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)\,}
is a reflection of
y
=
f
(
x
)
{\displaystyle y=f\left(x\right)\,}
through the x axis.
y
=
f
(
−
x
)
{\displaystyle y=f\left(-x\right)\,}
is a reflection of
y
=
f
(
x
)
{\displaystyle y=f\left(x\right)\,}
through the y axis.
y
=
|
f
(
x
)
|
{\displaystyle y={\begin{vmatrix}f\left(x\right)\end{vmatrix}}}
is a reflection of
y
=
f
(
x
)
{\displaystyle y=f\left(x\right)\,}
when y < 0, through the x-axis.
y
=
f
(
|
x
|
)
{\displaystyle y=f\left({\begin{vmatrix}x\end{vmatrix}}\right)}
is a reflection of
y
=
f
(
x
)
{\displaystyle y=f\left(x\right)\,}
when x < 0, through the y-axis.
y
=
f
−
1
(
x
)
{\displaystyle y=f^{-1}\left(x\right)\,}
is a reflection of
y
=
f
(
x
)
{\displaystyle y=f\left(x\right)\,}
through the line y = x. Note:
f
−
1
(
x
)
{\displaystyle f^{-1}\left(x\right)}
exists only if
f
(
x
)
{\displaystyle f\left(x\right)}
is bijective , that is, one-to-one and onto .
y
=
a
f
(
x
)
{\displaystyle y=af\left(x\right)\,}
is stretched toward the x-axis if
0
<
a
<
1
{\displaystyle 0<a<1\,}
and stretched away from the x-axis if
a
>
1
{\displaystyle a>1\,}
. In both cases the change is by a units.
y
=
f
(
b
x
)
{\displaystyle y=f\left(bx\right)\,}
is stretched away from the y-axis if
0
<
b
<
1
{\displaystyle 0<b<1\,}
and stretched toward the y-axis if
b
>
1
{\displaystyle b>1\,}
. In both cases the change is by b units.
y
=
f
(
x
−
h
)
{\displaystyle y=f\left(x-h\right)\,}
is a translation of f(x) by h units to the right.
y
=
f
(
x
+
h
)
{\displaystyle y=f\left(x+h\right)\,}
is a translation of f(x) by h units to the left.
y
=
f
(
x
)
+
k
{\displaystyle y=f\left(x\right)+k\,}
is a translation of f(x) by k units upwards.
y
=
f
(
x
)
−
k
{\displaystyle y=f\left(x\right)-k\,}
is a translation of f(x) by k units downwards.
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}\,}
, where y(t) is the final value,
y
0
{\displaystyle y_{0}}
is the initial value, k is the growth constant, t is the elapsed time.
k
=
−
ln
2
h
a
l
f
−
l
i
f
e
{\displaystyle k=-{\frac {\ln 2}{half-life}}}
, k for calculations involving half-lives.
If
y
=
e
k
x
{\displaystyle y=\operatorname {e} ^{kx}\,}
, then
d
y
d
x
=
k
e
k
x
{\displaystyle {\frac {dy}{dx}}=k\operatorname {e} ^{kx}}
If
y
=
ln
x
{\displaystyle y=\ln x\,}
, then
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)\,}
, then
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)}}}
, then
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)]\,}
, then
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}}}
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}}}
and n is even