Core 3 Flashcards
(23 cards)
Conditions of a function
1 to 1
Or
Many to 1
1 y value for a x value
Odd function
F(x) = -f(-x)
Rotational 180 degrees about the origin
Even function
F(x) = f(-x)
Reflection in the y axis
Composite functions
Fh(x)
Do h first then f
Inverses/ f-1
Symmetrical about y=x
Write as y=
Rearrange to make x the subject
Swap x and y
Arcsine
Arccosine
Arctan
Look at them
Modulus
|f(x)| reflection in the x axis
F(|x|) reflection of -ve x values in y
Transformations
F(x+c) - move left or right, opposite way to expected
F(x) + c - move up and down
F(ac) - horizontal squash scale factor 1/a
Af(x) - vertical stretch
Exponential
E^lnx = x Ln(e^x) = x
Logs
Lna + lnb = lnab
Lna - lnb = ln(a/b)
Klna = lna^k
Proof
Direct proof - use known facts to build up an argument
Proof by exhaustion - break down into cases then cover all situations
Proof by contradiction - suppose it’s false and prove that’s this can’t be true
Disproof by counterexample - find an example that doesn’t fit
Chain rule
Y=(x+2)^2
Dy/dx = 2(x+2)
Product rule
Y=uv
Dy/dx = u dv/dx + v du/dx
Quotient rule
Y= u/v Dy/dx = [ v du/dx - u dv/dx] / v^2
Differentiating exponential and logs
Y = e^2x Dy/dx = 2e^2x
Y = ln x Dy/dx = 1/x Y= ln 2x Dy/dx = 2/x
Differentiating trig
Sin x to cos x
Cos x to - sinx
Tan x to sec^2 x
Gradients
Gradient of tangent at a point
Dy/dx
D2y/dx2, bigger than 0 is minimum and smaller than 0 is a maximum
Rates of change
Rate = ?/dt
Implicit differentiation
Do x terms
Do y terms with dy/dx
Do xy terms by product rule
Integrating exponential and logs
Intergral of e^2x = 1/2 e^2x
Integral of 1/x = ln |x|
Integrating trig
Sinx to - cos x
Cos x to sinx
Integrating by substitution
U= X = Du/dx = , rearrange New limits Rewrite integral Integrate
Integrating by parts
U is always ln x or x
= uv - the integral of v du/dx
