Exam 2 Memorization

Integral of -\frac{1}{x\sqrt{x² - 1}}

arccsc x + C

Integral of ln(ax)

x ln(ax) - x

Double Angle Identity for \sin(2x)

2\sin(x)\cos(x)

Double Angle Identity for \cos(2x) (Three Answers)

1. \cos^2(x) - \sin^2(x)

2. 1 - 2\sin^2(x)

3. 2\cos^2(x) - 1

Double Angle Identity for \tan(2x)

\frac{2\tan(x)}{1 - \tan^2(x)}

Power Reduction Half Angle Identity for \sin²(x)

\frac{1}{2} - \frac{\cos(2x)}{2}

Power Reduction Half Angle Identity for \cos²(x)

\frac{1}{2} + \frac{\cos(2x)}{2}

Pythagorean Identity for sin and cos

\sin²x + \cos²x = 1

Pythagorean Identity for tan and sec

1 + \tan²x = \sec²x

Integral of tan(x)

\ln|\sec(x)| \quad \text{or} \quad -\ln|\cos(x)|

Integral of csc(x)

\ln|\csc(x) - \cot(x)| \quad \text{or} \quad -\ln|\csc(x) + \cot(x)|

Integral of sec(x)

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

Integral of cot(x)

\ln|\sin(x)|

Complete the Square Formula (ax²-bx + c given a, b find c)

\left( \frac{b}{2} \right)^2

d/dx ln(f(x))

f’(x) / f(x)

d/dx sin(x)

cos(x)

d/dx cos(x)

-sin(x)

d/dx tan(x)

sec²(x)

d/dx cot(x)

-csc²(x)

d/dx sec(x)

sec(x) tan(x)

d/dx csc(x)

-csc(x) cot(x)

d/dx arcsin(ax)

\frac{a}{\sqrt{1-(ax)²}}

d/dx arccos(ax)

-\frac{a}{\sqrt{1-(ax)²}}

d/dx arctan(ax)

\frac{a}{1+(ax)²}

\int \frac{dx}{x² + a²}

\frac{1}{a} \arctan(\frac{x}{a}) +C

d/dx arccot(ax)

-\frac{a}{1+(ax)²}

d/dx arcsec(ax)

\frac{a}{ax\sqrt{(ax)²-1}}

d/dx arccsc(ax)

-\frac{a}{ax\sqrt{(ax)²-1}}

Steps to solve \int \sin^m (x) \cdot \cos^n (x) dx

todo

Steps to solve \int \tan^m (x) \cdot \sec^n (x) dx (4 cases)

1. m and n are both even:

   1. take out one factor of \sec²(x) and turn it into 1 + \tan²(x)

   2. U-Sub for u=\tan(x), du=\sec²(x)dx

   3. Integrate in terms of u

2. m (tangent power) odd, n (secant power) even:

   1. Take out one \tan(x) from \tan^m (x)

   2. U-Sub for u=\tan(x), du=\sec²(x)dx

   3. Integrate in terms of u

3. n (secant power) odd, m (tan power) even:

   1. Take out one \sec(x) from \sec^n (x)

   2. U-Sub for u=\sec(x), du=\sec(x)\tan(x)dx

   3. Integrate in terms of u

4. Both m and n are odd:

   1. U-Sub for u=\tan(x)

   2. Integrate in terms of u

Partial Fraction Decomposition (PDF) Conditions

1. Rational Function - Must have polynomial on numerator and denominator

2. Proper Fraction - Degree of numerator MUST be less than degree of denominator (or else long divide)

3. Factorable Denominator - Denominator must factor into linear factors with degree of 1

PFD: (x-4)

Linear: \frac{A}{x-4}

PDF: (x+4)²

Repeated Linear: \frac{A}{x+4} + \frac{B}{(x+4)²}

PFD: (x²+4)

Irreducible Quadratic: \frac{Ax + B}{(x²+4)}

PFD: (x²+4)²

Repeating Irreducible Quadratic: \frac{Ax+B}{(x²+4)} + \frac{Bx+C}{(x²+4)²}

find x \to \infty \text{ and } -\infty for \arctan(x) =

\frac{\pi}{2} and -\frac{\pi}{2}

find x \to \infty \text{ and } -\infty for \operatorname{arccot}(x) =

0 and \frac{\pi}{2}

find x \to \infty \text{ and } -\infty for \operatorname{arcsec}(x) =

\frac{\pi}{2} and \frac{\pi}{2}

find x \to \infty \text{ and } -\infty for \operatorname{arccsc}(x) =

0 and 0

Recursive Sequence

uses a previous term to define the new/next term (Fibonacci Sequence a_{n+2} = a_{n+1} + a_{n} )

Explicit Sequence

Each new term is given by substituting the value of n into the formula (Ex: a_{n} = \frac{3^{n}}{n!} )

The limit of the sequence (a_{n} \text{ as } n \rightarrow \infty ) determines if the sequence ______ or ______

converges or diverges

Theorems that might help solve the limit of a sequence are the _________ and/or ________

squeeze theorem, limit laws, L'Hopital's rule, and/or factoring out the dominant term

What is the Squeeze Theorem?

For a_n \geq b_n \geq c_n and \lim_{n \to \infty} a_n = \lim_{n \to \infty } c_n = L then \lim_{n \to \infty} b_n = L

\lim_{x \to 0} \frac{\sin{x}}{x} =

1

\lim_{x \to \infty} \frac{\sin{x}}{x} =

0

\lim_{x \to 0} x \sin(1/x) =

0

\lim_{x \to 0} \sin(f(x)) =

f(x)

\lim_{x \to 0} \frac{1 - \cos(x)}{x²} =

1/2

\lim_{x\to\infty} -e^{-x}

0

Sum/Difference/Product/Quotent Law for Limits: \lim_{n \to \infty} (a_n \times \div \pm b_n) =

\lim_{n \to \infty} a_n \times \div \pm \lim_{n \to \infty} b_n

Constants/Exponential/Root Laws for Limits

1. Constant * limit

2. (limit)^power

3. root(limit)

L'Hopital's Rule

• If the limit is indeterminant \frac{0}{0} \text{ or } \frac{\infty}{\infty}

• Then: \lim \frac{f(x)}{g(x)} = \lim \frac{f’(x)}{g’(x)} (repeat until converges)

Factoring Out the Dominant Term (for a_n = \frac{f(x)^n}{g(x)^d} , where n and d are the degrees of the Polynomial)

Divide f(x) and g(x) by x^d, where d is the highest power of x present in the denominator. Then simplify and solve the limit.

• Ex: a_n = \frac{x}{\sqrt{4x² + 2}} \rightarrow \frac{\frac{x}{\sqrt{x²}}}{\sqrt{\frac{4x²}{4x²} + \frac{2}{4x²}}} = \frac{1}{2}

Convergence Definition

If the limit exists and is a finite number, it converges

Divergence Definition

The limit does not exist (or is infinite), or oscillates then it diverges

Sequence Limit Definition of e

\lim_{n \to \infty} (1 + \frac{r}{n})^{n} = e^{r+n} (when r = 1 answer is e)

Geometric Sequences

Sequence where each term is = previous term * a common constant ratio of r a_{n} = a_{n-1} \cdot r

How can you find the ratio in a geometric sequence?

r = any term / previous term

Convergence vs Divergence in Geometric Sequences/Series:

1. |r| < 1

2. |r| > 1

3. r = 1

4. r = -1

1. converges to zero

2. diverges to infinity

3. converges to a_1

4. Diverges due to occilation

Factorial Sequences can often be solved using the Recursive Definition n! = ? and (n+3)! = ?

1. n! = n \cdot (n-1) \cdot (n-2) \cdot …

2. (n + 3)! = (n+3)(n+ 2)! …

Monotonicity of sequences

• tells us if a sequence is increasing, decreasing, or neither

• helps us predict if a sequence is approaching a limit in a predictable way

To check if a sequence is increasing, find the _________ between terms. If _______ the sequence is increasing.

find the difference between terms. If positive the sequence is increasing.

(Ex: Sequence a_n = \frac{n}{n+1} \to a_{n+1} - a_n = \frac{n+1}{n+2} - \frac{n}{n+1} = \frac{1}{(n+1)(n+2)} which is positive, so increasing)

Growth Rates of Sequences Theorem

1. Logarithms grow slower than Polynomials

   1. \lim_{n \to \infty} \frac{\ln(n)}{n^p} = 0 for any p>0

2. Polynomials grow slower than Exponentials

   1. \lim_{n \to \infty} \frac{n^p}{a^n} = 0 for p > 0 and a > 1

3. Exponentials grow slower than Factorials

   1. \lim_{n \to \infty} \frac{a^n}{n!} = 0 for a > 1

4. Factorials grow slower than n^n

   1. \lim_{n \to \infty} \frac{n!}{n^n} = 0

If a sequence has an upper bound (M) and/or lower bound (m) (m \leq a_n \leq M) for all n, it is _____

bounded

• Can be bounded above (never goes above M)

• Can be bounded below (never goes below m)

• Can be bounded (stays between m and M)

Bounded Monotone Sequence Theorem: If a sequence is bounded and monotonic (always increasing or always decreasing) then the sequence will ______

the sequence will converge

Factorial Sequences will usually ______ unless counterbalanced

diverge unless counterbalanced

• Ex: a_n = n! diverges

• Ex: a_n = \frac{n!}{n^n} converges to 0 since n^n grows faster than n!

• Ex: a_n = \frac{n!}{2^n} diverges since power-tower

For a series (\sum{a_n}) to converge, the sequence must _______ __ _

converge to 0 (\lim_{n \to \infty} a_n = 0)

Infinite Geometric Series sum can be found with this formula:

S_{\infty} = \frac{a_1}{1-r} for |r| < 1

Finite Geometric Serries sum can be found with this formula:

S_n = a_1 \cdot \frac{1-r^n}{1-r}

In Telescoping Series, recognize _______ in terms to solve

repetition since terms will cancel each other out (S_n = \sum_{k=1}^{n} (\frac{1}{k} - \frac{1}{k+2}) \to (1 - \frac{1}{3}) + (\frac{1}{2} - \frac {1}{4}) + (\frac{1}{3} - \frac{1}{5})… = 3/2)

• Often can be found using partial fraction decomposition

For \int_1^{\infty} \frac{1}{x^p} dx when is this p-series convergent / divergent?

Convergent for p > 1, divergent otherwise

For \int_0^1 \frac{1}{x^p} when is this p-series convergent / divergent?

Convergent for p < 1, divergent otherwise

\sum_{n=1}^{\infty} \frac{1}{n!} =

e = 2.7…