Calculus Flashcards

Question 1

Integral of $\csc(x)\cot(x)$

Accepted Answer

$-\csc(x) + C$

Question 2

Integral of $\cot(x)$

Accepted Answer

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

Question 3

Integral of $	an(x)$

Accepted Answer

$-\ln|\cos(x)| + C$

Question 4

Divergence Test

Accepted Answer

If $\lim_{n 	o \infty} a_n 
eq 0$, then $\sum a_n$ diverges.

Question 5

P-Series Test

Accepted Answer

$\sum \frac{1}{n^p}$ converges if $p > 1$, diverges if $p \leq 1$.

Question 6

Ratio Test

Accepted Answer

If $\lim_{n 	o \infty} |\frac{a_{n+1}}{a_n}| < 1$, series converges absolutely.

Question 7

Arc Length Formula

Accepted Answer

$L = \int_a^b \sqrt{1 + (f'(x))^2} dx$

Question 8

Surface Area of Revolution (x-axis)

Accepted Answer

$S = \int_a^b 2\pi f(x) \sqrt{1 + (f'(x))^2} dx$

Question 9

Average Value of a Function

Accepted Answer

$f_{ave} = \frac{1}{b-a} \int_a^b f(x) dx$

Question 10

Simpson's Rule Approximation

Accepted Answer

$\frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + \dots + f(x_n)]$

Question 11

L'Hopital's Rule

Accepted Answer

If $\lim \frac{f(x)}{g(x)}$ is $\frac{0}{0}$ or $\frac{\infty}{\infty}$, it equals $\lim \frac{f'(x)}{g'(x)}$

Question 12

Euler's Formula

Accepted Answer

$e^{ix} = \cos(x) + i\sin(x)$

Question 13

Integral of $\sec^2(x)$

Accepted Answer

$	an(x) + C$

Question 14

Integral of $\csc^2(x)$

Accepted Answer

$-\cot(x) + C$

Question 15

Integral of $\sec(x)	an(x)$

Accepted Answer

$\sec(x) + C$

Question 16

Limit definition of derivative $f'(x)$

Accepted Answer

$\lim_{h 	o 0} \frac{f(x+h) - f(x)}{h}$

Question 17

Integral of $\sec(x)$

Accepted Answer

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

Question 18

Integral of $\csc(x)$

Accepted Answer

$-\ln|\csc(x) + \cot(x)| + C$

Question 19

Integration by Parts Formula

Accepted Answer

$\int u \, dv = uv - \int v \, du$

Question 20

Maclaurin Series for $e^x$

Accepted Answer

$\sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \dots$

Question 21

Maclaurin Series for $\sin(x)$

Accepted Answer

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

Question 22

Maclaurin Series for $\cos(x)$

Accepted Answer

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

Question 23

Maclaurin Series for $\frac{1}{1-x}$

Accepted Answer

$\sum_{n=0}^{\infty} x^n = 1 + x + x^2 + \dots$

Question 24

Fundamental Theorem of Calculus (Part 1)

Accepted Answer

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

Question 25

Fundamental Theorem of Calculus (Part 2)

Accepted Answer

$\int_a^b f(x) dx = F(b) - F(a)$

Calculus

Cards in this deck

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