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By the end of this module you will be expected to have learnt the following formulae:
y
=
−
f
(
x
)
{\displaystyle y=-f\left(x\right)\,}
y
=
f
(
x
)
{\displaystyle y=f\left(x\right)\,}
y
=
f
(
−
x
)
{\displaystyle y=f\left(-x\right)\,}
y
=
f
(
x
)
{\displaystyle y=f\left(x\right)\,}
y
=
|
f
(
x
)
|
{\displaystyle y={\begin{vmatrix}f\left(x\right)\end{vmatrix}}}
y
=
f
(
x
)
{\displaystyle y=f\left(x\right)\,}
y
=
f
(
|
x
|
)
{\displaystyle y=f\left({\begin{vmatrix}x\end{vmatrix}}\right)}
y
=
f
(
x
)
{\displaystyle y=f\left(x\right)\,}
y
=
f
−
1
(
x
)
{\displaystyle y=f^{-1}\left(x\right)\,}
y
=
f
(
x
)
{\displaystyle y=f\left(x\right)\,}
f
−
1
(
x
)
{\displaystyle f^{-1}\left(x\right)}
f
(
x
)
{\displaystyle f\left(x\right)}
bijective , that is, one-to-one and onto .
y
=
a
f
(
x
)
{\displaystyle y=af\left(x\right)\,}
0
<
a
<
1
{\displaystyle 0<a<1\,}
a
>
1
{\displaystyle a>1\,}
y
=
f
(
b
x
)
{\displaystyle y=f\left(bx\right)\,}
0
<
b
<
1
{\displaystyle 0<b<1\,}
b
>
1
{\displaystyle b>1\,}
y
=
f
(
x
−
h
)
{\displaystyle y=f\left(x-h\right)\,}
y
=
f
(
x
+
h
)
{\displaystyle y=f\left(x+h\right)\,}
y
=
f
(
x
)
+
k
{\displaystyle y=f\left(x\right)+k\,}
y
=
f
(
x
)
−
k
{\displaystyle y=f\left(x\right)-k\,}
e
ln
x
=
ln
e
x
=
x
{\displaystyle e^{\ln x}=\ln e^{x}=x\,}
y
(
t
)
=
y
0
e
k
t
{\displaystyle y\left(t\right)=y_{0}e^{kt}\,}
y
0
{\displaystyle y_{0}}
k
=
−
ln
2
h
a
l
f
−
l
i
f
e
{\displaystyle k=-{\frac {\ln 2}{half-life}}}
sec
θ
≡
1
cos
θ
{\displaystyle \sec \theta \equiv {\frac {1}{\cos \theta }}}
cosec
θ
≡
1
sin
θ
{\displaystyle \operatorname {cosec} \ \theta \equiv {\frac {1}{\sin \theta }}}
cot
θ
≡
1
tan
θ
≡
cos
θ
sin
θ
{\displaystyle \cot \theta \equiv {\frac {1}{\tan \theta }}\equiv {\frac {\cos \theta }{\sin \theta }}}
sec
2
θ
≡
1
+
tan
2
θ
{\displaystyle \sec ^{2}\theta \equiv 1+\tan ^{2}\theta }
cosec
2
θ
≡
1
+
cot
2
θ
{\displaystyle \operatorname {cosec} ^{2}\ \theta \equiv 1+\cot ^{2}\theta }
sin
(
A
±
B
)
=
sin
(
A
)
cos
(
B
)
±
cos
(
A
)
sin
(
B
)
{\displaystyle \sin(A\pm B)=\sin(A)\cos(B)\pm \cos(A)\sin(B)\,}
cos
(
A
±
B
)
=
cos
(
A
)
cos
(
B
)
∓
sin
(
A
)
sin
(
B
)
{\displaystyle \cos(A\pm B)=\cos(A)\cos(B)\mp \sin(A)\sin(B)\,}
tan
(
A
±
B
)
=
tan
(
A
)
±
tan
(
B
)
1
∓
tan
(
A
)
tan
(
B
)
{\displaystyle \tan(A\pm B)={\frac {\tan(A)\pm \tan(B)}{1\mp \tan(A)\tan(B)}}}
Note: The sign
∓
{\displaystyle \mp }
sin
2
A
≡
2
sin
A
cos
A
{\displaystyle \sin 2A\equiv 2\sin A\cos A}
cos
2
A
≡
cos
2
A
−
sin
2
A
≡
1
−
2
sin
2
A
≡
2
cos
2
A
−
1
{\displaystyle \cos 2A\equiv \cos ^{2}A-\sin ^{2}A\equiv 1-2\sin ^{2}A\equiv 2\cos ^{2}A-1}
tan
2
A
≡
2
tan
A
1
−
tan
2
A
{\displaystyle \tan 2A\equiv {\frac {2\tan A}{1-\tan ^{2}A}}}
Using radians r = amplitute α = phase.
r
=
a
2
+
b
2
{\displaystyle r={\sqrt {a^{2}+b^{2}}}}
a
sin
x
+
b
cos
x
=
r
⋅
sin
(
x
+
α
)
{\displaystyle a\sin x+b\cos x=r\cdot \sin(x+\alpha )\,}
where
α
=
arcsin
b
r
{\displaystyle \alpha =\arcsin {\frac {b}{r}}}
a
sin
x
+
b
cos
x
=
r
⋅
cos
(
x
−
α
)
{\displaystyle a\sin x+b\cos x=r\cdot \cos(x-\alpha )\,}
where
α
=
arccos
b
r
{\displaystyle \alpha =\arccos {\frac {b}{r}}}
If
y
=
e
k
x
{\displaystyle y=\operatorname {e} ^{kx}\,}
d
y
d
x
=
k
e
k
x
{\displaystyle {\frac {dy}{dx}}=k\operatorname {e} ^{kx}}
If
y
=
ln
x
{\displaystyle y=\ln x\,}
d
y
d
x
=
1
x
{\displaystyle {\frac {dy}{dx}}={\frac {1}{x}}}
If
y
=
f
(
x
)
.
g
(
x
)
{\displaystyle y=f(x).g(x)\,}
d
y
d
x
=
f
′
(
x
)
g
(
x
)
+
g
′
(
x
)
f
(
x
)
{\displaystyle {\frac {dy}{dx}}=f^{'}(x)g(x)+g^{'}(x)f(x)}
If
y
=
f
(
x
)
g
(
x
)
{\displaystyle y={\frac {f(x)}{g(x)}}}
d
y
d
x
=
f
′
(
x
)
g
(
x
)
−
g
′
(
x
)
f
(
x
)
{
g
(
x
)
}
2
{\displaystyle {\frac {dy}{dx}}={\frac {f^{'}(x)g(x)-g^{'}(x)f(x)}{\left\{g(x)\right\}^{2}}}}
d
y
d
x
=
1
d
x
d
y
{\displaystyle {\frac {dy}{dx}}={\frac {1}{\frac {dx}{dy}}}}
If
y
=
f
[
g
(
x
)
]
{\displaystyle y=f[g(x)]\,}
d
y
d
x
=
f
′
[
g
(
x
)
]
.
g
′
(
x
)
{\displaystyle {\frac {dy}{dx}}=f^{'}[g(x)].g^{'}(x)}
d
y
d
t
=
d
y
d
x
.
d
x
d
t
{\displaystyle {\frac {dy}{dt}}={\frac {dy}{dx}}.{\frac {dx}{dt}}}
∫
e
k
x
d
x
=
1
k
e
k
x
+
c
{\displaystyle \int \operatorname {e} ^{kx}\,dx={\frac {1}{k}}\operatorname {e} ^{kx}+c}
∫
1
x
d
x
=
ln
|
x
|
+
c
{\displaystyle \int {\frac {1}{x}}\,dx=\ln \left|x\right|+c}
For volumes of revolution:
V
x
=
π
∫
a
b
y
2
d
x
{\displaystyle V_{x}=\pi \int _{a}^{b}y^{2}\,dx}
V
y
=
π
∫
c
d
x
2
d
y
{\displaystyle V_{y}=\pi \int _{c}^{d}x^{2}\,dy}
Simpson's Rule
∫
a
b
y
d
x
≈
1
3
h
{
(
y
0
+
y
n
)
+
4
(
y
1
+
y
3
+
…
+
y
n
−
1
)
+
2
(
y
2
+
y
4
+
…
+
y
n
−
2
)
}
{\displaystyle \int _{a}^{b}ydx\approx {\frac {1}{3}}h\left\{\left(y_{0}+y_{n}\right)+4\left(y_{1}+y_{3}+\ldots +y_{n-1}\right)+2\left(y_{2}+y_{4}+\ldots +y_{n-2}\right)\right\}}
where
h
=
b
−
a
n
{\displaystyle h={\frac {b-a}{n}}}
![{\displaystyle y=f[g(x)]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eeede1246f16c924747c917c4a1b25a35b7990ce)
![{\displaystyle {\frac {dy}{dx}}=f^{'}[g(x)].g^{'}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67809fd87e9d64bf8436a6119694a540b6d60d99)
