H240/02 Pure and Statistics Solutions
Proof - ANS-Proof by deduction- 1. let nEZ 2. therefore...
Proof by exhaustion- trying all the possibilities
Proof by contradiction- assume the opposite
Solving equations: algebra techniques - ANS
Functions - ANS-ONE-TO-ONE: one value of x = one value of y
ONE-TO-MANY: one value of x = two values of y
MANY-TO-ONE: two values of x = one value of y
MANY-TO-MANY: two values of x = two values of y
Domain and Range:
Domain- interval of values on the x-axis
Range- interval of values on the y-axis
Composite Functions:
- fg(x) means "do g first then f" or f(g(x))
- g^2(x) means gg(x)
Inverse Functions:
- for a function to have an inverse, it has to be a ONE-TO-ONE function
To find the inverse:
1. Change the notation from f(x)=... to y=...
2. Switch the x and y values
3. Rearrange to make y the subject
4. Change y=... back to f(x)=...
The Modulus Function:
- the modulus is the distance from 0
|x| =....
SKETCHING: (the graph of y=|x|)
- sketch the graph of f(x)=x
- Modulus can't have negative values of y
- Therefore reflect the negative y values in the x-axis
, Sequence and Series - ANS-Types:
CONSTANT- the same
CONVERGENT- getting closer to a certain value
DIVERGING- doesn't get closer to a certain value
PERIODIC- repeating
OSCILLATING- surrounding a number
Finding the limit:
If it is converging, it has a limit
If it is diverging, it doesn't have a limit
ARITHMETIC:
nth term- Un= a+(n-1)d
GEOMETRIC:
nth term- Un= ar^(n-1)
Sum formulas are in the formula booklet
Differentiation: investigation of curves - ANS-dy/dx finds the gradient of the TANGENT
the NORMAL is the perpendicular of the tangent
dy/dx > 0 increasing function
dy/dx < 0 decreasing function
dy/dx = 0 stationary point
A stationary point where:
f''(x) > 0 is a local minimum
f''(x) < 0 is a local maximum
f''(x) = 0 is a local min, local max or a stationary point of inflection
Solving differential equations - ANS-1. Separate Variables
2. Integrate both sides (add +c on one side)
3. Plug in to find C
4. Solve for y
Vectors and geometry - ANS-Magnitude is the distance: |a| = √(x^2 + y^2)
Proof - ANS-Proof by deduction- 1. let nEZ 2. therefore...
Proof by exhaustion- trying all the possibilities
Proof by contradiction- assume the opposite
Solving equations: algebra techniques - ANS
Functions - ANS-ONE-TO-ONE: one value of x = one value of y
ONE-TO-MANY: one value of x = two values of y
MANY-TO-ONE: two values of x = one value of y
MANY-TO-MANY: two values of x = two values of y
Domain and Range:
Domain- interval of values on the x-axis
Range- interval of values on the y-axis
Composite Functions:
- fg(x) means "do g first then f" or f(g(x))
- g^2(x) means gg(x)
Inverse Functions:
- for a function to have an inverse, it has to be a ONE-TO-ONE function
To find the inverse:
1. Change the notation from f(x)=... to y=...
2. Switch the x and y values
3. Rearrange to make y the subject
4. Change y=... back to f(x)=...
The Modulus Function:
- the modulus is the distance from 0
|x| =....
SKETCHING: (the graph of y=|x|)
- sketch the graph of f(x)=x
- Modulus can't have negative values of y
- Therefore reflect the negative y values in the x-axis
, Sequence and Series - ANS-Types:
CONSTANT- the same
CONVERGENT- getting closer to a certain value
DIVERGING- doesn't get closer to a certain value
PERIODIC- repeating
OSCILLATING- surrounding a number
Finding the limit:
If it is converging, it has a limit
If it is diverging, it doesn't have a limit
ARITHMETIC:
nth term- Un= a+(n-1)d
GEOMETRIC:
nth term- Un= ar^(n-1)
Sum formulas are in the formula booklet
Differentiation: investigation of curves - ANS-dy/dx finds the gradient of the TANGENT
the NORMAL is the perpendicular of the tangent
dy/dx > 0 increasing function
dy/dx < 0 decreasing function
dy/dx = 0 stationary point
A stationary point where:
f''(x) > 0 is a local minimum
f''(x) < 0 is a local maximum
f''(x) = 0 is a local min, local max or a stationary point of inflection
Solving differential equations - ANS-1. Separate Variables
2. Integrate both sides (add +c on one side)
3. Plug in to find C
4. Solve for y
Vectors and geometry - ANS-Magnitude is the distance: |a| = √(x^2 + y^2)
