Maths Equations Flashcards
how to prove that there are two distinct real roots
b^2−4ac > 0
How to prove that there are equal roots
b^2−4ac = 0
How to prove that there are no real roots
b^2−4ac < 0
Factor theorem
If f(a) = 0, then (x - a) is a factor of f(x)
If f(b/a) = 0 then (ax - b) is a factor of f(x)
Gradient between 2 points
(y2-y1)/(x2-x1)
Midpoint
((x1+x2/2) , (y1+y2/2))
distance between two points
square root of (x2 - x1)^2 + (y2 - y1)^2
equation of a circle
(x - a)^2 + (y - b)^2 = r^2
where (a,b) is the centre and r is the radius
how to prove a function is increasing
dy/dx > 0
how to prove a function is decreasing
dy/dx < 0
prove there is a stationary point / turning point / maximums / minimums
dy/dx = 0
prove a point is a maximum
d^2y/dx^2 < 0
prove a point is a minimum
d^2y/dx^2 > 0
prove a graph is concave
d^2y/dx^2 < 0
prove a graph is convex
d^2y/dx^2 > 0
prove there is a point of inflection
d^2y/dx^2 = 0
(d^2y/dx^2 = 0 doesn’t guaruntee a point of inflection)
chain rule
dy/dx = dy/du x du/dx
the product rule
dy/dx = f’(x)g(x) + f(x)g’(x)
cosine rule
a^2 = b^2 + c^2 - 2bccosA
cosA = (b^2 + c^2 - a^2) / (2bc)
sine rule
a/sinA = b/sinB = c/sinC
Area of a triangle (trig)
(1/2)absinC
Arc length
rθ
(if θ is in radians)
Sector area
(1/2)(r^2)θ
(if θ is in radians)
converting radians to degrees
π radians = 180 degrees
tanx =
sinx/cosx
secx =
1/cosx
cosecx =
1/sinx
cotx =
1/tanx
cosx/sinx
sin2x =
2sinxcosx
cos2x =
cos^2x - sin^2x
2cos^2x - 1
1- 2sin^2x
tan2x =
2tanx / (1- tan^2x)
sin^2x =
1 - cos^2x
cos^2x =
1 - sin^2x
sec^2x =
tan^2x + 1
tan^2x =
sec^2x - 1
cosec^2x =
1 + cot^2x
cot^2x =
cosec^2x - 1
magnitude of vector a (when a = xi + yj +zk)
square root of x^2 + y^2 + z^2
unit vector in direction of a (when a = xi + yj +zk)
(1 / (square root of x^2 + y^2 + z^2)) x (xi + yj +zk)
loga(b) = c
a^c = b
loga(xy)
= loga(x) + loga(y)
loga(x/y) =
loga(x) - loga(y)
loga(x)^y
yloga(x)
