Pure - Single & Further Flashcards
1+cot2(x) = ?
cosec2(x)
1 + tan2(x) = ?
sec2(x)
sin(A ± B) = ?
sin(A)cos(B) ± sin(B)cos(A)
cos(A ± B) = ?
cos(A)cos(B) ∓ sin(A)sin(B)
tan(A ± B) = ?
tan(A) ± tan(B)/1 ∓ tan(A)tan(B)
d/dx tan(x) = ?
sec2(x)
sec2(x) + cosec2(x) = ?
sec2(x)cosec2(x)
What transformation is applied to go from z to z* on an Argand diagram?
reflection in the real (x) axis
What transformation is y=f(2x)?
Compression by a factor of 2 along x axis
What transformation is y=2f(x)?
Stretch by a factor of 2 along the y-axis
What transformation is y=-f(x)?
Reflection in the x-axis
What transformation is y=f(-x)?
Reflection in the y-axis
What transformation is y=f(x)+k?
shift upwards along y axis by k units
What transformation is y=f(x+k)?
shift leftwards along x-axis by k units
Quotient rule if y = u(x)/v(x)
dy/dx = u’(x)v(x) - u(x)v’(x)/v2(x)
Finite sum of an arithmetic series
Sn = (n/2)*(2a+(n-1)d)
Finite sum of a geometric series
Sn = a(1 - rn)/(1 - r)
Infinite sum of a geometric series
S∞ = a/1-r
given -1 < r < 1
Sum to n of r
n(n+1)/2
Sum to n of r2
n(n+1)(2n+1)/6
Sum to n of r3
n2(n+1)2/4
(f-1)’(x) = ?
1/f’(f-1(x))
When is a function even and what does this mean?
- f(x) = f(-x)
- reflective symmetry in the y-axis
When is a function odd and what does this mean?
- -f(x) = f(-x)
- rotational symmetry of π radians about the origin
What is the parts formula if you are integrating the function:
f(x) = u(x) * v’(x) dx ?
I = [u(x)v(x)] - (∫ v(x) * u’(x) dx)
What is the area scale factor when making the substitution u = g(x) ?
dx/du
What are the fractions for a partial fraction with distinct linear factors?
A/f(x) + B/g(x)
What are the fractions for a partial fraction with an irreducible quadratic factor?
(Ax + b)/m2x+k + c/bx+d
What are the fractions for a partial fraction with repeated factors e.g.
1/(x+1)(x-3)3 ?
a/x+1 + b/x-3 + c/(x-3)2 + d/(x-3)3
What are the derivatives of:
- arcsin(x/a)
- arccos(x/a)
- arctan(x/a) ?
- 1/sqrt(a2-x2)
- -1/sqrt(a2-x2)
- a/a2+x2
What are the small angle approximations for:
- sin(x)
- cos(x)
- tan(x) ?
- x
- 1-x2/2
- x
What is the general term that is summed in the MacLaurin Series?
f(n)(0)*xn/n!
What are the series expansions of:
- sin(x)
- cos(x)
- ex
- ln(1+x)
- (-1)n * x2n+1/(2n+1)!
- (-1)n * x2n/(2n)!
- xn/n!
- (-1)n * xn/n
What is the general term for the binomial expansion of (1 + x)n and for what values of x is it valid?
(1 + x)n = Σ nCk * xk (from k=0 to infinity)
where: nCk = n(n-1)(n-2)(n-3)…(n-k+1)/k!
What are Vieta’s relations for a polynomial with degree n and coefficients an, an-1, an-2, … a0?
sum of roots = -an-1/an
product of roots = (-1)n * a0/an
sum of products of pairs of roots = an-2/an
How to divide complex numbers in modulus-argument form?
Divide their moduli and subtract their arguments i.e.
r1⁄r2 * cos(θ1 - θ2) + isin(θ1 - θ2)
How to multiply complex numbers in modulus argument form?
Multiply moduli and add arguments.
Describe the locus of:
- |z - z1| = r
- arg(z - z1) = θ
- |z - z1| = |z - z2|
- |z - z1| = k*|z - z2| where k ≠ 1
- arg(|z - z1|/|z - z2|) = θ
- Circle, centre z1, with radius r
- (Half)line, starting at z1, with angle θ to the real axis
- Perpendicular bisector of line joining z1 and z2
- Circle defined by diameter which lies on the line produced by connecting z1 and z2
- major arc of circle through z1 and z2
Write:
- acosθ + bsinθ
- asinθ + bcosθ
in harmonic form
What is harmonic form useful for?
- Rcos(θ-α) if either a or b is negative, use cos(α) = a/R and sin(α) = b/R, ensuring they give the same value for α. If they don’t use the value of α which isn’t between 0 and π/2
- Rsin(θ+α)
Finding the range of sums of trigonmoetric identities, including minima and maxima.
What lines of symmetry exist for polar curves if r is a function of:
- cos only
- sin only
?
- symmetry about initial line (x-axis)
- symmetry about the line θ=π/2 (y-axis)
When should you convert a polar equation to cartesian in order to sketch it?
When r is a function of sec or cosec, because you can multiply both sides by cos or sin to get an x or y value.
What is the area of the section of the polar curve r = f(θ) bounded by the radii θ=α & θ=β?
1/2 * ∫ αβ r2 dθ
How to find the equations of tangents to a polar curve:
- perpendicular to the initial line?
- parallel to the initial line?
- substitute x = rcosθ into r = f(θ), make x the subject, differentiate x wrt θ and set derivative equal to zero.
- do the same but substitute y=rsinθ
How can we rewrite:
- sinAsinB in terms of cos?
- cosAcosB in terms of cos?
- sinAcosB in terms of sin?
- 1/2 * [cos(B-A) - cos(B+A)]
- 1/2 * [cos(B+A) + cos(B-A)]
- 1/2 * [sin(A+B) + sin(A-B)]
Given a first order DE of the form:
y’ + Py = Q
where P and Q are both functions of x, what is the integrating factor and how can it be used to solve the DE?
- Integrating factor, R = e ∫ Pdx
- Multiply all terms by R
- LHS becomes derivative of ye ∫ Pdx
- Re-write LHS as d/dx * ye ∫ Pdx
- Integrate wrt x (RHS becomes ∫ Qe ∫ Pdx dx)
For second order differential equations, what is the complementary function when the solutions to the auxiliary equation are:
- distinct and real (n1 & n2)?
- coincident and real (n)?
- distinct and complex (n1 ± in2)?
- y = Aen1x + Ben2x
- y = Aenx + Bxenx
- y = en1x[(A+B)cos(n2x) + i(A-B)sin(n2x)]
For inhomogenous second order DEs, what is the guess for the PI if the inhomogeneous bit (function purely in x) is:
- polynomial of degree n?
- exponential e.g. Qemx?
- trigonometric, of form asin(nx) or acos(nx)?
- polynomial of degree n
- Cemx (m stays same, coefficient in front of e doesn’t)
- y = Csin(nx) + Dcos(nx) (have to have both even if original function only had sin or only had cos)
How can we simplify 3x3 matrices to make finding the determinant easier?
Add the elements (or multiples of the elements) in a row or column to the corresponding elements in any other row or column with the aim of making as many of the elements in the top row equal to 0.
What is the determinant if two rows or columns are identical?
zero
What happens to the determinant if a row or column has all its values multiplied by a value k?
The determinant is also multiplied by the factor k.
What is the transpose of a 3x3 matrix?
The matrix obtained when you refelct a matrix about its principle (leading) diagonal - top left to bottom right. The determinant of the transpose is the same as the original.
What geometric arrangements can be formed when 3 simultaneous equations in 3 unknowns (x,y,z) are inconsistent?
- Triangular prism
- Two planes are parallel and separate, and both are intersected by the third.
- All three are parallel and separate.
- Two planes are conincident and one is parallel and separate.
What geometric arrangements are possible when there are solutions to 3 simultaneous equations in 3 unknowns (x,y,z)?
- All three planes are perpendicular and intersect at a single point (unique solution).
- Three planes intersect along a line, forming a sheaf with the line of intersection representing infinte solutions.
- The three planes are coincident (they are the same plane), giving infinite solutions.
How to find the component of a vector, a, in the direction of another vector, d, where a & d are separated by an angle θ
The component can be written as kd, for k > 0
Then k = a ∙ d/|d|2
What commutativity rule does the cross product follow?
a x b = -b x a
Anticommutative rule
How can we calculate the area of a triangle using the cross product?
1/2 * absinC = 1/2 * |a x b|
How can we find the shortest distance of a plane from the origin?
- Write the plane in the form:
r . n = d
where:
r is the vector equation of a plane
n is the direction vector which is normal to this plane
d = a . n where a is the position vector of a point A which lies on the plane
- Divide both sides (n & d) by the magnitude of the normal vector, n, so that n becomes a unit vecotr (magnitude 1) in the direction normal to the plane. Then d/|n| is the shorest distance of the plane from the origin.
For two planes written in the form r . n = d, what can we say if:
- |n| = 1 ?
- d has the same sign for two planes?
- d has opposite signs for the two planes?
- n is a unit vector and d is the distance between the plane and the origin
- the two planes are on the same side of the origin, so you subtract the smaller adjusted d from the larger adjusted d to find the distance between planes.
- the two planes are on opposite sides of the origin, so you add the two adjusted d values to find the distance between planes.
How to find the common line (line of intersection) of two non-parallel planes?
Cross product of the two normal vectors of each plane give the vector which is normal to both of these normals i.e. the direction vecotr of the line. Then find a point on the line by setting x, y or z equal to zero.
When you have a combination of dot and cross products e.g. a . b x c, what is the order of operations?
Cross product first always as you can’t cross a vector with a scalar (which is what you would get if you did the dot product first).
How can we tell if three vectors are coplanar?
a . b x c = a . (bc sinθ n̑) = abc sinθcosφ = abc sinθsinψ
where ψ=90º-φ = angle between a and the plane (because φ is the angle between a and n̑, the perpendicular to the plane)
therefore, if a . b x c = 0
a, b & c are coplanar
Give the formulae for the volume of:
- a cuboid
- a parallelpiped
- a tetrahedron
- a triangular prism
- a pyramid
in terms of dot and cross products.
- a . b x c
- a . b x c
- tetrahedron: 1/6 * a . b x c
- triangular prism: 1/2 * a . b x c
- pyramid: 1/3 * a . b x c
How to prove a series is convergent?
If |an+1/an| < 1 (strict) the series converges
Write the six hyperbolic functions, with sinh and cosh in exponential form.
- sinh(x) = 1/2 * (ex - e-x)
- cosh(x) = 1/2 * (ex + e-x)
- tanh(x) = sinh(x)/cosh(x)
- cosech(x) = 1/sinh(x)
- sech(x) = 1/cosh(x)
- coth(x) = 1/tanh(x)
Write the 5 hyperbolic identities
- cosh2(x) - sinh2(x) = 1
- coth2(x) -1 = cosech2(x)
- 1 - tanh2(x) = sech2(x)
- cosh(2x) = 2sinh2(x) + 1 = 2cosh2(x) - 1
- cosh(x) + cosh (y) = 2cosh[(x+y)/2]cosh[(x-y)/2]
- (& compound/double angle formulae but with sin replaced by sinh, cos replaced by cosh and tan replaced by tanh)
What are the derivatives of:
- sinh(ax)?
- cosh(ax)?
- tanh(ax)?
- a cosh(ax)
- a sinh(ax)
- a sech2(ax)
Give the logarithmic form and derivative of:
- sinh-1(x)?
- cosh-1(x)?
- tanh-1(x)?
- ln[x + √(x2 + 1)] , 1 / √(x2 + 1)
- ln[x + √(x2 - 1)] , 1 / √(x2 - 1)
- 1/2 * ln[(1+x)/(1-x)] , 1 / (1 - x2)
How to integrate fractions with polynomials when the degree of the numerator is the same as or higher than that of the denominator?
- Divide the numerator by the denominator:
- For same degree, write 1 + [f(x)/original denominator] and use inspection to find f(x)
- For higher degree, write xn-2 + [f(x)/original denominator] and use inspection to find f(x). n is the degree of the original numerator.
What is the reduction formula for ∫ sinnx dx and how is it useful?
Let ∫ sinnx dx = In
Then:
In = [(n-1)/n] In-2
Apply this until the integrand has sin0, which is easily antidifferentiated.
How can we find reduction formulae?
- Split integrand into the product of two parts, one which is easily integratable and one which is not.
- Find the antiderivative (A) of the easily integratable one and multiply this with the part that is not easily integratable (B). This product is called C.
- Differentiate the B and then multiply the result by A. this product is called D.
- The original integrand, I, is given by:
I = C - ∫D
Usually the integral we started with (before writing as a product) appears when ∫D is simplified, so we can write it as I, make I the subject and then solve.
What is the length of an arc of a polar curve between two lines given by θ = α and θ = β?
s = ∫αβ √ [r2 + (dr/dθ)2] dθ
When and how do we use improper integrals?
- When a limit of integration is ± ∞, or the integrand is undefined at one or both of the limits of integration, or at any point in between.
- If one of the limits is causing problems, replace it with a variable e.g. n, integrate and then take the limit as n approaches the problematic value.
- If a value inbetween the limits is causing problems, split the integral into two and repeat process as above for each new integral.
What order do the rows and column go when stating the order of a matrix, and which values have to be equal between two matrices if they can be multiplied?
rows x columns
Columns of first matrix have to be the same as rows of second.
For two matrices A & B, what is det(AB)?
det(A) x det(B)
How to find the inverse of a 3x3 matrix?
- Find the determinant.
- Find minor determinants for each of the 9 elements (determinant of 2x2 matrix formed by deleting row and column which element is in). Place each minor determinant value in the place of the element for which it was calulated, forming a 3x3 matrix.
- Counting from the top left entry horizontally rightwards, place a -ve sign infront of every other minor determinant (starting with no minus sign).
- Transpose the matrix by reflecting in the leading diagnoal.
When is the matrix T considered a linear transformation?
When T(ax) = aT(x) and T(ax + by) = aT(x) + bT(y)
What is the 2D rotation matrix?
cosθ -sinθ
sinθ cosθ
How to find an invariant line of a transformation T?
- Pick two points on the line y = mx + c e.g. (t , mt + c) and (T, mT + c)
- When you apply T to the point (t , mt + c), you should get (T , mT + c)
- Multiplying out, you get a set of simultaneous equations from which you can eliminate T and t to find the m and c which give an invariant line.
What is de Moivre’s theorem?
(cos θ + isin θ)n = cos nθ + isin nθ
when n is an integer (+ve or -ve).
*the plus on the LHS is important: if there is a -sinθ term, convert it to +sinθ*
If z = cosθ + isinθ, what are:
- zn + z-n
- zn - z-n
- 2 cos nθ
- 2i sin nθ
How to find nth roots using de Moivre’s theorem?
e.g. (a + bi)1/n
- Write in modulus-argument form.
- Apply de Moivre’s theorem.
- Find the n roots by adding/subtracting 2π from the numerator of the argument.
What is the exponential form of a complex number z = r(cosθ + isinθ)
reiθ
What is the mean value of a function, f(x), on [a,b] if
f : [a,b] → ℝ?
f = 1/(b-a) * ∫ab f(x) dx
What is the Newton-Raphson formula?
xr+1 = xr - [f(xr)/f’(xr)]
