Final Maths Flashcards
What is the formula for nCr?
r!-(n-r)!
What is (X+Y)^n ?
(n) (n-r) r
(r) x y
What is
*
———————–
(x+a)(x+b)^2
In partial fractions?
A B C
—– + ——— + ———
(x+a) (x+b) (x+b)^2
What is
*
———————–
(x+a)(x^2 + bx +c)
In partial fractions?
A Bx + C
—– + ——————–
(x+a) (x^2 + bx + c)
Sin2x=
2sinxcosx
Cos2x=
cos^2x - sin^2x
1 - 2sin^2x
2cos^2x - 1
Cos(-x)=
Cos(x)
Sin(x)=
-Sin(x)
What is it when f’‘(0) is less than 0?
Maximum
What is S’(T) and S’‘(T) ?
Speed and acceleration
Sin^2X=
1/2 (1-cos2x)
Cos2x=
-1/2 (1+cos2x)
What is an odd function?
When f(-x) = -f(x) or when the function has half turn symmetry.
What is an even function?
When f(x)=f(-x) or when the function is symmetrical across the y axis
What is the function used to describe the position of an asymptote?
X—> n +/- : {let x = (n +/- 0.1)} y —> +/- infinity
What is used to describe the domain?
{ X: XER, X () n}
How can you determine the range of a function?
The domain of the inverse
What are natural numbers (N)?
1,2,3,4,5
What are whole numbers (W)?
0,1,2,3,4
What are integer numbers (Z)?
-2,-1,0,1,2,3
What are rational numbers (Q)?
Numbers produced by a fraction
What are irrational numbers ?
Numbers which cant be written as a fraction
What are Real numbers (R)?
Amy number that isn’t complex
What are complex numbers (C)?
The combination of a real and an imaginary number
What is ill-conditioning?
When a small change in the equation leads to a significant change in the solution.
When is a matrix redundant?
000:0
When is a matrix inconsistent?
000:3
Derivative of sin(ax)
acos(ax)
Derivative of cos(ax)
-asin(ax)
Derivative of tan(ax)
asec^2(ax)
Derivative of sec(ax)
asec(ax)tan(ax)
Derivative of cosec(ax)
-acosec(ax)cot(ax)
Derivative of cot(ax)
-acosec^2(ax)
Derivative of e^(f(x))
f’(x) x e(x)
Derivative of Ln(f(x))
f’(x)/f(x)
Derivative of sin^-1 (x/a)
1
———— x(Chain rule factor)
(a^2 - x^2)^1/2
Derivative of cos^-1 (x/a)
- 1
———— x(Chain rule factor)
(a^2 - x^2)^1/2
Derivative of tan^-1 (x/a)
a
———— x(Chain rule factor)
(a^2 + x^2)
Derivative of lny
1/y dy/dx
Derivative of Log10x
1/xln10
Integral of ax^n
1
———- ax^(n+1) +C
n+1
Integral of (ax +b)^n
1
———- (ax+b)^(n+1) +C
a (n+1)
Integral of cos(ax + b)
1
—– sin(ax +b) +C
a
Integral of sin(ax + b)
-1
—– cos(ax +b) +C
a
Integral of Tan(x)?
lnsecx
Integral of e^(ax + b)
1/a x e^(ax + b)
Integral of sec^2(ax+b)
1/a tan(ax+b)
Integral of 1/(ax+b)
1/a ln(ax + b)
Integrate 1
————
(1 + x^2)
Tan^-1 (x) + C
Integrate 1
————
(1 - x^2)^(1/2)
Sin^-1(x) + C
Integrate 1
————
(a^2+ x^2)
1/a Tan^-1(x/a) +C
Integrate 1
————
(a^2 - x^2)^(1/2)
1/a Sin^-1(x/a) + C
Integrate fg’
fg- int of f’g
What are the two ways you can calculate the derivative of the inverse?
- Swap x and y values, rearrange so y is the subject and differentiate.
- Swap x and y values, differentiate by dx/dy. find the reciprocal of dx/dy = dy/dx. sunstitute a value for y
How do you find the area between two curves and the y axis?
int of f(y) by dy with the limits as y values
How do you find the volume of revolution around the x axis?
int of pie times (f(x))^2 dx with limits set as x values
What is the domain and range of sin^-1 x?
Domain: -90
What is the domain and range of cos^-1 x?
Domain: 0
What is the domain and range of tan^-1 x?
Domain: -90
How do you implicitly differentiate something?
d/dx = d/dy x dy/dy
What is a point of inflection?
When f’‘(x) = 0
How do you differentiate a parametric equation?
dy/dy = dy/dt x dt/dx
How do you find the second derivative of a parametric equation?
d/dt dy/dx x dt/dx
What is the conjugate of z= a + bi
a-bi
What does a complex number multiplied by the conjugate produce?
Whole number
How do you find the square root of a + bi?
Make it equal to a+bi and equate real and imaginary parts
How do you find a (argz)
Tan^-1 (b/a)
How do you find r the modulus of Z?
square root of a^2 + b^2
What are the quadrants of the cartesian diagram?
pie - theta 1 Theta 1
1 Re
————————————————-
-pie + theta 1 -Theta
1
1 im
How do you write Z in polar form?
r(cos (x) + isin (x) )
How do you find z1 x z2 ?
r1 x r2(cos (x1 + x2) + isin (x1 + x2))
How do you find z1 / z2 ?
r1 / r2(cos (x1 - x2) + isin (x1 - x2))
How do you find z^n ?
r^n(cos (nx) + isin (nx) )
How do you find sin 5x in sines?
Binomial expansion of (cos (x) + isin (x) )^5
Equate isin(x) with imaginary part of expansion
Factor out i
Sub in (1-sin^2 x) as cos^2 x
How do you find the geometrical interpretation of [z+5]=4
Sub in z = x + iy
Group real number with real part
Let the square root of x + 5+ iy = 4
Square both sides
What is the centre of the circle and radius with the equation x^2 + y^2 + 2gx + 2fy + C = 0
Centre = (-g,-f) Radius = The root of g^2 + f^2 - C
What is the centre of the circle and the radius of a circle with the equarion (x-a)^2 + (y-b)^2 = r^2
Centre = (a,b) Radius = r
What is 1 + tan^2 x ?
sec^x
What is the compound angle formula for sin(a+b)?
sinacosb + cosasinb
What is the compound angle formula for cos(a+b)?
cosacosb-sinasinb
What is the compound angle formula for sin(a-b)?
sinacosb - cosasinb
What is the compound angle formula for cos(a-b)?
cosacosb+sinasinb
How do you find the roots of a polar equation?
n root of r (cos((2pie x k)/n) + isin((2pie x k)/n))
What is the formula for a geometric sequence?
Un=ar^(n-1)
What is the sum of a geometric series?
a(r^n - 1)/r-1
What’s the sum to infinity of a geometric series?
a/(1-r)
Whats the formula for an arithmetic sequence?
un= a + (n-1)d
What’s the sum of an arithmetic sequence?
n/2 (2a + (n-1)d)
What’s the sum of an arithmetic series when you know the last term?
n/2 (a+l)
What’s the sum of sequential numbers?
n(n+1)/2
What’s the sum of sequential squared numbers?
n(n+1)(2n+1)/6
What’s the sum of sequential cubed numbers?
n^2(n+1)^2 / 4
What is the order of a matrix?
row x column
What is the transpose?
Replacing rows with columns
When is a matrix symmetrical?
When A = transpose
When is a matrix skew-symmetric?
When -A = transpose (main diagonal will consist of zero’s)
What is matrix A - Matrix B equal to ?
A + -(B)
What can A be re-written as with respect to transpose properties?
(A’)’
What can (A+B)’ be re-written as with respect to transpose properties?
A’ + B’
What can (AB)C be re-written as with respect to matrixes
A(BC)
What can A(B+C) be re-written as with respect to matrixes
AB + AC
What can (AI) be re-written as with respect to matrixes
A
How do you find A^3 with respect to matrixes?
A.A^2
What does it mean if AB=BA=I with respect to matrixes?
A is the inverse of B
What is the determinant of a 2x2 matrix?
AD-BC
What does it mean in the determinant of a 2x2 matrix is a)0 or b) not zero
a) Singular - matrix / no inverse
b) Non-singular - matrix / Inverse exists
How do you find the inverse of a 2x2 matrix?
1/ad-bc (d -c)
(-b a)
How can you find det (AB) ?
Det A x Det B
How do you find the determinant of a 3x3 matrix?
a(ei-fh) - b(ai-gf) + c(dh-ge)
How do you find the inverse of a 3x3 matrix?
1/det A x transpose of co-factors
How do you find solutions to equations using the inverse?
AX=B so X=A-1 x B
where x is the column matrix
What are the direction cosines?
cos (alpha) = x/|a|
cos (beta) = y/|a|
cos (y) = z/|a|
What is a unit vector?
A vector with the length of 1
How do you tell if two vectors are perpendicular?
a.b = 0
How do you find the angle between two vectors?
a.b / |a| x |b| = cos x
What is the cross product / vector product?
The determinant, sub top line as i j k
What is the area of a triangle?
1/2 absinc
What is the substitution of an area for a triangle?
|a x b| = |a| x |b| sinx
so area = 1/2 |a x b|
What is the area of a parallelogram?
|a x b|
What can a x b be re-written as with respect to vectors?
-b x a
What can a(b+c) be re-written as with respect to vectors?
ab + ac
What can (a+b)c be re-written as?
ac+bc
What is the result of a x a with respect to vectors?
0
What is a(bc) not equal to ?
(ab)c
How do you write in vector form?
r= i, j, k + lambda(i, j, k)
How do you write in parametric form?
x= a1 + lambda d1 y= a2 + lambda d2 z= a3 + lambda d3
How do you write in symmetric form? or cartesian form?
x-a1/d1 = y-a2/d2 = z-a3/d3 = lambda
How do you find the vectors perpendicular to both u and v ?
+/- (v x u)
What two types of vectors exist?
Identical
Parallel
Intersecting
Skew
How do you find if a vector intersects?
Put into parametric form
Simultaneous equations to work out the unknowns
Check for consistency with the third equations
How do you find if two vectors are skew?
Same method as intersecting but show that there is inconsistency with the solutions
What is the angle between two lines?
d1 . d2 / |d1| x |d2| = cos x
What is the vector equation of a plane?
r.n = a.n
What is the parametric equation of a plane?
r = a + lambda b + ewe c
Where a is a positive vector from the origin
b and c are non-parallel vectors, parallel to the plane
Lambda and ewe are scalars
How do you find the normal of a parametric equation of a plane?
bxc
What are the different ways two planes can exist together?
Parallel
Coinsident
Intersect on a line
How do you calculate where two planes intersect at a line?
we need a and d
a= a point on the line, sub x, y, or z as equal to zero then solve simultaneously for the other two values
d= n1 x n2
What is the angle between two planes?
n1 . n2 / |n1| x |n2| = cos x
What ways can three planes exist together?
Intersect at a point , intersect at a line (sub in parameter) and not intersect. All use gaussian elimination. Point will have a set of solutions. The line will have a a redundant matrix no intersection will have an inconsistent matrix
What ways can a plane and a line exist together?
Intersect (one solution), lie on the plane(many solutions 4=4) or parallel ( no solutions, n1.d1 = 0)
How do you find the angle between a line and a plane?
d1 . n1 / |d1| x |n1| = cos x (90-x)
How do you find out the integrating factor?
e^int of p(x) dx
What do you do once you have found the integrating factor?
Multiply both sides by it, giving
d/dx(I(x)y) = Int of I(x) x f(x)
What is the general solution of a homogenous equation when the values of n are real and not equal?
Y= Ae^mx + Be^mx
What is the general solution of a homogenous equation when the values of n are real and equal?
Y= Ae^mx + Bxe^mx
What is the general solution of a homogenous equation when the values of n are not real?
Y= e^p(x) (Acosqx + Bsinqx)
What is the particular integral when f(x) is linear?
ax+b
What is the particular integral when f(x) is quadratic?
ax^2 +bx +c
What is the particular integral when f(x) is a sine or cosine function?
psin(nx) + qcos(nx)
What is the particular integral when f(x) is exponential?
Ke^rx
What is the particular integral when f(x) is exponential and r is equal to a root of the equation?
Kxe^rx
What is the particular integral when f(x) is exponential and r is equal to both roots of the equation?
Kx^2e^rx
What is the particular integral when f(x) is a sum of two functions?
find each one separately then add to find overall yp
What is a power series?
The sum of anx^n where n starts at 0
What is the McClaurin series sum
f(r) (0) x^r / r! where r=0 and n=infinity
What is the expansion for e^x ?
1 + x + x^2/2! + x^3/3! + x^4/4! +….
What is the expansion for sinx?
x - x^3/3! + x^5/5! - x^7/7! +…..
What is the expansion for cosx ?
1 - x^2/2! + x^4/4! - x^6/6! +…..
When does ln(1+x) converge?
-1
When does 1/1+x converge?
|X|
When does than^-1 converge?
|X|
What functions do not have a M series?
Ln (x) + x^1/2
How do you find the series for two functions?
Expand both and multiply together
What is the expansion for tan^-1 x ?
x - x^3/3 + x^5/5 - x^7/7 +…..
What is the expansion for ln(x+1)
x - x^2/2 + x^3/3 - x^4/4 +….
How do you know if an iterative equation will converge?
If G’(x0)
What is the euclidean algorithm?
a=qb+r
What is the anti-clockwise rotational matrix?
cosx -sinx
sinx cosx
What is the clockwise rotational matrix?
cosx sinx
-sinx cosx
What is the reflection matrix?
cos2x sin2x
sinsx -sin2x
What is the scaling matrix?
(x 0)
0 x
How do you convert numbers to a different base?
Use division algorithm where the quotient is always the desired number. keep going until the divisor is 0
What is the equation for speed?
ds/dt = ((dx/dt)^2 + (dy/dt)^2)^1/2
