Formulas Flashcards
Modulus turning point
- Separate ONLY MODULUS PART into >0 and <0
- Solve for x, the answer on the other side
Finding f^-1 of a function
- MUST BE 1 TO 1 FUNCTION
- Let y = f(x)
- Solve for x
- Re substitute x for y
f(-x)
Reflection in y axis, so all negative x values became positive
-f(x)
Reflection in the x axis, so all negative Y values become positive
Order of transformations
REFLECTIONS FIRST e.g 2f(-x+5), REFLECT X VALUES FIRST
STRETCHES NEXT e.g 2f, so xY by 2 next
THEN TRANSLATIONS LAST y+7 or x+5 last ok
Gf(x)
Means g of f(x) so substitute f(x) wherever there in an x in G
Sec x
1/cosx
Sec x asymptotes
Wherever cos x = 0, so +-90, 270 etc
Cosec x
1/ sin x
Cosec asymptotes
Wherever sin x is 0, so +- 180, 360 etc
Cot x
1/tan x
Cot asymptotes
Wherever tan x =0, so +-180, 360, lines go other way
Sec ^2 x = ?
1 + tan^2 x = sec ^2 x
Cosec^2 x = ?
1 + cot^2 x
Arc sin arc cos and arc tan are what?
Sin^-1, cos^-1 and tan^-1
Range and domain of arc sin
Range: -pi/2 to pi/2
Domain: -1 to 1
Range and domain of arc cos
Range: 0 to pi
Domain -1 to 1
Range and domain of arctan
Range: -pi/2 to pi/2
Domain: all real numbers
Identity with arc sin and arc cos (hint: sum of them)
Arc sin + Arc cos = pi/2
Sin (A+B)
Sin (A- B)
SinACosB + SinBCosA
SinACosB - SinBCosA
Cos (A+B)
Cos (A - B)
CosACosB - SinASinB
CosACosB + SinASinB
Tan (A+B)
Tan (A- B)
TanA + Tan B / 1- TanATanB
Tan - TanB / 1 + TanATanB
Sin2A
2sinACosA
Cos2A
Cos^2 A - Sin^2 A
1 - 2Sin^2 A
2Cos^2 A - 1
Tan 2A
2TanA / 1- Tan^2 A
aSinx + bCosx
aCos x + bSin x
RSin(x+a)
RCos(x+a)
Getting into form RSin/Cos (x+a)
- First expand Rsin/cos (x+a) with addition formula
- Compare the formula and cancel out sinx and cosx so left with Rsina and Rcosa = …
- Divide Rsina by Rcosa to get tanA and arctan to get A
- Do Rsina^2 and Rcosa^2 equals wtv answer squared
- Factorise R out and sin^2 + Cos^2 equals 1
- Solve for R
When f(x) is e^x find f’(x)
The same: e^x
If f(x) is e^kx find f’(x)
K * e^kx
For graphs y = ax^n and y= ab^x against log y, log x and x respectively
- Take logs of both sides
- Compare to Y=mx +c to figure which represents each letter
Interpret the meaning of the constant … in the model
Y= the constant when the power is either 0 or 1, most likely 0
This is when the thing starts, so is the INITIAL value
Differentiate sin x
Differentiate sin kx
Cos x
Kcos kx
Differentiate cos x
Differentiate cos kx
-sin x
-Ksin kx
DON’T FORGET THE MINUS !
Y = ln x, differentiate
1/x
For differentiating complex things like if contains e and sin/cos
Use product rule:
1. Let u = one part and v = the other
2. Differentiate u and v
3. U’V + V’U equals the expression differentiated
Differentiate the function of a function y = f(g(x))
= f’(g(x))g’(x)
If they give you x = ….. and ask to differentiate
- Do dx/dy (basically just differentiate the y instead of x)
- Do 1/ans as dy/dx is just that flipped
Differentiate ln x
1/x
Quotient rule
Y’ = vu’ - uv’ / v^2
Product rule
Y’ = uv’ + vu’
Chain rule
Y’ = dy/du x dy/dx
Differentiate sin^2 x
2sinxcosx
Use chain rule:
Let u = sin x
Let v = u^2
Differentiate each
= 2u x cosx (sin differentiated)
= 2sinxcosx
