IB mathematics HL - Graph of the Reciprocal of a Function

Reciprocal of a Function

The following guidelines are useful in order to sketch the reciprocal of a function given the graph of the original function:


Where is positive or negative then is also positive or negative respectively.

Where has zero(s) then the reciprocal function has vertical asymptote(s) and vice versa.

Where has a horizontal asymptote at y=c then the reciprocal function has also horizontal asymptote at .

Where the original function is increasing then the reciprocal function is decreasing.

Where the original function is decreasing then the reciprocal function is increasing.

If the original function has a maximum at then the reciprocal function has a minimum at .

If the original function has a minimum at then the reciprocal function has a maximum at .

If the original function has a point of inflexion at then the reciprocal function has also a point of inflexion at .

http://www.ibmaths4u.com/viewtopic.php?f=3&t=408

Application calendar for all UCAS applications

Timetable for 2013 applications

http://www.ucas.com/members-providers/undergraduate/key-dates

New!! IB Maths Revision Notes by www.IBmaths4u.com

IB Maths Revision Notes - IB Mathematics HL, SL, Studies Revision Notes by www.IBmaths4u.com

Complex Numbers for IB Mathematics HL 
http://www.ibmaths4u.com/viewtopic.php?f=3&t=370


Mathematical Induction for IB Mathematics HL
http://www.ibmaths4u.com/viewtopic.php?f=3&t=370


Trigonometry for IB Mathematics HL
http://www.ibmaths4u.com/viewtopic.php?f=3&t=370

The Mathematics of Animal Behavior: An Interdisciplinary Dialogue

A very interesting article about  Mathematics of
Animal Behavior
Shandelle M. Henson and James L. Hayward

http://www.ams.org/staff/jackson/fea-henson.pdf

The Commutative Algebra

The Commutative Algebra of Singularities in Birational Geometry: Multiplier Ideals, Jets, Valuations, and Positive Characteristic Methods

Month: May 2013

Date: May 6--10

Name: The Commutative Algebra of Singularities in Birational Geometry: Multiplier Ideals, Jets, Valuations, and Positive Characteristic Methods

Location: Mathematical Sciences Research Institute, Berkeley, California.

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Description

The workshop will examine the interplay between measures of singularities coming both from characteristic p methods of commutative algebra, and invariants of singularities coming from birational algebraic geometry. There is a long history of this interaction which arises via the "reduction to characteristic p" procedure. It is only in the last few years, however, that very concrete objects from both areas, namely generalized test ideals from commutative algebra and multiplier ideals from birational geometry, have been shown to be intimately connected. This workshop will explore this connection, as well as other topics used to study singularities such as jets schemes and valuations.

Information

http://www.msri.org/web/msri/scientific/workshops/all-workshops/show/-/event/Wm9000

IB Mathematics HL– Calculus, Differentiation, Related rates, Rate of change


How can we solve the following related rates word problem?
“A 10-metres ladder is leaning against a vertical wall. The foot of the ladder slips away from the wall at the rate of 0.1 meter per minute. How fast is the top of the ladder sliding down the wall when the top is 4 meters above the ground?”
http://www.ibmaths4u.com/ib-maths-hl-rate-of-change-t258.html

The answer is at www.ibmaths4u.com

Math and Music Enthrall at MAA Distinguished Lecture

When St. Mary’s College of Maryland mathematics professor David Kung heard a public radio story about a presentation devoted to math and music, he took issue with the presentation’s title.

The seminar was called "Math and Music—Closer Than You Think," but, as Kung told a packed auditorium at the Carnegie Institution for Science on February 26, he already regarded the two disciplines as "epsilon apart."

From http://www.maa.org/dist-lecture/2013DL_SymphonicEquations.html

MYP Mathematics, gradient of a line, perpendicular & parallel lines

MYP Mathematics, gradient of a line, perpendicular & parallel lines

How can we find the slope of the line which is perpendicular to the line y=8x+2 ?

The answer is from www.ibmaths4u.com

If two lines are perpendicular their slopes are negative reciprocals and if two lines are parallel have equal slopes.


Regarding your question about the gradient of the line which is perpendicular to the line y=8x+2,
Let be the required slope then we have the following relation

IB Mathematics HL Option Group Theory

The fundamental theorems and propositions on Group Theory from www.ibmaths4u.com

Re: IB Maths HL Option: Sets, Relations and Groups

1.  is a group under addition modulo n. With identity 0 and the inverse of  is the 

2. The number of elements of a group is its order 

3. The order of an element g, which is denoted by , in a group G is the smallest positive integer n such that 

4. Cyclic group  and g is called a generator of 

5. The order of the generator is the same as the order of the group it generated.

6. Let G be a group, and let x be any element of G. Then,  is a subgroup of G.

7. Let  be a cyclic group of order n.
Then  if and only if gcd(n,k)=1.

8. Every subgroup of a cyclic group is cyclic.

9. Every cyclic group is Abelian.

10. If G is isomorphic to H then G is Abelian if and only if H is Abelian.

11. If G is isomorphic to H then G is Cyclic if and only if H is Cyclic.

12. Lagrange’s Theorem: If G is a finite group and H is a subgroup of G, then the order of H divides G.

13. In a finite group, the order of each element of the group divides the prder of the group.

14. A Group of prime order is cyclic.

15. For any  where e is the identity element of the group G.

16. An infinite cyclic group is isomorphic to the additive group 

17. A cyclic group of order n is isomorphic to the additive group  of integers modulo n.

18. Let p be a prime. Up to isomorphism, there is exactly one group of order p.

19. Let (G, *) be a group and . If , then  , where  is the identity element of the group G.

20. In a Cayley table for a group (G, *), each element appears exactly once in each row and exactly once in each column.

21. A group (G,*) is called a finite group if G has only a finite number of elements. The order of the group is the number of its elements.

22. A group with an infinite number of elements is called an infinite group.

23. If (G,*) is a group, then ({e},*) and (G,*) are subgroups of (G,*) and are called trivial.

24. Let G be a group and H be a non-empty subset of G. then H is a subgroup of G if and only if for all .

25. Let G be a group and H be a finite non-empty subset of G. then H is a subgroup of G if and only if for all .

26. Let  be a finite group of order n.
Then 

27. Let  be a finite cyclic group. Then the order of g equals the order of the group.

28. A finite group G is a cyclic group if and only if there exists an element  such that the order of this element equals the order of the group ().

29. Let G be a finite cyclic group of order n. Then froe every positive divisor d of n, there exists a unique subgroup of G of order d.

30. Let G be a group of finite order n. Then the order of any element x of G divides n and 

31.Let G be a group of prime order. Then g is cyclic.