Maths Small Problems Encountered

Question 1

What is a rational and irrational number?

Answer 1

Rational - includes any whole number, fraction or decimal that ends or repeats
Irrational - number that cannot be turned into a fraction (eg pi, root 10, root 2, 1.234536225)

Question 2

What is proof by contradiction?

Answer 2

Assumes the opposite is true, then contradict that with proof?

Question 3

Integrate cos3x

Answer 3

1/3 sin3x

Question 4

Differentiate lnx

Answer 4

1/x

Question 5

What is lne equal to?

Answer 5

1

Question 6

In integration by parts, is the natural log (ln) the u or dv/dx?

Answer 6

Always u

Question 7

What is an arithmetic sequence?

Answer 7

Goes up/down by same amount every time

Question 8

What is the equation for the nth term of an arithmetic sequence, and what does each letter represent?

Answer 8

a+(n-1)d

a - first term
n - number of terms
d - amount up/down by each time

Question 9

Find the interval of validity for (8-9x)^3, and the two ways of writing it

Answer 9

|x|<8/9
-8/9<x<8/9

Question 10

How do you find the area of a sector in degrees, and radians?

Answer 10

Degrees - (x/360)(π)(r)^2
Radians - (1/2)(r)^2(x)

Question 11

How do you find the length of a sector in degrees, and radians?

Answer 11

Radians - r(x)
Degrees - (x/360)(2πr)

Question 12

How do you convert from radians to degrees and vice versa?

Answer 12

Radians to degrees - x 180/π
Degrees to radians - x π/180

Question 13

Find the integral of cos^2x and sin(^2)3x

Answer 13

cos^2x = 1/4sin2x+1/2
sin(^2)3x = 1/2x-1/12sin6x

Question 14

Find the integral of 2sin3x

Answer 14

-2/3cos3x

Question 15

Which discriminant shows 2 real roots, 1 real root and no roots?

Answer 15

b^2-4ac > 0 - 2 roots
b^2-4ac = 0 - 1 root
b^2-4ac < 0 - no roots

Question 16

Volume of a sphere, rectangular prism and cylinder?

Answer 16

Cylinder - πr^2h
R. Prism - whl
Sphere - 4/3πr^3

Question 17

How do you show a function is always increasing?

Answer 17

The gradient (first derivative) is always bigger than zero

Question 18

How do you find x at an inflection point?

Answer 18

When second derivative is zero and using change of sign method

Question 19

What is the area/integral of a Cartesian equation?

Answer 19

integral of y(dx/dt)dt

Question 20

Small angle approx for cos1/2x and cos3x

Answer 20

1-1/8x^2
1-9/2x^2