Download Elementary Differential and Integral Calculus FORMULA SHEET

Elementary Differential and Integral Calculus
FORMULA SHEET
Exponents
xa · xb = xa+b ,
ax · bx = (ab)x ,
Logarithms
ln xy = ln x + ln y,
ax = ex ln a .
ln xa = a ln x,
(xa )b = xab ,
ln 1 = 0,
x0 = 1.
eln x = x,
ln ey = y,
Trigonometry
sin 0 = cos π2 = 0,
cos 0 = sin π2 = 1,
cos2 θ + sin2 θ = 1,
cos(−θ) = cos θ,
sin(−θ) = − sin θ,
cos(A + B) = cos A cos B − sin A sin B,
cos 2θ = cos2 θ − sin2 θ,
sin(A + B) = sin A cos B + cos A sin B,
sin 2θ = 2 sin θ cos θ,
sin θ
1
,
sec θ =
,
1 + tan2 θ = sec2 θ.
tan θ =
cos θ
cos θ
Inverse Functions
y = sin−1 x means x = sin y and − π2 6 y 6 π2 .
y = cos−1 x means x = cos y and 0 6 y 6 π.
y = tan−1 x means x = tan y and − π2 < y < π2 .
y = x1/n means x = yn .
y = ln x means x = ey .
Alternative Notation
arcsin x = sin−1 x, arccos x = cos−1 x, arctan x = tan−1 x, loge x = ln x.
Note: sin−1 x 6= (sin x)−1 , cos−1 x 6= (cos x)−1 , tan−1 x 6= (tan x)−1 .
However: sin2 x = (sin x)2 , cos2 x = (cos x)2 , tan2 x = (tan x)2 .
Lines
The line y = mx + c has slope m.
The line through (x1 , y1 ) with slope m has equation y − y1 = m(x − x1 ).
y − y1
y2 − y1
y2 − y1
and equation
=
.
The line through (x1 , y1 ) and (x2 , y2 ) has slope m =
x2 − x1
x − x1
x2 − x1
The line y = mx + c is perpendicular to the line y = m0 x + c0 if mm0 = −1.
Circles
√
The distance between (x1 , y1 ) and (x2 , y2 ) is (x1 − x2 )2 + (y1 − y2 )2 .
The circle with centre (a, b) and radius r is given by (x − a)2 + (y − b)2 = r2 .
Triangles
In a triangle ABC:
(Sine Rule)
b
c
a
=
=
;
sin A
sin B
sin C
(Cosine Rule) a2 = b2 + c2 − 2bc cos A.
4
Pascal’s Triangle
(x + y)2 = x2 + 2xy + y2 , (x + y)3 = x3 + 3x2 y + 3xy2 + y3 and so on.
The coefficients in (x + y)n form the nth row of Pascal’s triangle:
1
1 1
1
2
1 3
1
4
1
3
6
1
4
1
.............
and so on.
Quadratics
If ax + bx + c = 0, with a 6= 0, then x =
2
−b ±
√
b2 − 4ac
.
2a
Calculus
dy
du dv
dy
du
dv
If y = u + v then
=
+
.
If y = uv then
=
v+u .
dx{ dx dx } /
dx
dx
dx
du
u
dy
dv
If y = then
=
v−u
v2 .
v
dx
dx
dx
∫
∫
∫
∫
∫
dv
du
(u + v) dx = u dx + v dx.
u dx = uv −
v dx.
dx
dx
If y is a function of u where u is a function of x, then
∫
∫
dy
dy du
du
=
and
y
dx = y du.
dx
du dx
dx
Standard Derivatives and Integrals
∫
dy
xa+1
a−1
a
If y = x then
= a x ; and xa dx =
+ constant (a 6= −1).
dx
a+1
∫
dy
= cos x; and sin x dx = − cos x + constant.
If y = sin x then
dx
∫
dy
If y = cos x then
= − sin x; and cos x dx = sin x + constant.
dx
∫
dy
2
If y = tan x then
= sec x; and tan x dx = ln | sec x| + constant.
dx
∫
dy
x
x
If y = e then
= e ; and ex dx = ex + constant.
dx
∫
1
1
dy
= ; and
dx = ln |x| + constant.
If y = ln x then
dx
x
x
∫
dy
1
1
−1
If y = sin x then
; and √
dx = sin−1 x + constant.
=√
2
2
dx
1−x
1−x
dy
−1
−1
If y = cos x then
=√
.
dx
1 − x2
∫
dy
1
1
−1
If y = tan x then
=
; and
dx = tan−1 x + constant.
2
2
dx
1+x
1+x
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