MAT 1322 - CALC 2 Final

What is the Maclaurin Series

The Maclaurin Series is a special case of the Taylor Series, which represents a function as an infinite sum of terms calculated from the function's derivatives at a single point, specifically at zero. It is used to approximate functions that are infinitely differentiable at that point.

What is the taylor sereis

A Taylor series is an infinite sum of terms calculated from the values of a function's derivatives at a single point. It represents a function as a power series expansion around that point.

Theorem of Changing Power Sereis

Suppose f(x) = sum inf to a=0 a_n(x-c)^n

1) f’(x) = sum n=1 to inf na_n(x-c)^n-1

2) integral of f(x) = sim n=0 to inf a_n/n+1 (x-c)^n+1 + D

1/1-x as a power series

sum from n=0 to inf x^n, for |x| < 1.

e^x as a power series

sum from n=0 to inf \frac{x^n}{n!}, for all x.

sin(x) as a power series

sum from n=0 to inf
^n rac{(-1)^n x^{2n+1}}{(2n+1)!}, for all x.

General f(x) as a power series

f^n(c )/n! (x-c)^n for all x in the radius of convergence.

How to figure out a power series we don’t know

Replace x in a known series

What is the radius convergence of f’(x)

is the same as the radius of convergence of f(x).

What is the radius converegnce of intergal f(x)

is the same as the radius of convergence of f(x).

What happens to interval convergcne with f’(x)

May lose convergence at an endpoint

What happens to interval convergence with intergal of f(x)

May gain converegence

What is the binomial series

f(x) = (1+x)^k = sum of inf to n=0 (k chose n) x^n

How do you combine two know power series and find a taylor series

You can combine two known power series by adding or multiplying them term by term, ensuring to adjust the indices as necessary. The resulting series can be expressed as a single Taylor series centered at the same point.

What is a domain

All (x,y) ER² s.t f(x,y) is delievered as a real number

What is the range

All output values obtained by members of the domain

What is the formula for Ellipse

x²/a² + y²/b² = 1

What is the formula for a Hyperbolla

x²/a² - y²/b² = 1

What is the general formula for planes

f(x,y) = ax + by + c

What the Trace method for YZ

is a way to analyze the intersection of a plane with the YZ-plane by setting x to a constant value.

What is trace method for XZ

is a technique used to examine the intersection of a plane with the XZ-plane by setting y to a constant value.

What is the level curve or contour diagram

A level curve or contour diagram represents the set of points where a function of two variables takes on a constant value, helping visualize the function's behavior in a two-dimensional space.

What is the trace method for xy (level curve or contour diagram)

is a technique used to analyze the intersection of a surface with the XY-plane by fixing the value of the function.

Level curves are closer to each other

ROC increases faster

Level curves are further from eachother

ROC decreases slower.

Partial Derivatives

Freeze or fix one or the varibles (x or y), treat x or y as a fixed value

When is y a fixed value

It is a fixed value when calculating a partial derivative with respect to x, treating y as constant during the differentiation process.

WHne is x a fixed value

It is a fixed value when calculating a partial derivative with respect to y, treating x as constant during the differentiation process.

What are second partial derivatives

Second partial derivatives are derivatives of partial derivatives, calculated with respect to the same variable or different variables. They provide information about the curvature of a multivariable function.

Clairaut’s Theorem

states that the order of differentiation does not matter for continuous functions, meaning that mixed partial derivatives are equal if the mixed derivatives are continuous.

Length of a vector u(a,b) or v(a,b,c)

The length of a vector, denoted as ||u|| for u(a,b) or ||v|| for v(a,b,c), is calculated using the Pythagorean theorem, where ||u|| = √(a² + b²) for 2D vectors and ||v|| = √(a² + b² + c²) for 3D vectors. This represents the magnitude of the vector in Euclidean space.

Dot Product v*u

||u||*||v||*cos(fetha)

Cross Product

The cross product of two vectors, denoted as v × u, results in a vector that is perpendicular to both original vectors and has a magnitude equal to the area of the parallelogram formed by them. It is defined as |v||u|sin(theta), where theta is the angle between the vectors.

Equation of a line

n*(x-x0, y-y0)

Equation of a plane

ax+by+cz+d=0

Equation of the Tangent Plane

z = f(x0,y0) + fx(x-x0) + fy(y-y0)

Linear approximation of f(x,y,z) near (a,b,c)

P(x,y,z) = f(a,b,c) + fx(a,b)(x-a) + fy(a,b)(y-b) + fz(a,b)(z-c)

What is the directional derivative

of a function in the direction of a vector, measuring the rate of change at a point in that direction.

What is the gradient vector

Duf(x,y) = ▼f * uA vector that points in the direction of the greatest rate of increase of a function. It consists of the partial derivatives with respect to each variable.

Max value of Duf(x,y)

sqrt (fx²+fy²) = ||▼f||

▼f = fx,fy is orthogonal where

on the level curves

When ROC is increasing

level curves are closer

When ROC is zero

u is tangent to the level curve and there is no increase or decrease in the function's value.

When ROC is max

u is in the direction of ▼f henceDuf(x,y) = ||▼f||

What is a sequence

A sequence is an ordered list of numbers or terms that follow a specific pattern or rule, typically defined by a function or formula.

A sequence is a special type of ________.

Function

Notation of a sequence

a_1, a_2, a_3 ….. a_n

a_n = f(n), presented as a formula if possible

is the nth term of a sequence defined by the function f, where n is the position in the sequence.

Notation of sequence using curly brackets

{f(x)} or {f(x)} from n=1 to inf

When is a sequence considered convergent

if the limit approaches a unique number

When is a sequence considered divergent

The limit doesn’t exist

What are the two types of sequence divergence

1) limit approaches inf 2) limit approaches two different values at one a_n

lim(n → inf) (a_n+b_n)

lim (n→ inf) a_n + lim (n→ inf) b_n

lim(n → inf) (a_n-b_n)

lim (n→ inf) a_n - lim (n→ inf) b_n

lim(n → inf) Ca_n

C (lim (n→ inf) a_n)

lim(n → inf) (a_n*b_n)

(lim (n→ inf) a_n )(lim (n→ inf) b_n)

lim(n → inf) (a_n/b_n)

(lim (n→ inf) a_n )/(lim (n→ inf) b_n)

lim(n→ inf) (a_n)^p

(lim (n→ inf) a_n )^p

What is the sequence; the squeeze theorem

The squeeze theorem, states that if you have three sequences, a_n, b_n, and c_n, and if b_n is squeezed between a_n and c_n for all n (i.e., a_n ≤ b_n ≤ c_n) and both a_n and c_n converge to the same limit L as n approaches infinity, then the limit of b_n also converges to L.

Continuous Function applied to sequences

If

1) lim(n→ inf) a_n = L (converegent)

2) f is continuous at L

then

lim(n→ inf) f(a_n) = f(lim(n→ inf) a_n) = f(L)

What is a series

A series is the sum of the terms of a sequence. It can be finite or infinite, and its convergence depends on the behavior of the sequence's terms as they approach infinity.

What is an infinite Series

An infinite series is the sum of the terms of an infinite sequence, where the limit of the partial sums is evaluated to determine convergence or divergence.

What are the two ways to express a infinite series

1) Can be expressed as a nice formula

2)Can not be expressed as a nice formula

lim (k→ inf) ak

inf

lim (k→ inf) ak = L

inf

lim (k → inf) ak = 0

unknown

Theorem for infinte series

if lim(k→inf) ak dne 0 → sum (inf)(k=1) ak is diveregnt

Telescopic Series

A series where successive terms cancel out, simplifying the sum significantly. It often results in a finite limit of ½

Geometric Series Theorem

A theorem that states the sum of a geometric series can be calculated using the formula S = a / (1 - r), where a is the first term and r is the common ratio, provided that |r| < 1.

What is the Integral Test

A method used to determine the convergence or divergence of an infinite series by comparing it to an improper integral. If the integral converges, so does the series, and vice versa.

What is an automatic diverence using limits

if the limit doesnt approach 0

Mandatory requirements for integral test

Positive, Decreasing and Continuous

What is the Comparison test

A method for determining the convergence or divergence of an infinite series by comparing it to a second series that is known to converge or diverge. If the second series converges and the first is less than it, the first converges as well.

Rules for Comparison test

Positive Test

What is the limit test

A method used to determine the convergence or divergence of an infinite series by evaluating the limit of its terms as they approach infinity. If the limit is zero, the test is inconclusive; if the limit is non-zero or diverges, the series diverges.

Limit test rules

Positive terms

Rules on using the integral test

sum inf to k=1 1/k^p : converges if p > 1, diverges if p ≤ 1.

Remainder Estimate Error Bound

We known that for sum inf to =1 f(n), if

1. f(x) is positive

2. f(x) is continus

   1. f(x) is decreasing

      integral from 1 to inf f(x)dx <= sum inf to n=1 f(x) <= f(1)+ integral 1 to inf f(x)dx

General Rule (REEB)

integral from n+1 to inf f(x)dx <= sum inf to n+1 f(x) <= intergal n to inf f(x) dx

s = sum from k=1 to inf f(x) = Sn + sum k=n+1 to inf f(x) = f(1) + f(2) + f(3) … + f(n) + f(n+1) + f(n+2)

Alternating Series

A series sum an in which consecutive terms have opposite signs

When is something absolutely convergent

if sum n=1 to inf abs(an) is convergent, then sum n=1 to inf an is converegent

When is something conditionally convergent

if sum n=1 to inf abs(an) is divergent, but sum n=1 to inf an is converegent

Alternating series test

For sum n=1 to inf (-1)^(n-1) bn = b1-b2+b3-b4+…

bn >0 for all n>=1

if 1) lim n→ inf bn = 0

2) bn+1 < = bn for all n

then sum n+1 to inf (-1)^(n-1) bn is convergent

ASET

if bn>0 and s = sum n=1 to inf (-1)^(n-1)bn is the sum of an alternating series that satisfies

1) lim n → inf bn = 0

2) bn+1 <= bn

then abs(Rn) = abs(S-Sn) <= bn+1

where:

Sn = sum n=1 to inf (-1)^k-1bK → nth partial sum

Rn = sum k=n+1 to inf (-1)^k-1bk → remainder

using abs(S-Sn) <= bn+1 we can find an upper bound and a lower bound for s:

abs(S-Sn) <= bn+1

-bn+1 <= S - Sn <= bn+1

Sn - bn+1 <= S <= Sn + bn+1

Ratio Test

Suppose sum n=1 to inf is given:

Evaluate lim n→ inf abs (an+1/an) = L

1) if L<1, then sum n=1 to inf an is absoultely convergent and therefore convergent

2) if L>1, then sum n=1 to inf an is divergent

3) if L =1 inconclusive

Root Test

Suppose sum n=1 to inf is given:

Evaluate lim n→ inf sqrt[|an|]} = L

1. if L < 1, then sum n=1 to inf an is absolutely convergent

2. if L > 1, then sum n=1 to inf an is divergent

3. if L = 1, inconclusive.

How do you find a domain of a power series

The domain of a power series is found by determining the values of x for which the series converges. This often involves using the Ratio Test or Root Test to establish a radius of convergence.

What is the polynomial notation with infinitely many terms

A power series, typically expressed as ( \sum_{n=0}^{\infty} a_n (x - c)^n ), where ( a_n ) represents the coefficients and ( c ) is the center of the series.

Definition of a power series

A power series is an infinite series of the form ( \sum_{n=0}^{\infty} a_n (x - c)^n ), where ( a_n ) are coefficients and ( c ) is the center of the series.

Fundamental Theory of Calc

∫ a to b f(x) dx = f(b) - f(a)

Steps of improper integrals

1. Replace the problem number

2. Solve the indef intergal

3. Sub in values

4. Evaluate the limit as needed.

Formal rule: ∫ a to inf (1/x^p) dx

converges (1/p-1) if p > 1, diverges (inf) if p ≤ 1.

Formal Definition of an Improper Integral

An integral of the form ∫ a to inf f(x) dx where f(x) is unbounded or the interval is infinite.

Convergent

approaches a unique number when placed in a limit

Divergent

Limit doesn’t exist, therefore the integral does not converge to a finite value.

∫ -inf to inf f(x)dx

∫-inf to c f(x) dx + ∫c to inf f(x) dx where c is a chosen finite point.

Formal rule: ∫ 0 to 1 (1/x^p) dx

converges (1/1-p) for p < 1 and diverges (inf) for p ≥ 1.

Comparison theory

A method used to determine the convergence or divergence of an integral by comparing it to another integral with known behavior.

Comparison theory rules

1. If the higher is convergent, then the function is convergent

2. If the lower is divergent, then the function is divergent.

Order of subtraction when dealing top and down

top - down