Maths

Question 1

How do you find a limit of a simple function where the limit occurs as x→a?

Answer 1

Substitute in a+δ and evaluate the function, the limit will be where δ→0

Question 2

How do you resolve limits of x→∞ for product and quotient functions?

Answer 2

For product, find the value that each part tends towards, then find which one will dominate.

For quotient, divide the whole fraction by the largest power of x, the 1/x values tend towards zero meaning the remaining values can be evaluated.

Question 3

What is l’Hopital’s rule for limits?

Answer 3

The differential of a function will tend towards the same value as the function.

Question 4

How do you differentiate from first principles?

Answer 4

lim(Δx→0) of (f(x+Δx)-f(x))/Δx

Question 5

What is the derivative of tan(ax) and sec(ax)?

Answer 5

asec²(ax) and asec(ax)tan(ax) respectively

Question 6

What is the product rule and the quotient rule for differentiation?

Answer 6

d/dx(uv)=uv’+u’v

d/dx(u/v)=(u’v-uv’)/v²

Question 7

What are the two parts of the eigen operation?

Answer 7

The eigenfunction which repeats and the eigenvalue that changes

Question 8

What is a point of inflection?

Answer 8

A point where the graph changes from concave to convex, it does not have to be a stationary point.

Question 9

What is the difference between stationary and turning points?

Answer 9

Stationary points are points where f’(x)=0, turning points are where dy/dx changes sign on passing through the point.

Question 10

What 3 situations can occur when f⁽²⁾(x)=0? How can these be futher evaluated?

Answer 10

For stationary inflection points, f⁽¹⁾(x)=0 and f⁽³⁾(x)≠0

For not stationary inflection points, f⁽¹⁾(x)≠0 and f⁽³⁾(x)≠0

For turning points, f⁽¹⁾(x)=0, f⁽³⁾(x)=0 and f⁽⁴⁾(x)≠0

Question 11

How can functions with multiple variables be differentiated?

Answer 11

Using partial derivitives, treating all but one variable as constant.

For f(x,y)=xy (δf/δx)_y=δf/δx=f_x=y

Question 12

When does a function with two variables have a stationary point when x₀=x and y₀=y?

Answer 12

When the partial derivatives for each variable both equal zero:

f_x(x₀, y₀)=f_y(x₀, y₀)=0

Question 13

For two variable systems, what is the formula for D and what can it determine?

Answer 13

D=f_xx(x₀, y₀)f_yy(x₀, y₀)-f_xy(x₀, y₀)²

Local minimum when D>0 and f_xx(x₀, y₀)>0

Local maximum when D>0 and f_xx(x₀, y₀)<0

Saddle point when D<0

Question 14

How can the differential of a function with two variables be found?

How can this be used?

Answer 14

By taking the partial derivatives of each variable and adding them together.

When finding the relative error of a measurement of these functions, dP/P.

Question 15

What is the equation for integration by parts?

Answer 15

int(udv)=uv-int(vdu)

Question 16

When integrating something containing (1-x²)^½, what would be an appropriate subsitution?

Answer 16

x for cosu or similar as cos²u+sin²u=1

Question 17

What are differential equations and how are simple ones solved?

Answer 17

Differential equations are where an equation contains a differential such as dy/dx and are solved by multiplying out the factors so an intergration can be done on both sides.

Question 18

How do you solve first order linear differential equations?

Answer 18

The general form is dy/dx+yP(x)=Q(x)

The solution is y=R(x)^-1•int(R(x)Q(x)dx) where R(x)=e^int(P(x)dx)

Question 19

How do you solve second order differential equations with constant coefficients where Q(x)=0?

Answer 19

The general form is d²y/dx²+c₁dy/dx+c₂y=0

This form suggests that it is a function that is multiplied by a constant when differentiated such as e^ax, as such we substitute this in for y.

d²e^ax/dx²+c₁de^ax/dx+c₂e^ax=0

a²e^ax+ac₁e^ax+c₂e^ax=0

2 roots can be found from a²+c₁a+c₂=0, a₁ and a₂.

General solution: y=Ae^a₁x+Be^a₂x where A and B are constants based on the boundary conditions.

Question 20

How is the sum of the first n terms in a finite series represented?

Answer 20

For the series: a+ax+ax²+…..+ax^n-1

S_n=a(1-xⁿ/1-x)

Question 21

What is the ratio test for convergance or divergance?

Answer 21

As r→∞, a series converges if |u_r+1/u_r|<1 and diverges if |u_r+1/u_r|>1 where u_r and u_r+1 are the rth and (r+1)th terms in the series

Question 22

What is the Maclaurin series?

What is the Maclaurin series for y=e^x?

Answer 22

An expansion of the generic function f(x) about the point x=0, a specific type of the Taylor series.

First find the 1 to nth derivatives for a function. Then resovle them for x=0. Then multiply the result by x to the power of the derivative number and divide by the derivative number factorial

so (f⁽ⁿ⁾(0)/n!)xⁿ

y=e^x=1+x+(x²/2!)+(x³/3!)+…+(xⁿ/n!)+…

Question 23

What is the general form of the Taylor series?

Answer 23

f(x)=f(a)+f⁽¹⁾(a)(x-a)+(f⁽²⁾(a)/2!)(x-a)²+…+(f⁽ⁿ⁾(a)/n!)(x-a)ⁿ+…

Question 24

What is the general form for complex numbers?

Answer 24

z=x+iy

Where x is the real part of z, Re z, and y is the imaginary part of z, Im z.

Question 25

What is the complex conjugate?

Answer 25

z*=x-iy

Where the imagianary part of the number is reversed in sign.

Question 26

How do you divide complex numbers?

Answer 26

You multiply both sides of the fraction by the complex conjugate of the denominator. This gives a real denominator on the product.

Question 27

Draw and label parts of an Argand diagram.

What is the argument and modulus of z?

Answer 27

The argument is θ and the modulus is r.

Question 28

What is the polar form of complex numbers?

How can this be compressed?

Answer 28

z=r(cosθ+isinθ)

Using the Maclaurin series for cosθ and sinθ, the polar form can be rewritten as z=re^iθ

Question 29

What is Euler’s formula?

Answer 29

e^iθ=cosθ+isinθ

Question 30

What is the complex conjuate in polar form?

Answer 30

z=r(cosθ-isinθ)=re^-iθ

Question 31

What is the De Moivre theorem and where does it come from?

Answer 31

zⁿ can be expressed as rⁿe^inθ

This can then be reintroduced to trig functions to find the De Moivre theorem:

(cosθ+isinθ)ⁿ=cosnθ+isinnθ

Question 32

How do you find vector magnitudes and unit vectors?

Answer 32

Vector magnitudes are represented by |a|=(x²+y²+z²)^½ where x, y and z refer to the i, j and k directions.

Unit vectors are â=a/|a|, these are vectors divided by their own magnitude.

Question 33

What is the scalar product of two vectors and how can the angle between them be found?

Answer 33

a•b= the i, j and k parts of each vector multiplied by the same letter i, j and k parts.

The rule for the angles is a•b=|a||b|cosθ

Question 34

What is the vector product of two vectors?

Answer 34

axb where you multiply each i, j and k part by the different i, j and k parts of the other vector. The triangle of i→j→k→i decides if the result is the positive or negative version of the other unit. e.g ixj=+k, kxj=-i, therefore axb≠bxa