuous on the closed interval $[a,b]$
  • $f$ is differentiable on on interval $(a,b)$

  $⟹$then a number in the interval $(a,b)$ such $f(b)-f(a)=f^{\prime }(c)(b-a)$, same gradient

4.3: Derivatives and the shapes of graphs:

• Increasing/decreasing test

• if $f^{\prime }(x)$ of one interval (boundaries are critical points)’s sign is known , we can’t assume the 2 neighbour intervals’ signs because of inflexion point

• if $f^{\prime }(x)$ change from negative to positive then it’s local minimum point and same for maximum

• Second derivative, $f^{\prime \prime }(x)$: concavity

  • $f^{\prime \prime }(x)=0$ at inflection point, the point on a continuous curve where the graph change its concavity from CU to CD or vice versa
    • only when $f^{\prime }(x)$ change signs, from negative to positive or vice versa
      • meaning the graph of $f^{\prime \prime }x$ cross x-axis
  • $f^{\prime \prime }(x)>0$ then the curve concave upward
  • $f^{\prime \prime }(x)<0$ then the curve concave downward

• Draw graph by including:
  • $f^{\prime }(x)$ with critical points $f^{\prime }(x)=0$ and its intervals
  • $f^{\prime \prime }(x)$ with critical points $f^{\prime \prime }(x)=0$ and its intervals

4.4: Curve sketching:

• Domain
• Intercepts
• Symmetry, periodic
• Asymptotes, limits
• Interval of increase and decrease
• Local max and min points
• Concavity and points of inflection, includes all intervals of both differentiations
• Sketch, good luck

4.5: Optimisation problems:

• Form 1 big equation then differentiate implicitly
• In business and economics:
  • $R(x)=x.p(x)$ where $R(x)$ is revenue function and $P(x)$ is price function, x is the nb of products
  • Function of profit, $P(x)=R(x)-C(x)$ where $C(x)$ is cost function

4.6: Newton’s method:

• To find the roots of polynomials of degrees $>4$, using linear approximation
• The root =\lim\limits_{n\to \infin}x_n$$=\lim\limits_{m\to \infin}x_n$$=\lim\limits_{x\to \infin}x_n

$x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}$

$\text{Chapter 5: Integrals}$

5.1: Areas and distances:

• Area under graph by the sum of the areas of approximating rectangles:
  • Principle is as same as limits of area, the $f(x)$ value can be:
    • Left end point of the rectangle
    • Right end point of the rectangle
    • Take $f(x)$ as the midpoint of 2 of its endpoints, called sample points $x_n^{\ast }$
      • $x_n^{\ast }=\frac{x_n+x_{n+1}}{2}$, not mid-value ($\ne \frac{f(x_1)+f(x_2)}{2}$)

5.2: The definite integral:

• By Riemann sum, $f(x)$ is defined on $[a,b]$, then : $A\approx \underset{n\to \infty }{\lim }\sum _{i=1}^{n}f(x_i^{\ast })\Delta x$

  • where $\Delta x_i$ is separate subinterval length for each subinterval, need not to be equal
  • sample point, $x_i^{\ast }$ can be anywhere on the $i^{th}$ subinterval, need not to be mid point
  • If the limit exist then $A={\int }_a^{b}f(x)dx$

• Net change theorem: ${\int }_a^b\frac{dfx}{dx}dx=f(b)-f(a)$

  • The integral of rate of change of a function equals to the net change in the function
  • If the area has both positive and negative values then the integral gives net area

• $\sum _{i=1}^{n}i=\frac{n(n+1)}{2}$

• $\sum _{i=1}^n{i}^2=\frac{n(n+1)(2n+1)}{6}$

• $\sum _{i=1}^n{i}^3=[\frac{n(n+1)}{2}{]}^2$

• Definition of integral: If $f$ is integrable on $[a,b]$ then

  • ${\int }_a^{b}f(x)dx=\underset{n\to \infty }{\lim }\sum f(x_i).\Delta x$ where $x_i=a+i.\Delta x$

• Midpoint rule: the midpoint of each rectangle is used to compute

  • ${\int }_a^{b}f(x)dx\approx \sum _{i=1}^{n}f(\bar{x}).\Delta x$ where $\bar{x}=\frac{1}{2}(x_{i-1}+x_i)$

• Properties of the integrals: the area under graph is smaller than the large rectangle and bigger than the smaller rectangle

  • If $m\le f(x)\le M$ for $a\le x\le b$:

    $⟹m(b-a)\le {\int }_a^{b}f(x)dx\le M(b-a)$

5.3: Evaluate definite integrals:

• Evaluation theorem:
  • ${\int }_a^{b}f(x)dx=F(b)-F(a)$ where $F^{\prime }(x)=f(x)$
• Definite integral (without limits) results to a constant number
• Indefinite integral (without limits) is a function and requires to add C, arbitrary constant

$\int a^{x}dx\text{***}$

5.4: The fundamental theorem of calculus:

• If $f$ is continuous and $g(x)={\int }_a^{x}f(t)dt$ then $g(x)$ is a function of area under graph $f$
• The mean value theorem for integrals:
  • If $f$ is continuous on $[a,b]$ then there is a number, mean value, between a and b where $f(c)=\frac{1}{b-a}{\int }_a^{b}f(x)dx$
    • So mean value = area/domain

5.4: The substitution rule:

• $\int f[g(x)]g^{\prime }(x)dx=\int f(u)du$ where $u=g(x)$ then use $g^{\prime }(x)$ to cancel out term in $f(x)$
  • Where the limits in terms of $x$ is changed to limits in term of $u$

$\text{Chapter 6: Techniques of integration}$

6.1: Integration by parts

6.2: Trigonometric integrals and substitution:

• Try to write an integrand involving powers of sine and cosine in a form where we have only one sine factor (and the remainder of the expression in terms of cosine) and vice versa

  • Then use ${\cos }^2(x)+{\sin }^2(x)=1$

• Half-angle formulae:

  • $\cos 2x=1-2{\sin }^{2}x=2{\cos }^{2}x-1$
  • $\sin 2x=2\cos x\sin x$

• $\int \tan xdx=\ln |\sec x|+c$

• $\int \sec xdx=\ln |\sec x+\tan x|+c$

• Integrate by substitution:

  • To transform difficult integral to simpler integral
  • The new integrand must be rewritten in terms of the new variable, u.
  • Either cancel out some terms of x when substitute u in or rewritten entire integrand in term of u
  • Convert limits of $x$ to limits in term of $u$ for definite integral

• Inverse substitution

6.3: Partial fractions:

• Case 1: $\frac{px+q}{(ax+b)(cx+d)}=\frac{A}{ax+b}+\frac{B}{cx+d}$
• Case 2: $\frac{p{x}^2+qx+r}{(ax+b{)}^3{x}^2}=\frac{A}{(ax+b{)}^3}+\frac{B}{(ax+b{)}^2}+\frac{C}{ax+b}+\frac{D}{x^2}+\frac{E}{x}$
• Case 3: non factorisable $\frac{p{x}^2+qx+r}{(ax-b)(c{x}^2+d)}=\frac{A}{ax-b}+\frac{Bx+C}{c{x}^2+d}$, one degree lower
• Case 4: repeat non factorizable $\frac{p{x}^2+qx+r}{(ax-b)(c{x}^2+dx+e{)}^n}=\frac{A}{ax-b}+\frac{Bx+C}{(c{x}^2+bx+e{)}^1}+.+\frac{Yx+Z}{(c{x}^2+bx+e{)}^n}$

6.5: Approximate integration:

• Types of area approximation:
  • Left-end points
  • Right-end points
  • Trapezoidal Rule
  • Mid-point
  • Simpson’s rule
• Trapezoidal rule: (trapezium rule)
  • ${\int }_a^{b}f(x)dx=\frac{\Delta x}{2}[f(x_0)+2f(x_1)+2f(x_2)+...+f(x_n)]$
    • Use sum of y-values, different from Riemann’s sum
• Error bounds:
  • For $|f^{\prime \prime }(x)|\le K$ and $a\le x\le b$:
    • Error for trapezoidal: $|E_T|\le \frac{K(b-a{)}^3}{12n^2}$
    • Error for mid-point approximation: $|E_M|\le \frac{K(b-a{)}^3}{24n^2}$
• Simpson’s rule: Approximate a curve by another parabola

  • Any 3 consecutive points on the curve can form a parabola form $A{x}^2+Bx+C$

  • Take left, middle and right end points and find the Area, we have

    ${\int }_a^{b}f(x)dx\approx \frac{\Delta x}{3}[1f(x_0)+4f(x_1)+2f(x_2)...+2f(x_{n-2})+4f(x_{n-1})+1f(x_n)]$

    • Start with 1, then alternate between 4 and 2 then end with 1

  • Error bound

    • For $|f^{4^{\prime }}(x)|<K$ and $a\le x\le b$:
      • Error for Simpson’s rule: $|E_S|\le \frac{K(b-a{)}^5}{180n^4}$

6.6: Improper integrals:

• Improper Type 1:

  • If the limit exists then improper integral converges, otherwise diverges
  • ex: ${\int }^{\infty }{}_1\frac{1}{x}dx=\underset{a\to \infty }{\lim }{\int }_1^a\frac{1}{x}dx=\underset{a\to \infty }{\lim }\ln (a)⟹$diverges

• Improper Type 2: discontinuous integrands

  • If the limit exists then improper integral converges otherwise diverges

  • If there is vertical asymptotes then split the integral:

    ${\int }_a^{b}f(x)dx={\int }_a^{c}f(x)dx+{\int }_c^{b}f(x)dx$

• Comparison theorem: for $f(x)>g(x)>0$ for $x>a$
  • If ${\int }_a^{\infty }f(x)dx$ is convergent, then ${\int }_a^{\infty }g(x)dx$ converges
  • If ${\int }_a^{\infty }g(x)dx$ is divergent, then ${\int }_a^{\infty }f(x)dx$ diverges

$\text{Chapter 7: Applications of integration}$

7.1: Area between curves:

• The area of region bounded by curve $f(x)$ and $g(x)$ given $f(x)>g(x)$ is
  • $={\int }_a^b[f(x)-g(x)]dx$
• Area bounded by area to the left and the y-axis
  • $={\int }_c^d[f^{-1}(y)-g^{-1}(y)]dx$

7.2: Volumes:

• Definition of volume given the equation of area:
  • $V=\underset{\Delta \to 0}{\lim }\sum _{i=1}^{n}A(x_i^{\ast })\Delta x={\int }_a^{b}A(x)dx={\int }_a^b\pi f^2(x)dx$
  • Find $A(x)$ first, which a function of area by cross-slide, generally $A(x)=\pi y^2$

• Volume of revolution by cross-section:
  • $V=\pi \int f^2(x)dx$
  • If revolves around $x-$axis, find each and minus
  • If revolve about $x=-1$, not about the axis
    • Translate the curve so that the line of revolution is seen as the axis
  • If revolves around y-axis, $V=\pi \int x^2.dy$

7.3: Volumes by cylindrical shells method:

• Volume revolving around y-axis, opposite:
  • $V=\underset{n\to \infty }{\lim }\sum _{i=1}^n(2\pi {\bar{x}}_i).[f({\bar{x}}_i)]\Delta x={\int }_a^{b}2\pi xf(x)dx$
• Volume revolving around x-axis:
  • $V={\int }_a^{b}2\pi y{f}^{-1}(y)dy$

7.4: Arc length:

• $L=\underset{n\to \infty }{\lim }\sum _{i=1}^n|P_{i-1}.P|$, sum of $n$ straigh lines’ lengths
  • $|P_{i=1}.P|=\sqrt{(\Delta x{)}^2+(\Delta y{)}^2}$
• $L={\int }_a^b\sqrt{1+(\frac{dy}{dx}{)}^2}dx$
• If $x=f(y)$ then $L={\int }_a^b\sqrt{1+[\frac{dx}{dy}{]}^2}dy$

7.5: Area of a surface of revolution:

• $S={\int }_a^{b}2\pi y\sqrt{1+(\frac{dy}{dx}{)}^2}dx$
  • $ds=\sqrt{1+(\frac{dy}{dx}{)}^2}.dx$ or $ds=\sqrt{1+(\frac{dx}{dy}{)}^2}.dx$

7.6: Applications:

• Work:

  • $F=m.a=m\frac{d^{2}s}{d{t}^2}$ and $W=F.d$
    • $lb$ is force in pounds, the weight not mass

  $⟹W(\text{from a to b})={\int }_a^{b}f(x)dx$ where $f(x)$ is force function

  • Hooke’s law: $F=kx$ where $x$ is the extension in meters

  $⟹W(\text{to stretch from a m to b m)}={\int }_a^{b}kxdx$

• Hydrostatic pressure and force:

• Center of mass

• Separable equations:
  • Separate into $h(y).\frac{dy}{dx}=g(x)$ and integrate both sides
  • General solution and particular solution using initial-value
• Mixing problems:
  • $y(t)$ is amount of salt at time $t$, then $y^{\prime }t$ is the rate of salt going in - rate of going out
• Direction field:

$\text{Chapter 8: Series}$

8.1: Sequences:

• Notation $\{a_n\}$ or $\{a_n{\}}_{n=1}^{\infty }$
• Fibonacci sequence = $\{1,1,2,3,5,8,13,12,....\}$
• $\underset{n\to \infty }{\lim }{a}_n=L$ then the sequence converges, otherwise diverges

• Theorem: if $\underset{n\to \infty }{\lim }{a}_n=0⟺\sum _{n=1}^{\infty }{a}_n$ is convergent

• Theorem: if $\underset{n\to \infty }{\lim }{a}_n\ne 0⟺\sum _{n=1}^{\infty }{a}_n$ is divergent

• A sequence is monotonic if it is either increasing or decreasing

  • Use squeeze theorem

• Proof that a sequence is always decreasing by showing that $a_{n+1}<a_n$

• Bounded sequence: bounded below and or above making $a_n$ cant go pass that

  

8.2: Series:

• Sum of first n terms, $S_n=\sum _{i=1}^n{a}_i$
  • For geometric sequence: $a_n=a_1.r^{n-1}$
    • $S_n=\frac{a(1-r^n)}{1-r}$, where $|r|<1$
    • $S_{\infty }=\frac{a}{1-r}$
• Term test: If ${\sum }_{n=1}^{\infty }{a}_n$ converges, $\lim a_n=0$
  • otherwise $\lim a_n\ne 0$ or doesnt exist then series diverges
• Sum of 2 convergent series converges

8.3: The integral and comparison tests:

• Integral test: given $f(x)$ is continuous, positive and globally decreasing on $[1;\infty ]$
  • if ${\int }_1^{\infty }f(x)dx$ is convergent then $\sum a_n$ is convergent
  • if ${\int }_1^{\infty }f(x)dx$ is divergent then $\sum a_n$ is divergent
  • *** in the exam
• Comparison test: $\sum a_n$ and $\sum b_n$ are positive sequences with similar mostly similar terms with smaller power terms are different (Example : $\frac{\ln k}{k}>\frac{1}{k}$ or $\frac{5}{2n^2+4n+3}<\frac{5}{2n^2}$)
  • If every term of $a_n<b_n$ and $\sum b_n$ is convergent then $\sum a_n$ also converges
  • If every term of $a_n>b_n$ and $\sum b_n$ is divergent then $\sum a_n$ also diverges
• Limit comparison test: $\sum a_n$ and $\sum b_n$ are positive sequences
  • $\underset{n\to \infty }{\lim }\frac{a_n}{b_n}=0$, the series converges
  • If$\underset{n\to \infty }{\lim }\frac{a_n}{b_n}=$ a finite constant, then both sequences either diverge or converge

8.4: Other convergence tests:

• Alternating series test: $\sum _{n=1}^{\infty }(-1{)}^{n-1}{b}_n$ has terms alternately positive and negative:

  • Is convergent if $b_{n+1}\le b_n$, without sign, for all $n$ and $\underset{n\to \infty }{\lim }{b}_n=0$
  • When $\sum _{n=0}^{\infty }(-1{)}^n.[f(x){]}^n$ only converges whe f(x)<1

• Absolute convergence: if $\sum |a_n|$ is convergent then $\sum a_n$ is also convergent

• The ratio test: $\underset{n\to \infty }{\lim }|\frac{a_{n+1}}{a_n}|=L$, next term divided by the current term

  • If $L<1$ then the series converge, if $L>1$ then the series diverge
  • If $L=1$ then no conclusion drawn

• The root test:

  

8.5: Power series: ( use ratio test)

• $\sum _{n=0}^{\infty }{c}_n{x}^n=c_0+c_{1}x+c_2{x}^2+c_3{x}^3+....$, an infinite polynomial
• Power series centred at a: $\sum _{n=0}^{\infty }{c}_n(x-a{)}^n=c_0+c_1(x-a)+c_2(x-a{)}^2+....$
  • Use ratio test: $|\frac{a_{n+1}}{a_n}|=\frac{c_{n+1}(x-a{)}^{n+1}}{c_n(x-a{)}^n}<1$ in order to be convergent
  • Only 3 possibilities it can converge:
    1. When $x=a$
    2. The series converges for all $x$
    3. A range $|x-a|<R,R$ is radius of convergence

8.6: Representing functions as power series:

• Use the sum to infinity backward, $\frac{1}{1-x}=1+x+x^2+x^3+....=\sum _{n=0}^{\infty }{x}^n$
  • Example: Express $\frac{1}{1+x^2}=\frac{1}{1--x^2}=\sum (-x^2{)}^n$
    • Use this if the form is similar to $\frac{1}{1\pm f(x)}$
• Find a power series of $x^3/(x+2)$:
  • $=x^3.\frac{1}{2(1--x/2)}=\frac{x^3}{2}.\sum (-\frac{x}{2}{)}^n=\sum \frac{(-1{)}^n}{2^{n+1}}{x}^{n+3}$
  • As $=\frac{x^3}{2}\sum (\frac{-x}{2}{)}^n$, the series converges at $|-x/2|<1$

• Convert from differentiation, integration: derive/integrate on equation to get another serie

not in the exam

8.7: Taylor and Maclaurin series:

• Taylor series at a:
  • $f(x)=\sum _{n=0}^{\infty }\frac{f^n(a)}{n!}(x-a{)}^n=f(a)+\frac{f^{\prime }(a)}{1!}(x-a)+\frac{f^{\prime \prime }a}{1!}(x-a)+...$
    • where $|x-a|<R$
• Maclaurin series: $f(0)$ from Taylor$=f(0)+\frac{f^{\prime }(0)}{1!}x+\frac{f^{\prime \prime }(0)}{2!}{x}^2+...$
  • Taylor series but $a=0$
• $e=\sum _{n=0}^{\infty }\frac{x^n}{n!}$
• $T_n,n^{th}$ degree Taylor, slide 60,