(Edexcel, stuff not given in exam & unashamedly C3-orientated)
( b^2 - 4*a*c ) bit of your quadratic formula :) If the answer is:
- greater than zero ( >0 ), there are two real roots!
- equal to zero ( =0 ), there is one repeated root!
- less than zero ( <0 ), there are no real roots.
Logs??
loga(xy) = loga(x) + loga(y)
loga(x^n) = n*loga(x)
loga(x/y) = loga(x) - loga(y)
2 π radians = 360° (using r = radius and θ [theta] = angle)
r * θ = arc length
0.5 * r^2 * θ = area of sector
sin(- θ) = -sin(θ) = sin(θ+180) = cos(θ+90)
sin(θ+90) = cos(θ) = cos(-θ)
cos(θ+180) = -cos(θ)
tan(-θ) = -tan(θ)
tan(θ+180) = tan(θ)
1 / cos(θ) = sec(θ)
1 / sin(θ) = cosec(θ)
1 / tan(θ) = cot(θ)
sec^2(θ) = 1 + tan^2(θ)
cosec^2(θ) = 1 + cot^2(θ)
sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos^2(θ) - sin^2(θ) = 2cos^2(θ) - 1 = 1 - 2sin^2(θ)
tan(2θ) = [2tan(θ)] / [1-tan^2(θ)]
e is a long number that is a special snowflake because it differentiates to itself.
- loge() even has a special name ln(). Natural log or “lun”, if you’re cool.
- when y = e^x, x = loge(x) = ln(x)
- when e^ln(x) = x, ln(e^x) = x
But it gets even better! Let’s see that differentiation!!
FUNCTION —————————– DERIVATIVE
f(x) —————————– f’(x)
e^(k*x) —————————– k * e^k*x
ln(x) —————————– 1/x
sin(k*x) —————————– k * cos(k*x)
cos(k*x) —————————– -k * sin (k*x)
Product rule??
when you’ve got to differentiate a function multiplied by another function
First, split up those functions into u & v
- so y = u * v
- find dy/dx, ensure that you have u & v clear then differentiate them separately to find du/dx and dv/dx
- dy/dx = u * [dv/dx] + v * [du/dx]
Chain rule??
(there’s a fancy way of doing it…but I don’t really use that)
SO you gotta differentiate something that’s already nicely put together with some weird exponent/index/power stuck on the end? no problem
dy/dx = n * f(x)^[n-1] * f’(x)
Integration??
AND DON’T FORGET YOUR CONSTANT + C
f(x) —————————– ∫ f(x) dx
e^(k*x) —————————– [1/k] * e^(k*x)
1/x —————————– ln|x|
sin(k*x) —————————– [-1/k] * cos(k*x)
cos(k*x) —————————– [1/k] * sin(k*x)
f(x+a) ~ slide horizontally to the left by a
f(x) + a ~ slide up by a
f(ax) ~ stretch horizontalyl by 1/a
a * f(x) ~ stretch vertically by a
y = |f(x)| ~ ensure nothing goes beneath the x axis. The part of the curve below the x axis should be reflected in the x - axis.
y = f(|x|) ~ reflect in y - axis. Sketch half of the normal y = f(x) graph [for x>0] and then reflect it in the y - axis.
Hope this is helpful and good luck with your exams!!
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