AP Calc BC Formulas

Sorry for not using "c" as the value for unknown x-values of f, knowt doesn't allow me to write f(c)

$\frac{d}{dx}$ $tan(u)$ =

$$sec^{2}(u)\frac{du}{dx}$$

$\frac{d}{dx}$ $sec(u)$

$$sec(u)tan(u)\frac{du}{dx}$$

$\frac{d}{dx}$ $csc(u)$

$$-csc(u)cot(u)\frac{du}{dx}$$

$\frac{d}{dx}$ $cot(u)$

$$-csc²(u)\frac{du}{dx}$$

$\int$ $tan(x)dx$

$$\ln|sec(x)| + C$$

$\int$ $sec(x)dx$

$$\ln|sec(x)+tan(x)| + C$$

$\int$ $\frac{1}{1+x²}dx$

$$arctan(x) + C$$

$\int$ $\frac{1}{\sqrt{1-x²}}dx$

$$arcsin(x) + C$$

$\int$ $\frac{-1}{\sqrt{1-x²}}dx$

$$arccos(x) + C$$

What are the three components of continuity?

1: $f(a)$ exists

2: $\lim_{x\to a} f(x)$ exists

3: $\lim_{x\to a} f(x)$ = $f(a)$

What is the mean value theorem?

If:

1: The function is continuous on [a,b]

2: The function is differentiable on (a,b)

3: There must be a value where f’(c ) = $\frac{f(b)-f(a)}{b-a}$

$\frac{d}{dx}$ $\log_a(u)$ = ?

$\frac{1}{x \ln(a)}$ * $\frac{du}{dx}$

$\frac{d}{dx}$ $a^u$ = ?

$a^u$ × $\ln(a)$ × $\frac{du}{dx}$

Limit definition of an integral?

$\lim{n\to\infty}$ $\sum_{k=1}^{n}$ f($x_k$)(∆$x_k$)

$\int$ $a^xdx$

$$\frac{a^x}{lna}+C$$

f’(a) exists and is undefined, what is x = a?

a critical value

f’(a) exists and equals 0, what is x = a?

a critical value

f’’(a) exists and is negative, what is f(x) at a?

concave down

f’’(a) exists and is positive, what is f(x) at a?

concave up

What are the conditions that MVT, IVT, and EVT require?

ALL theorems require the function f(x) to be continuous within a closed interval

What is the first fundamental theorem of calculus?

$\int^b_a$ $f’(x) dx$= $f(b) - f(a)$

What is the second fundamental theorem of calculus?

$\frac{d}{dx}$ $\int^x_a$ $f(t)dt$ = f(x)

OR, for chain rule…

$\frac{d}{dx}$ $\int^{g(x)}_a$ $f(t)dt$ = $f(g(x))g’(x)$

What is the average value of f(x) on [a,b]?

$f_{avg}$ = $\frac{1}{b-a}$ $\int^b_a$ $f(x)dx$

How to find $v(t)$ using $s(t)$ ?

$$s’(t)$$

How to find $a(t)$ using $s(t)$ ?

$$s’’(t)$$

What is speed?

$||v(t)||$ or $\sqrt{(\frac{dx}{dt})²+(\frac{dy}{dt})²}$

What value must a function go to for L’Hopital’s Rule to apply?

$\frac{0}{0}$ or $\frac{∞}{∞}$

What does L’Hopital’s Rule do to functions?

makes $\frac{f(x)}{g(x)}$ turn into $\frac{f’(x)}{g’(x)}$

Can L’Hopital’s Rule be applied continuously until the limit reaches a real value?

yes

$\int$ $udv$ = ?

$uv$ - $\int$ $vdu$

What represents the area between two functions?

$A$ = $\int^b_a$ [upper - lower]$dx$

OR

$A$ = $\int^d_c$ [right - left]$dy$

How to find the volume of the horizontal axis of rotation using the disk method?

$V$ = π$\int^b_a$ $f\left(x)\right)^2dx$

How to find the volume of the vertical axis of rotation using the disk method?

$V$ = π$\int^d_c$ $\left(f\left(y)\right)^2\right)$ $dy$

How to find the volume of the horizontal axis of rotation using the washer method? (assume rotation around the x-axis)

$V$ = π$\int^b_a$ [(upper)²-(lower)²]$dx$

The “upper” portion represents the solid portion of the washer while the “lower” portion represents the missing volume due to the hole of the washer

How to find the volume of the vertical axis of rotation using the washer method? (Assume rotation around only the y-axis)

$V$ = π$\int^d_c$ [(right)²-(left)²]$dy$
Same applies to here with the horizontal axis of rotation but it’s just flipped around.

Arc length of a line formula

$L$ = $\int^b_a$ $\sqrt{1+(\frac{dy}{dx})²}$

OR
$L$ = $\int^d_c$ $\sqrt{1+(\frac{dx}{dy})²}$

The function $f(x)$ is derivable at $a$ , what does the $nth$ Taylor polynomial for $f$ at $a$ ?

$P_n(x)$ = $f(a)$ + $f’(a)(x-a)$ + $\frac{f’’(a)}{2!}(x-a)^{2}$ + $\frac{f³(a)}{3!}(x-a)³$ + …

What are the conditions for $\sum_{n=1}^{∞}$ $a_n$ converging/diverging?

Converging: no conditions

Diverging: $\lim_{x\to\infty}a_{n}$ =/= 0

What are the conditions for $\sum_{n=1}^{∞}$ $ar^n$ converging/diverging?

Converging: |r| < 1

Diverging |r| ≥ 1

IF it converges, converges to $\frac{a_1}{1-r}$

What are the conditions for $\sum_{n=1}^{∞}$ $\frac{1}{n^p}$ converging/diverging?

Converging: p > 1

Diverging: p ≤ 1

What are the conditions for $\sum_{n=1}^{∞}$ $(-1)^{n-1}a_n$ converging/diverging?

Converging:

1: $a_n$ > 0

2: terms are decreasing ($b_{n+1}$ < or equal to $b_n$)

3: the limit of $a_n$ going to ∞ is 0

Diverging: fails minimum of one term above

When does a summation using ratio test converge or diverge?

If L < 1, converge

If L > 1, diverge

If L = 1, inconclusive

Formula for cross-section

$V$ = $\int^b_a$ $Adx$
$A$, in this instance, means the area formula for any shape (square, triangle, semicircle, etc). So, the integral is “slicing” up the shapes to generate a 3D solid that represents that particular 2D shape area.

Maclaurin series for sin(x)

$\sum_{n=0}^{∞}$ $\frac{_{(-1)^{n}x^{2n+1}}}{(2n+1)!}$

Maclaurin series for cos(x)

$\sum_{n=0}^{∞}$ $\frac{(-1)^nx^{2n}}{(2n)!}$

Maclaurin series for $e^x$

$\sum_{n=0}^{∞}$ $\frac{x^n}{n!}$

Inhibited growth model function

$\frac{dy}{dt}$ = $k(M-y)$ where y < M

$k$ is a constant and > 0

M is a constant and > 0

M is the upper value

Inhibited decay model function

$\frac{dy}{dt}$ = $k(y-M)$ where y > M

$k$ is a constant and < 0

M is a constant and > 0

Logistic model for inhibited growth of a population (using carrying capacity)

$\frac{dP}{dt}$ = $kP(1-\frac{P}{M})$
OR

$\frac{dP}{dt}$ = $\frac{k}{M}P(M-P)$

What is the solution to a logistic differential equation? This is also called the logistic curve.

$P(t)$ = $\frac{M}{1+ae^{-kt}}$

What does $k$ represent in a logistic growth/curve equation?

maximum growth rate of a population

What does $M$ represent in a logistic growth/curve equation?

carrying capacity

What type of function is $\frac{dP}{dt}$ = $k$?

linear

What type of function is $\frac{dP}{dt}$ = $kt$?

Quadratic

What type of function is $\frac{dP}{dt}$ = $kP$?

Uninhibited exponential

What type of function is $\frac{dP}{dt}$ = $k(M-P)$?

inhibited exponential

What type of function is $\frac{dP}{dt}$ = $kt(A-t)$?

Cubic

What type of function is $\frac{dP}{dt}$ = $kP(M-P)$?

Logistic growth

When does $P(t)$ have an inflection point?

At $P = \frac{M}{2}$

$P(t)$ has two horizontal asymptotes, where are they?

$P$ = 0 and $M$

What does $a$ mean in the equation$P(t) = \frac{M}{1+ae^{-kt}}$

$\frac{M-P_0}{P_0}$ , represents the ratio of the initial population to the carrying capacity minus the initial population.

Rate of change $A$ with respect to $t$ is proportional to $A$ itself, how can this be modeled?

$\frac{dA}{dt}$ = $kA$

If $A_0$ is known, what can $kA$ be integrated to?

$A$ = $A_0e^{kt}$

Arc length of a polar function

$s$ = $\int^β_α$ $\sqrt{r²+(\frac{dr}{dθ})²}$

REMEMBER: α and β do NOT represent x-values. These are RADIAN values for $r(θ)$.

Area of a polar function

$A$ = $\frac{1}{2}$ $\int^β_α$ $r²dθ$

Area of a trapezoid

$A$ = ($\frac{1}{2}$)($h$)($b_1$+$b_2$)

What are the three requirements for the integral test?

$a_n$ must be able to be integrated

if $a_n$ = $f(n)$, then $f(n)$ must be positive for all $n$

$f(n)$ must be decreasing for all $n$

0 ≤ $a_n$ ≤ $b_n$, and $b_n$ converges, what can we determine for $a_n$

$a_n$ must also converge

0 ≤ $c_n$ ≤ $b_n$ and $c_n$ diverges, what can be determined about $b_n$?

$b_n$ must also diverge

$\lim_{n\to\infty} \frac{a_n}{b_n}$ is finite and $b_n$ converges, what can be determined for $a_n$?

$a_n$ converges

$\lim_{n\to\infty} \frac{a_n}{b_n}$ is finite and $b_n$ diverges, what can be determined for $a_n$?

$a_n$ must diverge as well.

The alternating series for $a_n$ converges but |$a_n$| diverges, what is this called?

conditional convergence

The alternating series for $a_n$ converges and |$a_n$| converges, what is this called?

absolute convergence

$\int$ $\frac{1}{ax+b}$ $dx$ = ? (Assume $a$ and $b$ are non-zero, real constants)

$\frac{1}{a}$ $\ln(ax+b)$ + C

Lagrange error bound formula

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

What does $f^{n+1}(u)$ mean in the Lagrange error bound?

the “worst case scenario” for the next order derivative to the Taylor Polynomial $T_n(x)$

What is the alternating series error?

|$f_n(x)$ - $T_n(x)$| ≤ |$a_{n+1}$|

What does |$a_{n+1}$| mean in the alternating series formula?

The error of the actual function minus the taylor polynomial is less than or equal to the next omitted term

Arc length of a parametric function

$s$ = $\int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} dt$

What is the EVT?

If:

1: $f(x)$ is continuous on [a,b]

2: $f(x)$ must have an absolute max/min

What is the IVT?

If:
1: $f(x)$ is continuous on [a,b]

2: $f(x)$ will take on every y-value between $f(a)$ and $f(b)$

What does the maclaurin series for sin(x) look like when expanded?

x - $\frac{x³}{3!}$ + $\frac{x^5}{5!}$ - $\frac{x^7}{7!}$ + $\frac{x^9}{9!}$ - …

What does the maclaurin series for cos(x) look like when expanded?

1 - $\frac{x²}{2!}$ + $\frac{x^4}{4!}$ - $\frac{x^6}{6!}$ + $\frac{x^8}{8!}$ - …

What does the maclaurin series for $e^x$ look like when expanded?

1 + $x$ + $\frac{x²}{2!}$ + $\frac{x³}{3!}$ + $\frac{x^4}{4!}$ + …

$\frac{d}{dx}$ $\sqrt{u}$ =

$$\frac{1}{2\sqrt{u}}\frac{du}{dx}$$

Assume $f(x)$ is above $g(x)$ for (a,b), how do you find the volume of the solid of the area between $f(x)$ and $g(x)$ if rotated around y = n?

$V$ = π$\int^b_a$ ([$f(x)$-n]²-[$g(x)$-n]²)$dx$

Assume $f(y)$ is to the right of $g(y)$ for (c,d), how do you find the volume of the solid of the area between $f(y)$ and $g(y)$ if rotated around x = n?

$V$ = π$\int^d_c$ ([$f(y)$-n]²-[$g(y)$-n]²)$dy$

Maclaurin series for $\ln(1+x)$

$\sum_{n=1}^{infty}$ $\frac{(-1)^{n+1}x^n}{n}$

A function is increasing for a riemann sum estimation from [a,b]. Will a left riemann sum be an over/underestimate?

underestimate

A function is increasing for a riemann sum estimation from [a,b]. Will the right riemann sum be an over/underestimate?

overestimate

A function is decreasing for a riemann sum estimation from [a,b]. Will a left riemann sum be an over/underestimate?

overestimate

A function is decreasing for a riemann sum estimation from [a,b]. Will a right riemann sum be an over/underestimate?

underestimate

What is the area of an equilateral triangle? (Know this for unit 8)

$\frac{\sqrt{3}}{4}k²$ where $k$ is the function at hand that you’re evaluating, which is also the length of one side of the equilateral triangle.

Integral to find the volume underneath $f(x)$ is the cross-sectional area is done in semicircles perpendicular to the x-axis.

$\int_{a}^{b}\frac{π}{2}\left(\frac{f(x)}{2}\right)^2$ $dx$ ($f(x)$ , in this instance, represents the diameter of the semicircles. The area function would therefore be: $A = \frac{π}{2}(\frac{k}{2})²$

What are the bounds to a convergent p-series summation?

$\frac{1}{1-p}$ < $\sum_{n=1}^{∞}$ $\frac{1}{n^p}$ < $1+\frac{1}{1-p}$

The improper integral $\int^∞_1$ $f(x)dx$ converges. Then, the sum of the series must be bounded by what?

$\int^∞_1f(x)dx$ < $\sum_{n=1}^{∞}a_n$ < $a_1+\int^∞_1f(x)dx$