Pure Year 2 Flashcards
What is important to remember about sectors and small angle approximations?
Angles in radians NOT degrees
Shift
Setup
2
2
Arc length formula
r{theta}
theta must be in radians
Sector area formula
0.5r²{theta}
theta must be in radians
Area of a segment formula
(0.5r²{theta} - 0.5r²sin{theta}
theta must be in radians
When theta is small, 9{theta}²-2{theta}+1 is what?
1
theta is small so approximately 0, meaning that the first 2 terms become 0 so just leaves 1
What is a mapping?
Transforms one set of numbers into a different set of numbers
It is a function if every input has a distinct output
Is a one-to-one mapping a function?
Yes
Each input has one distinct output
Is a many-to-one mapping a function?
Yes
Each input has a distinct output
Is a one-to-many mapping a function?
No
Not every input has a distinct output
Is y=1/x a function
No
No value at 0, doesn’t map anywhere
Can be made a function by restricting domain to x≠0
Domain…
The set of X values for which the function is valid
Writen as x=a
Range…
The set of Y values that the function can take
Written as a<=f(x)<=b
How do you deal with partial fractions?
Multiply up
Knock out brackets by substituting values in to make some coefficients 0
What is a piecewise-defined funtion?
A function which consists of more than one part
When drawing:
●=less/more than or equal to
○=less/more than
What is an inverse function?
The mathematical opposite of the original function
Only exists for one-to-one functions
Y becomes X, X becomes Y, reflected in line Y=X
How do you get an inverse function?
Replace f(x) with y
Swap y and x
Rearrange to make x the subject
Swap y for f(x)
Check f(x) and not any other function notation e.g g(x)
When do inverse functions exist
One-to-one functions
What is the domain of an inverse?
The range of the ordinary function
What is the range of an inverse function?
The same as the domain of the normal function
If f(x)=sinx, the inverse function can be written as…
f-¹(x)=sin-¹x=arcsinx
Sketch the graph of y=arcsinx
State the range and domain
X Intercept: 0
Y Intercept: None
Coordinates: (-1,-90) (0,0) (1,90)
If f(x)=cosx, the inverse function can be written as…
f-¹(x)=cos-¹x=arccosx
if f(x)=tank, the inverse function can be written as…
f-¹(x)=tan-¹x=arctan
What is the modulus of a number?
It is the non negative numerical value
Also known as the absolute value
Denoted as |x|
What is the difference between y=f(|x|) and y=|f(x)|?
y=f(|x|) will not have a negative value input, meaning that negative values of x will become the positive values of x. The first quadrant is reflected in the y axis into the second quadrant and the fourth into third
y=|f(x)| will not go below the x axis since y cannot be negative. All negative in the third quadrant will reflect up into the second, and all negative in the fourth reflected up into the first
Sketch the graph of y=arccosx
State the domain and range
X intercept: 1
Y intercept: 90°
Key coordinates: (-1,180) (0,90) (1,180)
Asymptotes: ?
Reflected in the y=x axis
Sketch the graph of y=arctanx
State the range and domain
Online image
How do you access modulus feature on calculator?
Shift
Abs (absolute)
What’s important to remember once you have solutions for a modulus function?
Sub back in to check no phantom solutions (signs not equal once put back in on either side of equation)
How do you deal with solving modulus equations?
Draw the graphs look for intersections
Make modulus positive, solve, check for phantom solutions
Make modulus negative, solve, check for phantom solutions
What are reciprocate functions called?
Cosecant=Cosec
Secant=Sec
Cotangent=Cot
What is the rule to remember reciprocal functions?
Look at the third letter for each, will match what is on the bottom
Formula for Cosecant?
Cosec(x)=1/sin(x)
What is the formula for Secant?
Sec(x)=1/cos(x)
Formula for Cotangent?
Cot(x)=1/tan(x) OR cos(x)/sin(x)
Sketch the graph of y=cosec(x) in range (range two pie to negative two pie)
Image online
Sketch the graph of y=sec(x) in range negative two pi and two pi
Get an image online
Sketch the graph of y=cot(x) in the range negative two pi to two pi
Get an image from online
Method for solving equation involving reciprocal functions?
Change
Flip
Solve
Solve sec(x)=0
1/cos(x)=0
cos(x)=1/0
1/0 = no solutions
Solve cosec(x)=0
1/sin(x)=0
sin(x)=1/0
1/0 = no solutions
Solve cot(x)=0 in range 0° to 360°
1/tan(x)=0
tan(x)=1/0
1/0 = asymptotes, meaning 90°, 270°
2 trig functions from year 12
tan(x)=sin(x)/cos(x)
sin²(x)+cos²(x)=1
2 trig identities from year 13, derived from sin²(x)+cos²(x)=1
1+tan²(x)=sec²(x)
1 with a tan is sexy
1+cot²(x)=cosec²(x)
1 in a cot is cosy
How can you derive the two trig identities from sin²(x)+cos²(x)
Divide by cos
Divide by sin
Where do the addition formula appear?
Page 6 of formula booklet
Double angle formula for sin2x
2sinxcosx
Double angle formulas for cos2x
cos²x-sin²x
2cos²x-1
1-2sin²x
Double angle forumla for tan2x
2tanx
______
1-tan²x
When is dy/dx used
Gradient or rate of change
Tangent or normal
Increasing or decreasing
Stationary point or turning point
When is d²y/dx² used
Nature of a stationary point
Checking if minimum or maximum
Finding minimum or maximum
Differentiate tan(x)
sec²(x)
Differentiate x^n
nx^(x-1)
Differentiate e^x
e^x
Differentiate 4e^2x
8e^2x
Differentiate e^sin(x)
cos(x)e^sin(x)
Differentiate ln(x)
1/x
Differentiate ln(x²+1)
2x/(x^2+1)
Differentiate a^x
a^xln(a)
Differentiate 2^3x
3(2^3xln(2))
Differentiate sin, cos, -sin, -cos
cos
-sin
-cos
sin
Bracket rule for differentiating (f(x))^n
n(f(x))^n-1
Steps for product and quotient rule
Identify u and v
Differentiate u and v
Substitute into formula
Product rule formula
u’v+uv’
Quotient rule formula
(u’v-uv’)/v^2
dx/dy …
reciprocal of dy/dx
When is a function concave
If and only if f’‘(x)<0 for every x-value in given interval
When is a function convex
If and only if f’‘(x)>0 for every x-value in given interval
What is a point of inflection
Point at which a curve changes from being concave to convex or vice versa
f’‘(x) changes sign
Does not have to be a stationary point
How do you prove a point of inflection
d^y/dx^2
Equal to 0
Opposite signs either side very close to point
If y=u/v
Then dy/dx=…
(u’v-uv’)/v²
How do you put something in harmonic form
Rewrite the RHS using addition formula Compare coefficients Find tan a Use pythagoras to find R Write out your function in full
How do you find exact values
Draw a triangle
Lable OAH
Check signs
Signs for sin(x) from 0 to 360
P
P
N
N
Signs for cos(x) going from 0 to 360
P
N
N
P
Signs for tan(x) going from 0 to 360
P
N
P
N
Explain a cast diagram
C bottom right (cos)
A top right (all)
S top left (sin)
T bottom left (tan)
Starting at A, draw arrow anticlockwise
Letter in each quadrant is what is positive for that range
Harmonic motion
asin(x)+bcos(x)
R sin(x+a)
acos(x)+bsin(x)
R cos(x-a)
asin(x)-bcos(x)
R sin(x-a)
acos(x) - bsin(x)
R cos(x+a)
How do you integrate e^x
e^x + c
How do you integrate e^4x
0.25 e^4x + c
How do you integrate e^0.25x
4e^0.25x + c
How do you integrate
Sin
Cos
-Sin
-Cos
-Cos
Sin
Cos
-Sin
How do you integrate sin(2x)
-0.5cos(2x) + c
How do you integrate cos(0.5x)
2sin(0.5x) + c
How do you integrate a^x
a^x/ln(a)
How do you integrate 2^x
2^x/ln(2) + c
Integrate 2^3x
2^3x/3ln(2) + c
Reverse Bracket Rule
(f(x))^n = (f(x))^n+1/(n+1)(f’(x))
Integrate (2x+1)⁵
(2x+1)⁶/12 + c
How can you integrate a complex fraction using Y12 methods
Split the fraction into a polynomial by multiplying up by the denominator ^-1
When do you use ln to integrate a fraction
If the numerator is a multiple of the derivative of the denominator
Integrate 3/(1+3x)
3ln(1+3x) ÷ 3 = ln(x) + c
Integrate 3/x
3ln(x) + c
Integrate x/(3x²+1)
xln(3x²+1)/6x + c = ln(3x²+1)/6 + c
When do you use algebraic division in integration
If you have an improper fraction with a numerator of same power or higher than the denominator
When do you use partial fractions for integration
If the denominator has a higher power than the numerator
AND the denominator can be factories I multiple brackets
How do you simplify integrating trig
Use identities
Use double angles
Check if then in the book
Relationship between product rule and integration by parts
Opposites
Integration by parts formula
Page 7
uv - ∫ vu’ dx
Acronym for labelling u then v in integration by parts
Logs
Algebra
Trig
Exponentials
Integrate ln(x)
1/x + c
Integrate (lnx)^2
u=(lnx)^2
v=x
xln(x)^2 - 2xln(x) + 2x + c
Integration by substitution method
Chose the substitution u Differentiate du/dx Rearrange for dx Substitute, replacing dx and cancelling Sort limits if applicable Simplify Integrate Substitute back out if necessary
What is the Reimann Sum
Adding together rectangles to estimate the area under a curve
More rectangles means more accurate the estimate
ALWAYS DO RECTANGLES UNDER CURVE, DO NOT CROSS LINE
What happens during the Riemann Sum as the number of rectangles increases
The value for area approaches a limit which is the actual value of the integral
How do you prove that increasing number of triangles is more accurate for Riemann Sum
Draw a curve in double positive axis (1st quadrant)
Draw one rectangle
Repeat with 2 and 3
Conclude with; more triangles means more of the area under the graph is filled by rectangles
Definition of a cartesian equation
Give a direct relationship between x and y eg y=(x+1)^2
Definition of a parametric equation
x and y are defined in terms of a third variable known as the parameter
eg x=t-1 and y=t^2
How do you go from a parametric equation to a cartesian equation without trig
Rearrange one to get t
Substitute t into the other equation
How do you go from a parametric equation to a cartesian equation with trig
Chose an appropriate identity
Rearrange the parametric equation
Substitute into identity
For a parametric equation x=p(t) and y=q(t) with cartesian equation y=f(x)
Domain of f(x)
Range of f(x)
Domain = Range of p(t) Range = Range of q(t)
Rule for differentiating parametric equations
(dy/dt)/(dx/dt) = (dy/dt) x (dt/dx) = dy/dx = y’/x’
Formula to integrate parametric equations
∫ y dx/dt dt
How do you integrate parametric equations
Swap limits from x to t
Get dx/dt
Plug into formula
Segment
Area created by joining two ends of an arc with a straight line
Where are small angle formulas for trig found
Page 6 formula booklet
Integrating trig
Swap for an identity so easier
Cant have extra x on bottom
Integrating trig with sin or cos squared
Use double angle
Integrate lnx
by parts
u=lnx
v=x
xlnx-x+c
Continous function
No vertical asymptote
Drawn without taking a pen off the page
When will Newton raphson fail
Stationary point chosen to start with
Derivative will be 0
Can’t divide by zero since its invalid
Tangent to the x axis means it won’t intersect the x axis
Start point close to stationary point
Gradient of tangent is so small it intercepts the x axis a long way from x0
