Calculus

This is by no means complete

1Odd and Even Functions¶

1.1Even Functions¶

• An even function is such that $f(-x) = f(x)$.
• An example is $\cos{x}$.
• Even functions are symmetric over the y-axis.
• For an even function, $\inf_{-a}^{a} f(x) dx = 2\int_{0}^{a} f(x) dx$

1.2Odd Functions¶

• An odd function is such that $f(-x) = -f(x)$.
• An example is $\sin{x}$.
• For an odd function, $\int_{-a}^{a} f(x) dx = 0$

2Surface Area:¶

$A=∫2πx~ds$
(rotated about y-axis)
$A=∫2πy~ds$
(rotated about x-axis)

3Arc Length:¶

$ds=\sqrt{1+\frac{dx}{dy}^2}~dy$
$ds=\sqrt{1+\frac{dy}{dx}^2}~dx$

3.1For parametric:¶

(Use the same initial equations)
$ds=\sqrt{\frac{dy}{dt}^2+\frac{dx}{dt}^2}~dt$

3.1.1Horizontal and vertical tangents:¶

if $\frac{dx}{dt}=0$ and $\frac{dy}{dt}\neq0$ it is a vertical tangent
if $\frac{dy}{dt}=0$ and $\frac{dx}{dt}\neq0$ it is a horizontal tangent
the $t$ value you get from the equation must be substituted in the respective parametric equation to get the tangent at that point.

4Polar Curves:¶

$r=f(θ)$
$x=f(θ) cos⁡(θ)$
$y=f(θ) sin⁡(θ)$
$0≤θ≤2π$

4.1for area:¶

$A=\frac{1}{2}\int{(f(θ))^2 dθ}_{\theta_1}^{\theta_2}$
$(A=1/2 ∫▒〖r^2 dθ)$