0
Mathematical Analysis
Mathematical analysis is, simply put, the study of limits and how they can be manipulated. Starting with an exhaustive study of sets, mathematical analysis then continues on to the rigorous development of calculus, differential equations, model theory, and topology. Topics including real and complex analysis, differential equations and vector calculus can be discussed in this category.
2,575 Questions
What is the sequence for 0112358?
That's the famous Fibonacci sequence, where every term is the sum of the previous two.
What is the area of the semicircle if the length of the arc of a semicircle is 10 and pi feet?
It is 27.49 units.
When Resultant velocity is calculated by using which two math functions?
The answer depends on what information you start with. For example, if you are given acceleration then you might integrate whereas if you are given displacement, you might differentiate.
What is shapes of distribution?
It is any shape that you want, provided that the total area under the curve is 1.
Is this domain (1 1) (2 2) (3 3) (3 6) (4 4) (5 5) (6 6) and this range 1 2 3 4 5 6 a function?
Whether or not is is a function depends on how the mapping is defined. If, for example the mapping is f(x, y) = x, where the coordinates of points in 2-d space are mapped to their abscissa or g(x, y) = y, where the coordinates of points in 2-d space are mapped to their ordinates then they are functions.
I am not entirely sure what you mean by "3 11". If this is supposed to be a fraction, the denominators are already the same, so you just need to add the numerators.
17, 8, 1, 15, 10, 6, 3, 13, 12, 4, 5, 11, 14, 2, 7, 9, 16
What Is The Importance Of Limits And Continuity?
Those are among the most fundamental concepts in calculus; they are used to define derivatives and integrals.
The remainder is 0.
If A has a remainder of 1 when divided by 3, then A = 3m + 1 for some integer m
If B has a remainder of 2 when divided by 3, then B = 3n + 1 for some integer n
→ A + B = (3m + 1) + (3n + 2)
= 3m + 3n + 1 + 2
= 3m + 3n + 3
= 3(m + n + 1)
= 3k where k = m + n + 1 and is an integer
→ A + B = 3k + 0
→ remainder when A + B divided by 3 is 0
-------------------------------------------------------------------------
From this, you may be able to see that:
- if A when divided by C has remainder Ra; and
- if B when divided by C has remainder Rb; then
- (A + B) divided by C will have remainder equal to the remainder of (Ra + Rb) divided by C
let the number of bottles in the first group be B
Then
- the 2nd group has 1st + 20 = B + 20 bottles;
- the 3rd group has 2nd × 2 = 2 × (B + 20) = 2B + 40 bottles;
- the 4th group has 3rd + 10 = 2B + 40 + 10 = 2B + 50 bottles.
The total number of bottles is the sum of the groups which is:
bottles = B + B + 20 + 2B + 40 + 2B + 50 = 6B + 110
But there are 1700 bottles, we can solve for B:
6B + 110 = 1700
→ 6B = 1590
→ B = 265
→ the groups contain:
- B = 265 bottles
- B + 20 = 265 + 20 = 285 bottles
- 2B + 40 = 530 + 40 = 570 bottles
- 2B + 50 = 530 + 50 = 580 bottles
The groups contain 265, 285, 570 and 580 bottles respectively.
What irrational number has the decimal representation that begins 0.232323232?
23/99, which is rational.
What fraction of whole numbers is equal to 17100 and that has a denominator that is a power of 10?
17100 = 17100/1
= 17100/1 × 10/10 = 171000/10
= 17100/1 × 10²/10² = 1710000/100
= 17100/1 × 10³/10³ = 17100000/1000
...
= 17100/1 × 10ⁿ/10ⁿ
There is no upper limit to this sequence: n = 0, 1, 2, 3, ....
These numbers can be put in a one-to-one relationship with the counting numbers 1, 2, 3, ...
The counting numbers can also be put in a one-to-one relationship with the whole numbers
Therefore the sequence of fractions can be put in a one-to-one relationship with the whole numbers.
Therefore the fraction of whole numbers which are in the sequence is 1.
-----------
Let's try another approach:
As 10 to some power is positive, there is no way we can change a negative number to make it positive by dividing by 10 to some power, therefore the solution numbers are all positive.
As negative whole numbers make up half of the whole numbers, the fraction of whole numbers which are equivalent to 17100 when divided by a power of 10 is at most ½.
The fraction of the ½ of the whole numbers which are positive can now be considered:
After the nth whole number has been found which matches the criteria that is it equivalent to 17100 when it is divided by a power of 10, in total there are n numbers matching out of a total of 17100 × 10ⁿ⁻³ numbers (the value of the nth number); thus:
The first number which meets the criteria is 171/10⁻² → the fraction is 1/171
The second number to meet the criteria is 1710/10⁻¹ → the fraction is 2/1710
The third number to meet the criteria is 17100/10⁰ → the fraction is 3/17100
The fourth number to match the criteria is 171000/10 → the fraction is 4/171000
The fifth number to match the criteria is 1710000/10²→ the fraction is 5/1710000
→ For the nth match, the fraction is: n/(17100 × 10ⁿ⁻³) = (1/17100) × n/10ⁿ⁻³
This gives us a sequence of fractions:
1/17100 × 1/10⁻², 1/17100 × 2/10⁻¹, 1/17100 × 3/10⁰, 1/17100 × 4/10¹, 1/17100 × 5/10², ...
= 100000/17100000, 20000/17100000, 3000/17100000, 400/17100000, 35/17100000, 6/17100000, ....
Consider terms n and n+1:
U{n} = (1/17100) × n/10ⁿ⁻³ = (1/17100) × 10 × n/10ⁿ⁻²
U{n+1} = (1/17100) × (n+1)/10ⁿ⁻²
U{n} - U{n+1} = (1/17100) × 10 × n/10ⁿ⁻² - (1/17100) × (n+1)/10ⁿ⁻²
= (10n - (n+1))/(17100 × 10ⁿ⁻²)
= (9n - 1)/(17100 × 10ⁿ⁻²)
As n ≥ 1,
9n - 1 ≥ 9×1 - 1 = 8
→ term n - term n+1 ≥ 8 > 0
→ term n is larger than term n+1
for all n ≥ 1
Each of these terms is less than 1, and each term of the sequence is smaller than the previous one and so as n increases the value of the each term (the fraction of whole numbers which meet the criteria) tends towards 0.
→ The fraction of all whole numbers which equate to 17100 when divided by a power of 10 is as near enough to zero as make no odds.
ie the fraction is so small it is effectively none of the whole numbers.
-----------------------
This is a problem of dealing with the infinite.
How do you do an equation where there are variables on both sides?
You try to bring all instances of the variable to one side. Here is an example:5x + 5 = 3x - 2
Subtracting 3x on both sides:
2x + 5 = -2
Subtracting 5 on both sides:
2x = -7
If A and B are multiples of C, then A + B is also a multiple of C:
If A is a multiple of C then A = mC for some integer m
If B is a multiple of C, then B = nC for some integer n
→ A + B = mC + nC
= (m + n)C
= kC where k = m + n and is an integer
→ A + B is a multiple of C
What is the unit rate for 8580 words in 2 h 45 min?
If you divide the number of words by the number of minutes, you'll get a rate of words per minute.
4d - 1
It is -7.
The answer will depend on whether the interest is calculated on the monthly balance or annual balance. On an annual basis, it will be approx 290.
What multiplied by itself equals 4.84?
The idea is to take a calculator, and calculate the square root of 4.84.
Point: (1, 4)
Slope: -3
Equation: y = -3x+7
