maths Flashcards
algebraic division
divide into it and then subtract
partial fractions
type 1 - distinct linear factors
type 2 - repeated linear factors
type 3 - irreducible quadratic factors
differentiation - the product rule
f’g + g’f
differentiation - the quotient rule
(f’g - g’f) / g^2
differentiate sinx
cosx
differentiate cosx
-sinx
what does secx equal?
1/cosx
what does cosecx equal?
1/sinx
what does cotx equal?
1/tanx
sin^2x + cos^2x
= 1
tan^2x + 1
= sec^2x
1 + cot^2x
= cosec^2x
differentiate tax
sec^2x
differentiate secx
secxtanx
differentiate cosecx
-cosecxcotx
differentiate cotx
-cosec^2x
differentiate e^x
e^x
differentiate ln(x)
1/x
how can extrema be found?
Using the second derivative
f’‘(x) < 0
max turning point
f’‘(x) > 0
min turning point
f’‘(x) = 0
May be a point of inflexion - check with nature table
binomial theorem
use pascals triangle and label a and b
factorials
3! = 3 x 2 x 1
what does 0! =
1
^nCr - known as binomial coefficient
n! / r!(n-r)!
binomial theorem - what is the general term?
(n r) x^n-r y^r
rule 1 - what does (n r) equal?
(n n-r)
(n r-1) + (n r)
(n+1 r)
when do vertical asymptotes occur?
when the denominator equals zero.
what are the two non-vertical asymptotes?
horizontal and slant
horizontal asymptotes
y = 0 or y = ai
slant asymptotes
divide out: y= mx + c
inverse functions calculate
change y and x
when is there an inverse function
when there is a one-to-one correspondence between x and y. If there is two x values for on y value, restrict the domain.
how to draw the modulus function
reflect all negative y values in the x axis
what is an even function?
f(-x) = f(x)
what is an odd function?
f(-x) = -f(x)
what is a redundant equation?
when at least two equations are equivalent - there is an infinite number of solutions.
what is an inconsistent equation?
parallel lines - there are no solutions.
Gaussian elimination steps
express system in augmented matrix form, reduce it to upper triangular form and then perform back substitution.
when is a set of equations said to be ill-conditioned?
When small changes in the coefficients produce relatively large changes in the solution.
integrate sinx
-cosx
integrate cosx
sinx
what does cos^2x equal?
1/2 (1+cos2x)
what does sin^2x equal?
1/2 (1-sin2x)
what can cos2x equal?
2cos^2x - 1
or
1 - 2sin^2x
integrate e^x
e^x
integrate 1/x
lnx)
integrate sec^2x
tanx
integrate f’(x)/f(x)
ln(f(x))
how do you integrate improper fractions?
divide out
integrating by substitution steps
find ‘u’
differentiate ‘u’
multiply by ‘dx’
substitution and definite integrals
where the variable changes you must change the limits too.
how do you find the area between curve and y axis
it can only be done if x is expressed as a function of y.
what is the formula for volume of revolution?
integral (b a) pi y^2 dx
what does ‘i’ equal?
root -1
what does ‘i^2’ equal?
-1
what are complex numbers of the form?
z = x +iy
complex numbers: what do we write x as?
x = Re(z)
complex numbers: what do we write y as?
y = Im(z)
what is the complex conjugate?
z = x - iy
What is the modulus of a complex number?
The measure of magnitude of z and is written as (z).
What is the argument of a complex number?
The size of the angle, between the x axis and the line representing the complex number on the Argand diagram.
What is the formula for the argument of z?
the square root of (x^2 + y^2)
What is the formula for the argument of z?
tan-1(y/x)
What is the formula for polar form of a complex number?
z = r(cosx + isinx)
What is a locus?
A set of points.
How do you multiply two complex numbers in polar form?
Add the arguments and multiply the moduli.
How do you divide two complex numbers in polar form?
Take away the arguments and divide the moduli.
use de moivre’s theorem to show z^n.
Raise the modulus to that power. Multiply the argument by that power.
using de moivres theorem, how many roots would a cubic equation have?
three - they would be 120 degrees apart.
what would you do if you find the fourth roots of unity?
find fourth roots of one, four solutions 90 degrees apart.
write 1 in polar form. z = cos0 + isin0
what do you get when you solve an irreducible quadratic with no real roots?
the roots are complex conjugates of each other.
differentiate inverse sinx
1/square root of (1 - x^2)
differentiate inverse cosx
-1/square root of (1 - x^2)
differentiate inverse tanx
1/(1 + x^2)
differentiate inverse sin(x/a)
1/square root of (a^2 - x^2)
differentiate inverse cos(x/a)
-1/square root of (a^2 - x^2)
differentiate inverse tan(x/a)
a / (a^2 + x^2)
when a function is complicated by powers, products and quotients of several factors, how can it be made easier?
By taking logarithms before differentiating.
parametric equations: what does f’‘(x) equal?
d/dt (dy/dx) dt/dx
what is an arithmetic sequence?
When the number separating the terms is a constant.
What is the equation for the nth term of an arithmetic sequence?
Un = a + (n-1)d
what is a in the equation for the nth term?
the first term
equation for sum of the first n terms of an arithmetic sequence?
Sn = 1/2n (2a + (n-1)d)
What is the equation for the nth term of a geometric sequence?
Un = ar^(n-1)
equation for sum of the first n terms of a geometric series?
Sn = (a(1-r^n)) / (1-r)
when r<1, what is the sum to infinity of a geometric series?
a/(1-r)
what can 1/(1-x) be interpreted as?
The sum to infinity of a geometric series with a=1 and r=x. r must be <1.
sigma 1
n
sigma r
1/2n (n+1)
sigma r^2
1/6n (n+1)(2n+1)
sigma r^3
1/4n^2 (n+1)^2
when trying to integrate a proper fraction, what do you do?
change it into partial fractions
how can improper fractions be simplified?
through algebraic division
integral of f’g dx equals?
fg - integral fg’ dx
what do you use when trying to integrate something you cannot?
use the dummy variable, multiply it by one and then use integrating by parts.
difference between a general solution and a particular solution?
general solutions include the constant c.
when the differential equation is of the form dy/dx = f(y), what can you do?
you can separate the variables, integral 1/f(y) dy = integral 1dx
proof by induction: step 1
prove that the statement is true for n=1
proof by induction: step 2
assume it is true for n=k
proof by induction: step 3
consider n=(k+1)
proof by induction: step 4
prove that it is true for n=(k+1)
proof by induction: conclusion
thus if it is true for n=k then it is also true for n=(k+1) but since it is true for n=1, then by induction it is true for all n in the set of natural numbers.
what can the greatest common divisor of a and b be denoted by?
(a,b)
euclidean algorithm: if a = qb + r then…
(a,b) = (b,r)
If the gcd of two numbers is 1, what are the numbers said to be?
Co-prime or relatively prime.
What is Maclaurin series?
f(0) + f’(0)x/1! + f’‘(0)xx/2! + f’’‘(0)xxx/3!…
Maclaurin series for e^x, when is it valid?
For all x in the set of real numbers.
Maclaurin series for sinx/cosx, when is it valid?
For all x in the set of real numbers.
Maclaurin series for ln(1+x), when is it valid?
-1<x<1
Maclaurin series for (1+x)^n, when is it valid?
-1<x<1
Maclaurin series for tan-1x, when is it valid?
-1<x<1
What is the transpose of a matrix? (A’ or A^T)
interchanging rows and columns.
What does (AB’) equal?
B’A’
AI = IA =…
A
A’A = I if …
orthogonal
How do you find detA of a 2x2 matrix?
ad - bc
How do you find the inverse of a 2x2 matrix?
1/ad-bc (adjA)
What happens in detA is 0?
The matrix has no inverse and is said to be singular.
What does A^-1A equal?
I
How do you find the inverse of a 3x3 matrix?
Use Gaussian elimination for A and I and make all ones in A and then resultant is inverse.
1st order linear differential equation form:
dy/dx + P(x)y = Q(x)
1st order linear differential equation IF:
e^integral of p(x)dx
2nd order linear differential equation form:
ad2y/dx2 + bdy/dx + cy = Q(x)
2nd order linear differential equation, what is it called whenn q(x) = 0?
The equations are homogeneous.
2nd order linear differential equation, what is it called whenn q(x) doesn’t equal 0?
The equations are non-homogeneous.
Find complimentary function (solution to corresponding homogeneous equation)
Use the auxiliary equation.
How do you disprove a statement?
Give one counter example to disprove a conjective.
Direct proof: What do you do if n is even?
Let n = 2k , n in the set of natural numbers.
Direct proof: What do you do if n is odd?
Let n = 2k-1 , n in the set of natural numbers.
Proof by contradiction: What do you do initially?
Assume the opposite, eg., if n^2 +1 is odd, then n is even -> if n^2 +1 is odd, then n is odd.
Proof by contrapositive: what is the contrapositive of p->q?
not q implies not p.
What does rational mean?
It can be written as a fraction.
What would you do to prove something is rational?
Let it = m/n where m and n share no common factor, m, n in the set of integers.
What is the scalar product/dot product formula?
a.b. = detAdetBcos(angle)
How do you find the angle between two vectors?
cos(angle) = a.b./ detAdetB
What is the vector product/cross product?
a x b = ndetAdetBsin(angle)
What does it tell you if det(a x b) = 0?
The two vectors are parallel.
What does a x b =?
-(bxa)
What is the cartesian equation of a plane?
ax + by + cz = k
What is the angle between two planes the same as?
The angle between the two normals.
What is the vector equation of a plane?
r = a + tb + uc
What are parametric equations of a plane?
x = a + tb + uc
y = …
z = …
To find the equation of a plane, what do you need?
A point on the plane and two vectors parallel with the plane.
For the equation of a line in space, what do you need?
A point on the line and a direction vector (a vector parallel to the line)
The angle between a line and a plane: Two steps:
Find the acute angle between the normal and the line, subtract this value from 90.
Intersection between line and plane: Steps
Sub x, y and z of line into plane equation to find t. Sub t to get coordinates of intersection.
Intersection of two lines: Steps
Show they are not parallel, express parametric equations, find x and y, sub these into 3rd equation to find intersection point or if they are skew.
Intersection of two planes: steps
Cross the two normals, z = 0, find x and y,
Intersection of three planes: steps
use Gaussian elimination
