Calculus BC Stuff

Maclaurin Series of e^x

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots

Maclaurin Series of \sin(x)

\sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots

Maclaurin Series of \cos(x)

\cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots

Maclaurin Series of \frac{1}{1+x}

\frac{1}{1+x} = \sum_{n=0}^{\infty} (-1)^n x^n = 1 - x + x^2 - x^3 + \cdots \quad \text{for } |x| < 1

Taylor Polynomial Approximation

P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(c)}{k!} (x - c)^k = f(c) + f'(c)(x - c) + \frac{f''(c)}{2!}(x - c)^2 + \cdots + \frac{f^{(n)}(c)}{n!}(x - c)^n

Lagrange Error Bound

|R_n(x)| \leq \left| \frac{\max_{x \leq z \leq c} \left| f^{(n+1)}(z) \right| (x - c)^{n+1}}{(n+1)!} \right|

Alt. Series Error Bound

\text{If } \sum_{n=1}^{\infty} (-1)^{n+1} a_n \text{ converges, then } |S - S_n| = |R_n| \leq a_{n+1}

Euler’s Method

y_{n+1} = y_n + y' \, \Delta x

Exponential Growth (Solution to \frac{dy}{dt} = ky)

y(t) = C e^{kt} , C = Initial Value

Derivative of Logistic Function

\frac{dy}{dt} = ky\left(1 - \frac{y}{L}\right)
Where L is the Limiting Value

Point of Inflection for Logistic Function

(t, \frac{L}{2})
Where L is the Limiting Value
Max Growth Rate occurs here too

Derivative of Parametric Equation
x(t), y(t)

\frac{dy}{dx} = \frac{y'}{x'} \,, x’≠0

Second Derivative of Parametric Equation
x(t), y(t)

\frac{d²y}{dx²} = \frac{(\frac{dy}{dx})’}{x’}
where \frac{dy}{dx} = \frac{y’}{x’}

Arc Length for f(x) from a to b

L = \int_a^b \sqrt{1 + f’(x)^2} \, dx

Arc Length in Parametric Form for x(t), y(t) from a to b

L = \int_a^b \sqrt{(x'(t))^2 + (y'(t))^2} \, dt

Convert rectangular x, y to polar form

x = rcos\theta
y = rsin\theta

Convert polar r, \theta to rectangular

tan\theta = \frac{x}{y}

r² = x² + y²

Area of Polar Region r(\theta) from a to b

A=\frac{1}{2}\int_{a}^{b}r²d\theta

Integration by Parts (\int{udv})

\int{udv} = uv - \int{vdu}

Area Under a Curve

A = \int_a^b{[upper - lower]}

A = \int_a^b{[right - left]}

Volume - Disk Method (No Hole)

V = \pi\int_a^b{(R)²dx}

where R is the distance to the axis of rotation

Washer Method (with hole)

V = \pi\int_a^b{(upper)² - (lower)²dx}

V = \pi\int_a^b{(right)² - (left)²dx}

Fundamental theorem of calculus, pt. 1

\int_a^x{f’(t)dt} = f(x)

Fundamental theorem of calculus, pt. 2

\int_a^b{f’(t)dt} = f(b) - f(a)

Derivative of polar equation

x = f(\theta)cos(\theta)

y = f(\theta)sin(\theta)

\frac{dy}{dx} = \frac{y’(\theta)}{x’(\theta)}

Ratio Test for Integral of Convergence (Power Series)

\sum_{n=1}^{\infty}a_n

Ratio Test:

lim < 1 → converges to an interval

= 0 → all values x

= infinity → only to center (x = c)

Average rate of change from a to b

\frac{f(b) - f(a)}{b - a}

Average value of function from a to b

\frac{\int_a^b{f(x)dx}}{b-a}

Trapezoid Area (For Riemann Sums)

A = \frac{b1+b2}{2} * h