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Friday, December 27, 2013

Theoretical Remarks #2 - some insight into the function $\int_{a}^{x}f(t)dt$

   Let us first recall the proposition mentioned (without proof) in yesterday's post:
Proposition: Let a real function $f$, with domain $D_{f}$ and continuous on an interval $\Delta \subseteq D_{f} \subseteq  \mathbb{R}$ and let $a \in \Delta$ be a fixed point of $\Delta$. Then the function $F(x)=\int_{a}^{x}f(t)dt$ is an antiderivative function of $f$. In other words:
$$
F'(x) = \big( \int_{a}^{x}f(t)dt \big)' = f(x)
$$
for all $x \in \Delta$.

   If $x \in \Delta$ is a variable point,
then the value of the definite  integral  $F(x)= \int_{a}^{x}f(t)dt$ represents the area of the shaded  region shown in the figure:
Remark: Notice that the function $F(x)$ is defined on any interval $\Delta \subseteq D_{f}$ in which:
  • $a \in \Delta$
  • f is continuous on $\Delta$
and (according to the preceding proposition) is differentiable on that $\Delta$.

   Considering $h$ to be infinitesimal, we can now compute
$$
\begin{array}{c}
\Delta F(x) = F(x+h)-F(x) = \int_{a}^{x+h}f(t)dt - \int_{a}^{x}f(t)dt = \\
     \\
 = \int_{x}^{x+h}f(t)dt = E(\Omega) \approx h \cdot  f(x)
\end{array}
$$
In the above, the change $\Delta F(x)$ in the value of the function $F(x)=\int_{a}^{x}f(t)dt$ when $x$ changes to $x+h$ is denoted by $E(\Omega)$ and equals the area of the shaded strip $\Omega$, displayed in the next figure:
Consequently, $f(x) \approx \frac{E(\Omega)}{h}$ and since $h$ is considered to be infinitesimal, we can write:
$$
f(x)=\lim_{h \rightarrow 0}\frac{F(x+h)-F(x)}{h}=F'(x) = \frac{d \big(\int_{a}^{x}f(t)dt \big)}{dx}
$$
Remarks:
(1). The above should not be considered to be a rigorous proof. It should rather be taken as an intuitive  line of thinking, aiming to shed some light into "what is really going on" inside the function $\int_{a}^{x}f(t)dt$
(2). Let a real function $f$, continuous on an interval $\Delta \subseteq  \mathbb{R}$ and let $a \in \Delta$ be a fixed point of $\Delta$. Let another real function $g(x)$ with domain $D_{g}$ and differentiable (and thus: continuous) on an interval $\Delta_{1} \subseteq D_{g}$. We can then consider the composite function $G=F \circ g$:
$$
G(x) = \int_{a}^{g(x)}f(t)dt
$$
The domain $D_{G}$ of $G$ will be:
$$
D_{G} = \{ x \in D_{g} \ , \ g(x) \in \Delta \}
$$
   The composite function $G=F \circ g$ is differentiable on $\Delta_{2} = \Delta_{1} \cap D_{G}$. Its derivative can be computed by combining the above differentiation rule with the chain rule of differentiation (for differentiating composite functions). We readily get the following formula:
$$
\Big( \int_{a}^{g(x)}f(t)dt \Big)' = f\big( g(x) \big) \cdot g'(x)
$$
for all $x \in \Delta_{2}$.

Thursday, December 26, 2013

Theoretical Remarks #1 - indefinite integrals, antiderivatives and the function $\int_{a}^{x}f(t)dt$

   Suppose we are given a continuous, real function $f(x)$ defined on an interval $\Delta \subseteq \mathbb{R}$ and let $a \in \Delta$ be a fixed point.
   Any other function $F(x)$, with domain $D_{F}= \Delta$ will be called an antiderivative function of $f$ if
$$
F'(x)=f(x)
$$
(notice that $F$ is by definition differentiable (and thus continuous) in $\Delta$.)
   The above definition implies that: the antiderivative function is not uniquely determined, but rather there is a family of functions satisfying the above relation. Actually, any other function $G(x)$, $D_{G}= \Delta$ with the property
\begin{equation} \notag
G'(x)=F'(x)=f(x)
\end{equation}
will also be an antiderivative function. In such a case $F(x)$ and $G(x)$ will differ by a constant:
\begin{equation} \notag
G(x)=F(x)+c
\end{equation}
for some $c \in \mathbb{R}$. (this comes from a well known theorem of elementary calculus). We can thus now lay the following
   Definition: We will call antiderivative or indefinite integral of $f$, and we will denote it by $\int f(x)dx$ the set of all functions satisfying the above property, thus:
\begin{equation} \notag
\begin{array}{r}
\int f(x)dx = \{F | F'(x)=f(x), \ x \in \Delta  \} = \\
     \\
= \{G(x)+c |\textrm{for all } c \in \mathbb{R} \} \ \ \ \ \ \ \ \
\end{array}
\end{equation}
where in the last equality $G$ is an antiderivative function (actually any antiderivative function) of $f$.

   Now it can be proved (the proof can be found on standard calculus texts and i will -hopefully- post it here later) that:
Proposition: one of these functions belonging in the above set (thus, one of the antiderivative functions of $f$ or equivalently: one of the indefinite integrals of $f$) is the function
\begin{equation} \notag
\int_{a}^{x} f(t)dt
\end{equation}
In other words: $(\int_{a}^{x} f(t)dt )'=f(x)$ for all $x \in \Delta$.

Remarks:
(1). Notice that the above proposition readily implies that $\int_{a}^{x} f(t)dt$ is differentiable (and thus continuous) for any $x \in \Delta$. Of course the $\ ' \ $ symbol indicates differentiation with respect to the variable $x$.
(2). Thus: the definite integral $\int_{a}^{x} f(t)dt $ with variable upper limit of integration, is an antiderivative function of $f$. Consequently, we can write
\begin{equation} \notag
\int f(x)dx = \int_{a}^{x} f(t)dt + c
\end{equation}
for all $c \in \mathbb{R}$.
(3). What the above proposition actually tells us is that: any function $f$ which is continuous on an interval $\Delta \subseteq \mathbb{R}$, has an antiderivative function given by $\int_{a}^{x} f(t)dt $ for $a,x \in \Delta$.
(4). It is worth noticing the meaning of the number $a \in \Delta$: Varying the value of $a \in \Delta$ produces different antiderivative functions (because the variation of $a \in \Delta$ simply alters the value of the constant of integration $c$). However, we cannot hope that varying the value of $a \in \Delta$ "covers" all possible antiderivatives of a given (continuous function $f$). In other words, this means that: although the above theorem tells us that  $\int_{a}^{x} f(t)dt$ is an antiderivative function of $f$, not all antiderivative functions of $f$ can necessarily be expressed as $\int_{a}^{x} f(t)dt$ for some $a \in \Delta$. This can be clearly seen in the following example:
   If $f(x)=2x$ and $\Delta = \mathbb{R}$, then for  any real $a$ we have
\begin{equation} \notag
\int_{a}^{x} 2tdt = [t^{2}]_{a}^{x}=x^{2}-a^{2}
\end{equation}
But, the family of functions $\{F(x)=x^{2}-a^{2}, \ x \in \mathbb{R} | \textrm{for all } a \in  \mathbb{R} \}$ does not include for example the function $G(x)=x^{2}+1$, which is an obvious antiderivative function of $f(x)=2x$.





Tuesday, December 24, 2013

Question of the week #4

This week's posting will have to do with  integral calculus and more specifically with integral equations and antiderivatives of continuous functions (recall that the antiderivative is another name for the indefinite integral).
Let a continuous real function $f$ satisfying $f(x)=e^{\int_{0}^{x}f(t)dt}$ for all $x<1$. Find the formula of the function $f$.


Question of the week #3 - the answer

Question of the week #3: Given a complex number $z$, determine its locus, given that $w=\frac{i}{z^{2}+1}$ belongs on the real axis (i.e. $w$ is a real number)

Solution: Substituting $z=x+iy$ with $x,y \in \mathbb{R}$ and denoting by $\bar{z}=x-iy$ the complex conjugate, we have:
$$
\begin{array}{c}
  w \in \mathbb{R} \Leftrightarrow w=\bar{w} \Leftrightarrow \frac{i}{z^{2}+1}=\overline{\frac{i}{z^{2}+1}} \Leftrightarrow \\
   \\
  \Leftrightarrow \frac{i}{z^{2}+1}=-\frac{i}{\bar{z}^{2}+1} \Leftrightarrow \bar{z}^{2}+1= -z^{2}-1 \Leftrightarrow \\
   \\
 \Leftrightarrow \bar{z}^{2}+z^{2}+2=0 \Leftrightarrow  \\
   \\
   \Leftrightarrow x^{2}-y^{2}-2ixy+x^{2}-y^{2}+2ixy+2=0 \Leftrightarrow \\
      \\
 \Leftrightarrow 2x^{2}-2y^{2}+2=0 \Leftrightarrow y^{2}-x^{2}=1
\end{array}
$$

which is an isosceles hyperbola:
with the vertices $A(0,1)$ and $B(0,-1)$ excluded, because for these values we have $z=i$ and $z=-i$ respectively, thus $z^{2}+1=0$.


Saturday, December 21, 2013

Revision Exercises (with solutions) on IB Math SL material

   Here are some revision exercises (most of them come with their solutions) on the material of IB Math SL (mainly). They are of course suitable for HL courses as well.
   You can practise on



I hope that these will be of help, particularly in view of the upcoming midterms.

Marry Christmas to everybody !!


Monday, December 16, 2013

Question of the week #3

This week's Question comes from complex numbers:
Given a complex number $z$, determine its locus, given that $w=\frac{i}{z^{2}+1}$ belongs on the real axis (i.e. $w$ is a real number)
You can check out the pdf version here
Waiting till next week, for your ideas and thoughts.

Sunday, December 15, 2013

Question of the week #2 - the answer

Question of the week #2

   Last week's question was dealing with two different standard forms of the hyperbola equation.
The answer to the question, is that these two different forms are equivalent descriptions and this can be shown by a counterclockwise rotation $(x,y)\rightarrow(x',y')$ of the planar coordinate system, through an angle $φ=π/4 (rad)$.
   Here are some more details (in the example that follows $a > 0$):

Some resourses for IB Further Math SL

For those courageous students around the world, struggling with Further Math SL here are some notes and a book !

 

Sunday, December 8, 2013

Question of the week #2

This week's question comes form $2d$ analytic geometry, and deals more specifically with the coordinate equations of the hyperbola:

"In a given coordinate system $(x,y)$ the equation $y=\frac{a}{x}$, $a \in \mathbb{R}$ represents an hyperbola. Show that under a suitable change of coordinates i.e. under a suitable transformation $(x,y)\rightarrow(x',y')$ the same hyperbola becomes $x'^{2} - y'^{2}=2a$"

check out the pdf version here.

Waiting again for your thoughts, ideas and answers till next week!

Question of the week #1 - the answer

Question of the week - #1

   ok, so here we have the answer to last week's question on the continuity of the derivative function at a given point ...
   The answer is in general negative! A function may be differentiable at a point of its domain, with the derivative being discontinuous at that point! Here is a counterexample. you can provide a proof on your own by directly applying the definition of the derivative at $x=0$, but if you find the computation cumbersome, here are some more details ;)

Enjoy !!


Sunday, December 1, 2013

Some e-notes for the android

 
Those of you using smartphones or tablets running android can download some free e-notes with material relevant to the A-levels (but is also useful for the IB's math courses).
   Take a look at A-level math, #1 and also at A-level math, #2

They cover only a portion of the necessary material but are still particularly useful, although far from complete ! They also have the advantage that they can be carried at your mobile device (tablet or smartphone) and be readily available anytime you need them!

Beware: the purpose of distributing such material is strictly educational! Please do not use them for cheating in tests or exams !
video

Saturday, November 30, 2013

online Encyclopedias of Mathematics

You can find a wealth of information on Mathematics of all levels at the following locations:
Also, you can find interesting (and much more than mathematical) topics at:
Finally, it will be very instructive to gain some insight into an experimental "Computational Knowledge Engine" (whatever this means to the present!)

Polynomials and equations

Here are some notes on the theory of polynomials and on various forms (polynomial, rational, irrational, eq. with absolute values) of equations.
You should test your understanding trying to workout the problems and exercises either here or there !

 

Straight line: notes and homework

Quadratics: notes and some homework

Here are some notes on quadratics and some homework as well.
Enjoy!
 

Some background knowledge

those of you (parents or students) who are wondering what is the expected prerequisite level, for some student aiming to start an A-level or an IB HL-SL course, should take a look at the following: Background material
 

Thursday, November 28, 2013

Question of the week - #1

A question which is addressed to all those studying the fundamentals of differential calculus, and more specifically the definition of the derivative:

"Let a function $f(x)$ which is differentiable (and thus continuous) at a point $x=a$ of its domain. Is the derivative function $f '(x)$ necessarilly continuous at $x=a$ ?"

I am waiting for your ideas and answers.
I will post the solution next week, together with the names of those who will have communicated me correct answers (with proofs or counterexamples!) 

Indefinite Integrals - tables

A "computational" tool !! --> Indefinite integrals - tables
 

Mathematics Workbook - UoC

Contains 30 review exercises, indicative of the prerequisite level for someone intending to study Mathematics in a top University: Mathematics Workbook - UoC 

Notes for an introduction to Pure Mathematics (in Greek)

A general introduction to undergraduate level mathematics, focusing the viewpoint of pure mathematics (not for the faint-hearted !!)  Σημειώσεις για μια εισαγωγή στα Καθαρά Μαθηματικά

 

Fermat's last theorem (in Greek)

A very nice and interesting paper, with an appendix focusing on the historical development of the topic.
Unfortunately the text is in Greek only ;)    -->  Το τελευταίο θεώρημα του Fermat
(From: University of Athens, School of Mathematics)

 

Monday, November 25, 2013

Trigonometry I - class notes


Homework 18 (on functions)


Homework 17 (on functions)


Functions - class notes


Welcome !

   I would like to welcome everybody on my new blog.
   The primary purpose of this blog will be to post various material related to the curriculum of A-level mathematics, IB Math HL, SL and Further Math, and mathematics for grades K11-K12 or equivalent. We will post classnotes, material for deepening into the theory (notes, remarks, various proofs of propositions and theorems which are unlikely to be found on standard textbooks), solved exercises of escalating difficulty (in order to cover even the needs of the most demanding students), tests, collections of questions and exercises for practise, projects , etc.
   It is hoped that the material will be of interest to the demanding students and to instructors/teachers/ tutors offering classes at that level.

   The topics discussed here will cover a range such as:  
  • Quadratics (discriminant, roots, factorization, Vietta's forulae, graph),  
  • Straight line equation
  • elementary Number theory (Euclidean division, primes, prime factorizations, LCM, GCD), 
  • Polynomials (Euclidean division, long division, Horner's scheme, factor and remainder theorem, theorems on roots of polynomials, multiplicities and geometric interpretation),  
  • Partial fractions,  
  • arithmetic-geometric series,
  • Mathematical induction
  • Axioms of Real numbers (binary relations and their properties: reflexive, (anti)symmetric, transitive, ordering relations and equivalence relations, N, Q, R. R as an ordered field: intervals and bounds, completeness of real numbers and the "nested sphere theorem")
  • Exponentials and Logarithms (completeness of the reals and the definition of an irrational exponent, laws of logarithms, change of base, Exp and Log functions, the number e), 
  • Binomial theorem, Pascal's triangle 
  • Combinatorics (multiplication principle, inclusion-exclusion principle, permutations with or without repetition, arrangements with or without rep., combinations with or without rep., applications),    
  • Linear algebra (matrices, algebra of matrices, determinants, mxn linear systems of equations and their geometrical interpretation)
  • $2d$ and $3d$ Coordinate and geometry and Vector geometry (straight line equation in $2d$ and $3d$, equation of a plane, relative positions between two lines, two planes, a line and a plane, distance between a point and a line or a plane, Conics: circle, ellipse, parabola, hyperbola, coordinate, parametric, vector equations),  
  • Functions (graphs, "$1-1$", onto, inverse, composite,  monotonicity, linear, quadratic, polynomial, rational, irrational, exponential, logarithmic),  
  • Trigonometry ( radian measures: rad and deg, geometry of a circle, trigonometric ratios in right triangles, oriented angles, trigonometric circle, trigonometric ratios for oriented angles, elementary identities, reduction to the first quadrant, trigonometric functions sinx, cosx, tanx, cotx, secx, cosecx, graphs, trig. equations, general solution formulae, sine-cosine laws, solution of triangles, addition forumale, double angle formulae, asinx+bcosx, transformation formulae from sums to products and vice versa),  
  • Sequences (Convergent and Cauchy sequences, criteria of convergence, limits, application: e as limit of sequences),
  • Limits (abstract definitions, various computational methods),  
  • Continuity (Bolzano and Intermediate value theorem, extreme value theorem,),  
  • Differential and Integral Calculus (Differentiation, basic derivatives, products-quotients, tangents, chain rule, differentiation of the inverse function, implicit differentiation, Rolle's  and mean value theorem, local extrema, turning points, curve sketching, Integration, Indefinite integral - Antiderivative, basic integrals, integration by factors and by substitution, Definite integral, Fundamental theorem of integral Calculus, areas and volumes,  applications, mean value theorem of integral Calculus), 
  • Complex numbers (axiomatic description, Argand's diagram, operations and their geometrical interpretation, trigonometric form, exp form, De Moivre's theorem),  
  • Vectors and Vector spaces 
  • Groups, Rings, Fields,  
  • Statistics (descriptive and inferential),  
  • Probability (intuitive and axiomatic definition, sample spaces, events, mutually exclusive and independent events, Bayes' theorem, expected values, discrete and continuous distributions, gaussian, binomial, geometric, hypergeometric),  
  • Series expansions (Taylor, McLauren, ratio of convergence, criteria of convergence),  
  • Differential Equations,  
  • Arithmetic methods and approximations
  • (axiomatic) Euclidean Geometry,  
  • Mathematical Logic, .... etc
   There will also frequently appear posts related to the syllabus of the Greek Panhellenic Exams (corresponding to the three classes A' ,B' ,C' of the Greek Lyceum). These will often appear in greek (however, I will try to provide english translations for the most interesting of these).
   Finally, anything (ranging from Pure to Applied Mathematics) with sufficient interest for mathematically-oriented students/teachers/researchers may appear from time to time (under suitable headings).
   You are welcome to send me questions, problems, ideas etc related to the above material or with mathematics in general! I will try my best to respond to possible questions -as soon as possible- and to adopt best practise ideas in order to help anybody taking or giving such courses and to anyone inerested in mathematics generally.

So, keep on reading and thinking!