C4 Flashcards
sin2A = ?
2sinAcosA
cos2A = ?
cos²A - sin²A
2cos²A - 1
1 - 2sin²A
tan2A = ?
2tanA/(1 - tan²A)
sin4x = ?
2sin2xcos2x
d/dx sinax = ?
acosax
d/dx cosax = ?
-asinax
d/dx sinx = ?
cosx
d/dx cosx = ?
-sinx
d/dx sin(ax + b) = ?
acos(ax+b)
d/dx cos(ax + b) = ?
-asin(ax +b)
How would you differentiate sin²x?
Chain rule of (sinx)²
to obtain 2⋅(d/dx sinx)⋅(sinx)²⁻¹
= 2⋅cosx⋅sinx
= sin2x
How would you differentiate cos²x?
Chain rule of (cosx)² to obtain 2⋅(d/dx cosx)⋅(cosx)²⁻¹ = 2⋅-sinx⋅cosx = -2sinxcosx = - sin2x
How do you make parametric equations cartesian?
Eliminate the parameter
dy/dx of a parametric equation?
dy/dx = dy/dt ⋅ dt/dx = g’(t)/f’(t)
Implicit: d/dx y = ?
dy/dx
Implicit: d/dx y² = ?
2y⋅dy/dx
Implicit: d/dx xy = ?
u = x, v = y u' = 1, v' =dy/dx
∴ d/dx xy = vu’ + uv’
∴ d/dx xy = x⋅dy.dx + y
∫ sinax dx = ?
-1/a cosax + c
∫ cosax dx = ?
1/a sinax + c
∫ sec²ax dx = ?
1/a tanax + c
∫ kf’(x)/f(x) dx = ?
kln|f(x)| + c
Formula for Integration by parts?
∫ uv’ dx = uv - ∫ vu’ dx
where u is a value that will differentiate to a constant, or is lnx
Steps for Integration by substitution?
|> u = f(x)
|> Make obvious substitution (change f(x) for u)
|> Differentiate u = f(x) and rearrange for dx
|> Rearrange u = f(x) for x=
|> Integrate with respect to u
|> Change limits
|> Replace u with f(x)
a/b ⋅ c/d = ?
ac/bd
a/b ÷ c/d = ?
ad/bc
Binomial Expansion formula?
(1 + x)ⁿ = 1 + nx + n(n-1)/2! ⋅ x² + n(n-1)(n-2)/3! ⋅ x³
What form must binomial expansions be in?
MUST be in the form (1 + x)ⁿ
When do binomial expansions converge?
Converge as |x| < 1
How would you write (3x² - 1)/(x-1)(2x-1)² as partial fractions?
= A/x-1 + B/(2x-1) + C(2x-1)²
let x = 1 [cancels (x-1)], A = 2
let x = 1/2 [cancels (2x-1)], -1/2⋅C = -1/4, C = 1/2
let x = 0 [takes note of repeated factor], -1 = A + B - C. -1 = 2 + B - 1/2 B = -5/2
Magnitude of unit vectors?
1
Length of unit vectors?
1
magnitude/length of unit vectors?
1
Where do position vectors start from?
The origin
-AB-> = ?
-OB-> − -OA->
for x = ai + bj + ck, |x| = ?
√(a² + b² + c²)
magnitude/length
Features of a vector line?
have a position vector, a direction vector and a parameter
in the form: position vector + parameter ⋅ direction vector
When are vector lines parallel?
Direction vectors are multiples of one another
When are vector lines skew?
If you can prove they are not parallel, and do not intersect
where do the vector lines r = (ai + tpi) + (bj + tqj) + (ck + trk) & r = (di + smi) + (ej + snj) + (ek + sok) intersect?
a + tp = d + sm
b + tq = e + sn
c + tr = f + so
Equation for scalar product?
(-a->) ⋅ (-b->) = |a||b|cosθ
where θ is the angle between the vectors
If lines are perpendicular, (-a->) ⋅ (-b->) = ?
0
because cos90 = 0
How to calculate the shortest distance between a vector and a point?
(-a->) ⋅ (-b->) = 0
