Unit 2 Flashcards
What is i equal to
the root of -1
What is the form of complex numbers
a+bi
How to add or subtract complex numbers
To add or subtract complex numbers gather the real parts together and the imaginary parts together
How to multiply complex numbers
To multiply complex numbers multiply out brackets
How to divide complex numbers
To divide times the division by the complex conjugate of the denominator
How to find the square roots of complex numbers
let a + bi = the root square a+bi to remove the root on the other side multiply the left side out correlate the left to right in terms of complex and nornaml numers Then solve to find both a and b the answer is thus the a and b values in the form a + bi
How to find the roots of a cubic or quartic when one complex root is given
If the root given is non-real then its conjugate is also a root of the polynomial we now know too roots and thus the factors are (z-(the first )) and (z-(the second)) Multiply them together Divide the original cubic by the product using long division the result of such can then be manipulated to find the final root meaning what is z equal too for the quartic the roots for the result will have to be find via a different method as there will be two
What is the quadratic formula
x= -b+- the root of b^2 - 4ac divided by 2a
Formula to find the nth term of an arthimetic sequence
un = a + (n-1)d
Formula to find the nth term of a geometric sequence
un = ar^n-1
How to find the sum to infinity of a geometric series
is r >1 then the sum to infinity is undefined if r<1 then the sum to infinity can be found using the equation s(infinity sign) = a divided by 1-r
What is a
The first term of a series
How to find the maclaurian expension of sin2x
find the expansion of sinx and substiutie the end answer x with 2x
How to expand composite functions such as e^xcos3x
find the expansion of e^x find the expansion of cos3x multiply them together
What is the expansion of e^x
1 + x + x^2/2! + x^3/3! + x^4/4! + …
How to find The sum of (ar + b) when r =1
an/2(n+1)+ bn
How to find the sum when r = k+1
find the sum when r =1 - the sum when n =k
How to prove a function is even
show that f(-x) = f(x)
How to prove a function is odd
show that f(-x)= -f(x)
How to find stationary points
The first derivative equal to 0
What order to sketch curve
- Mark roots 2. Turning points 3. Vertical asymptotes 4. y-intercept
How to prove that a function is even
show that f(-x) = f(x)
How to prove that a function is odd
f(-x) = -f(x)
What does -f(x) do
reflection of the graph f(x) in the x-axis
graph of f(-x)
reflection of the graph of f(x) in the y-axis
graph of f(x) +- a
+ a shifts the graph up a units - a shifts the graph down a units
graph of kf(x)
k>1 then the graph is strectched vertically k<1 the graph will be compressed vertically
f(x +- b)
f(x + b) will shift the graph left b units f(x - b) will shift the graph right b units
f(cx)
c>1 then the graph f(x) will be stretched horizontally c<1 then the graph will be compressed horizontally
What is the formula for volume of revolution around the x-axis
integral of b a limit pi x y^2 dx
What is the formula for volume of revoltuion around the y-axis
integral of b a pi * x^2 dy
What must be true for matrix multiplcation to take place
for A x B to be possible, the number of columns of A must match the number of rows of B
what makes a matrix orthogonal
if A’A= I
what makes a matrix symmetric
if A’ = A, the matrix will be symmetrical about the leading diagonal
What makes a matrix skew-symmetric
If A’ = -A there can only be zeros in the leading diagonal
what makes matrix B the inverse of matrix A
if AB = BA = I thus only square matrixes can be inverse
the determinant is denoted by det (A) or |A|, how is this calculated
is defined as |A| = ad - bc
What makes a matrix non-singular and invertible
if |A| doesnt equal 0
What makes a matrix a singular
if |A| = 0
What is the General term of the binomail expansion
How do you expand (n
r)
What are the four forms of auxillary equations
Distinct real roots: y = Ae^m1x + Be^m2x
Repeated real roots: Y = (A + Bx)e^mx
Imaginary roots: Y = Acosβx + Bsinβx (For imaginary roots in the form +/- βi)
Complex roots: Y = e^αx(Acosβx + Bsinβx) (For imaginary roots in the form α +/- βi)
