Things

Question 1

Centre of mass of a semicircle

Answer 1

4r/3π

Question 2

Centre of mass of solid cone

Answer 2

1/4 h

Question 3

Centre of mass of hollow cone

Answer 3

1/3 h

Question 4

Centre of mass of a solid hemisphere

Answer 4

3/8 r

Question 5

Centre of mass of a hollow hemisphere

Answer 5

1/2 r

Question 6

Even function

Answer 6

f(x)=f(-x)

Question 7

Odd function

Answer 7

f(-x)=-f(x)

I.e. rotational symmetry of 2

Question 8

d/dx(tan x)

Answer 8

sec^2(x)

Question 9

d/dx(cosec x)

Answer 9

-cosec x cot x

Question 10

d/dx(sec x)

Answer 10

sec x tan x

Question 11

d/dx(cot x)

Answer 11

-cosec^2(x)

Question 12

S tan x dx

Answer 12

ln(sec x) + c

Question 13

S cosec x dx

Answer 13

-ln(cosec x + cot x) + c

Question 14

S sec x dx

Answer 14

ln(sec x + tan x) + c

Question 15

S cot x dx

Answer 15

ln(sin x) + c

Question 16

SinCos -+ identities

Answer 16

cos(π-α) = -cos(α)
cos(α-π) = -cos(α)
cos(-α) = cos(α)

sin(π-α) = sin(α)
sin(α-π) = -sin(α)
sin(-α) = -sin(α)

Question 17

Double angle formulae

Answer 17

sin(2A)=2sinA cosA
cos(2A)=cos^2(A)-sin^2(A)
=2cos^2(A) - 1
=1 - 2sin^2(A)

Question 18

Half angle formulae

Answer 18

Use double angle formulae and replace A with θ/2

Question 19

PF denominator with linear factor

Answer 19

(ax+b) -> A/(ax+b)

Question 20

PF denominator with irreducible quadratic factors

Answer 20

(ax2 + bx + c) -> (Ax + B)/(ax2+bx+c)

Question 21

PF denominator with a repeated factor

Answer 21

(ax+b)^2 -> A/(ax+b) + B/(ax+b)^2

Question 22

PF improper

Answer 22

When degree of numerator is greater than or equal to degree of numerator, quotient is polynomial of order top - bottom
Order4/order3 -> Ax + B + other fractions from other rules

Question 23

Integration by parts

Answer 23

S uv’ = uv - S u’v

Question 24

Integrating odd powers of sinx and cosx

Answer 24

Split sin^3(x) into sinx and sin^2(x), then convert the latter into 1-cos^2(x). Expand brackets and separate integrals.

Question 25

Integrating even powers of sinx and cosx

Answer 25

Use double angle formula of cos to turn them into cos(2x)

Question 26

Even powers of secx and cosecx

Answer 26

Use tan sec and cot cosec identities and differentials. Split into sec^2(x)sec^2(x)

Question 27

Odd powers of secx and cosecx

Answer 27

Write sec^(n-1)xsecx and then integrate by parts

Question 28

Dividing polynomials

Answer 28

To divide say x2 + 4x - 5 by (x+3), Set it identically equivalent to (x+3)(ax+b)+r Where ax+b is the quotient and r is the remainder

Question 29

Remainder theorem

Answer 29

When f(x) is divided by (αx-β) the remainder is f(β/α)

Question 30

Special sums

Answer 30

``` Σr = n(n+1)/2 Σr^2 = n(n+1)(2n+1)/6 Σr^3 = n^2(n+1)^2/4 ```

Question 31

Newton Raphson iteration

Answer 31

x(n+1) = xn - f(xn)/f’(xn) Limitations include a slow convergence speed when the tangent is shallow near the root, and the fact that some starting points fail to find the nearby root, and find other roots instead. A divide by zero error can occur, which causes the method to fail.

Question 32

Binomial expansion

Answer 32

(1+x)^n=1+nx+n(n-1)/2!x^2… | Valid for |x|<1

Question 33

Maclaurin expansion

Answer 33

f(x)=f(0) + f’(0)x + f’’(0)x^2/2! +…

Question 34

How to determine if g(x) converges

Answer 34

If 1

Question 35

Plotting a complex conjugate

Answer 35

Reflects in real axis

Question 36

Modulus and argument of z^n

Answer 36

arg(z^n)=narg(z) | |z^n|=|z|^n

Question 37

Modulus and argument of wz

Answer 37

|wz|=|w||z| | arg(wz)=arg(w)+arg(z)

Question 38

Rotation with matrix

Answer 38

Clockwise (cosθ sinθ) (-sinθ cosθ)

Question 39

Inverse matrix

Answer 39

1/(ad-bc) (d -b) | (-c a)

Question 40

(AB)^-1

Answer 40

B^-1 * A^-1