Final Exam Flashcards
(88 cards)
1
Q
Pre-Calculus Formulas
1. Volume of Rectangular Prism
2. Volume of Sphere
3. Surface Area of a Sphere
4. Area of a Circle
5. Circumfrence of a Circle
6. Pythagorean Theorem
7. Quadratic Formula
8. Trig Definitions: tan, cot, sec, csc
9. Exact values for sin/cos(np/4, np/6)
10. Concavity, incr/decr behavior, rates of change, differentiation rules
11. Derivatives of Standard Functions
A
- Volume of Rectangular Prism= width x length x height
- Volume of Sphere=4/3pir^3
- Surface Area of a Sphere=4pir^2
- Area of a Circle= pir^2
- Circumfrence of a Circle: 2pir
- Pythagorean Theorem=a^2+b^2=c^2
- Quadratic Formula: -b+/-sqrt(b^2-4ac)/2a
- Trig Definitions: tan, cot, sec, csc=sin/cos, cos/sin, 1/cos, 1/sin
- Exact values for sin/cos(np/4, np/6): special triangles + ACTS
- Concavity, incr/decr behavior, rates of change, differentiation rules: first, second derivative, quotient, product, reciprocal
- Derivatives of Standard Functions: x^n, arctan, arcsin, arccos, cos, sin, tan, e^x, 2^x, logx, sqrt(x)
2
Q
Improper Integral + How to Solve
A
Integral having infinite limit of integration (horizontal asymptote) or an unbounded integrand (vertical integrand), can’t apply FTC to infinity, use limits that may equal a number (converge) or not (diverge)
- Sketch graph + Find VA & HA, edit integral accordingly
- FTC
- Evaluate Limit in terms of FTC (sketch graph)
- Check/Do It by splitting p-test or comparison test or limit comparison test
3
Q
What functions can the Fundamental Theorem of Calculus be used?
A
- FTC can be used to evaluate continuous functions & those with finite number of jump discontinuities by subdividing interval into continuous function.
- Analytical Function, not an empirical (data) or numerical function
- Antiderivatie can be found based on elementary functions
4
Q
Converge vs Diverge
A
Improper Integral Converges: limit exists as a number → integral is this number
- Function approaches bounds fast enough to enclose an area
Improper Integral Diverges: all other cases → integral diverges to +,-, oscillating, etc. ,
- Function approaches bounds too slow, area is never enclosed
5
Q
Define + Solve Infinite Domain
A
Possesing a never ending domain (horizontal asymptote)
ex. InfinityS0 e^(-x) dx
1. Draw + limit
2. FTC
3. Solve limit
6
Q
Define + Solve Unbounded Integrand (Singularity)
A
Integral that doesn’t posses an upper or lower bound (vertical asymptote)
infS-inf
ex. 4S0 1/x dx
1. Draw + Separate Integral if VA exists + limit
2. FTC
3. Solve limit
7
Q
Define + Solve 2 Infinities
A
Must be split at an intermediate point
Bounds are inf S -inf
1. Draw + limit
2. FTC
3. Solve limit
8
Q
Define + Solve Infinity at middle point
A
Must be split at that point
1. Draw + limit
2. FTC
3. Solve limit
9
Q
When does a split integral converge and diverge? To what value?
A
Integral converges to the sum of both if both integrals converge.
All other cases it diverges.
10
Q
P-Test
A
- Split integral at 1
- Evaluate
Converges:
- 1S0 1/x^p p<1
- 1Sinfinity 1/x^p p>1
Diverges:
- 1S0 1/x^p p>1
- 1Sinfinity 1/x^p p<1
Always Diverges: infinty S 0 1/x^p
11
Q
P-Test Proof
A
See doc
12
Q
Comparison Test
A
Evaluating difficult/impossible improper integrals if f and g are integrable
Smaller than a convergent (finite), must converge (finite):
g(x)>f(x), g(x) converges –> f(x) converges
Larger than a divergent (infinity), must divergent (infinity):
g(x)<f(x), g(x) diverges –> f(x) diverges
Drawing
13
Q
Limit Comparison Test + When to Use + Inconclusive
A
a S infinity
- g(x)=dominating/higher order of f(x)
e^x>x^n>logx
1/logx>1/x^n>1/e^x - lim x->infinity f(x)/g(x) exists/finite/similar in speed
- Evaluate g(x)
- g(x) c/v –> f(x) c/v
Useful: quotient of polynomyials, g(x)=highest terms or replace sin(x)>-1, cos(x)<1
Inconclusive: comparison doesn’t work or hard to cancel out terms
14
Q
Absolute, Relative, Max Error
A
A=approximation, Q=True Value=(Q-A)+A
Absolute Error: A-Q=Q-A
Relative Error/Percentage Change: (A-Q)/Q
Upper Bounds/Threshold/Maximum Absolute Error/Worst Case Scenario: if f a 4x differentiable function
- Trapezoid: M/12 (b-a)^3/n^2>abs. error (M=max value f’‘(x) when x[a,b])
- Simpson: L/180 (b-a)^5/n^4>abs. error (L-max value f’’’‘(x) when x[a,b]) –> must be even # over 4
15
Q
Why error Riemann>Trapezoid>Simpson
A
- S=n4=1/180>T=n2=1/12 error gets smaller faster as denominator is larger/tends to 0 faster, larger the denominator smaller the error
- However: T=(b-a)3>S=(b-a)5 error gets larger faster as numerator is larger+4th derivative can behave worse than 2nd derivative
- M & L measure how oscillating the function/how hard to approximate
- More accurate/precise approx. more data points required, two consecutive intervals, 3 points
- Higher the degree, the better the approximation
16
Q
When Trapezoidal>Simpson
A
- Less than 4x differentiable, f degree below 2
- n<4 or odd
17
Q
Compute Spreadsheet R, T, S
A
See doc
18
Q
Key Idea+Formulas of Numerical Integration
A
Replace rectangles by geometric objects that better approximate the function within each small interval(s)
The higher the degree of approximation, the more accurate/better it is, as it accounts for more info/points and precision
Change x = b-a/n
xi= a+1(change x)
19
Q
Riemann Right Endpoint +
Left Endpoint
A
Degree 0 (Constant), least accurate, least information. Rectangles under curve.
Right: Ixj=1nf(xj)=x(0f(x0)+f(x1)+f(x2)+…f(xn))
Coefficient Pattern: 0,1,1,1,1… 1
Left:
Ixj=1nf(xj)=x(f(x0)+f(x1)+f(x2)+…0f(xn))
Coefficient Pattern: 1,1,1,1… 0
20
Q
Trapezoidal Rule
A
Degree 1 (Linear), more accurate than Riemann, more info required
Ix(12f(x0)+f(x1)+…f(xn-1)+12f(xn))=x2(f(x0)+2f(x1)+…2f(xn-1)+f(xn))
Coefficient Pattern: ½,1,1,1… 1,½ = 1,2,2,2…2,1
21
Q
Simpson`s Rule
A
Degree 2 (Quadratic), most accurate, most information required
Splitting approximate into pairs of intervals using quadratic function
Ix13(f(x0)+4f(x1)+2f(x2)+4f(x3)+…+2f(xn-2)+4f(xn-1)+f(xn))
Coefficient Pattern: 1,4,2,4…4,2,4,1
N must be even and above 4
22
Q
How to find area bounded b/w curves?
A
- X/Y Starting Function: plot graph if possible and switch x or y in order to make function easily integratable
- Find Meeting Points+Bounds of Enclosed Area: f(x)=g(x), separate integrals at those points and f(x)=larger-smaller
- Evaluate: if undrawable, evaluate absolute value, area should always be positive
23
Q
Partial Fractions
A
Goal is to reverse back large quotient into many simple small quotient with constant numerator & linear denominator which are easily integratable.
p(x)q(x)=p(x)a(x)b(x)Linear=Aa(x)+BConstantsb(x)+… Important: 1xdx=lnx+C
Find A & B: p(x)=Ab(x)+Ba(x), sub in x that cancels out one variable and isolate for the other
24
Q
When to use each method of integration
A
- Algebra First to Simplify: ex+exdx=exeex
- SUB: function beside its derivative, complicated function inside a simple function
- IBP: 2 functions being multiplied, log, arctan
- Partial Fraction: quotient composed of factorable functions
- Trig Substitution: trig definitions (tan, cot, sec, csc), trig identities with trig powers (cos5(x)sin7(x)dx=cos5(x)(1-cos2(x))3sin(x)dx)
- Trig Integration: more complicated to less complicated (14+sqrt(x^2)dx)
25
Integration by Parts
u dv=uv-Sv du
Function u a simple derivative: xn,log(x), arctan(x)
Function dv a simple antiderivative: ex,sin(x), cos(x), 1
Not every function can use this method: xex2, can change nothing or complexify
26
Volume of a Cone
V=R^2piH/3
Proof: see doc
27
Volume Integral + What each part of integral means
V=HS0 A(h)dh
28
Volume of a Solid Revolution y=x^2, y=sqrt(x), within eachother
See doc
29
Need to know Trig Info + Identites
See doc
30
Substitution
G'(u)=g(u)g(u)du=G(u)+C
Indefinite: Factor, simplify, separate into simple parts, manipulate to reveal effective substitution, Declare u, Transform integral entirely in terms of u, Solve indefinite integral by obtaining anti derivative in terms of u
Sub x back into u so expression returns to original terms of x
Definite: compute antiderivative in terms of u & apply FTC2 using u & replaced limits of integration u(a) & u(b)
31
Distance --> Speed --> Acceleration through geometry + integration + derivation
See doc
32
6 Properties of Integrals equation + graph
1. Linearity: homogeneity, additivity
2. Zero-Interval
3. Interval Addition
4. Reverse Interval
5. Comparison
6. Even & Odd Function Integration
See doc
33
Function Table
See doc
34
Fundamental Theorem of Calculus 1
FTC1 connects integration and differentiation and definite integrals to indefinite integrals (antiderivatives), area function is an antiderivative of its integral.
A'(x)=f(x)
Proof: see doc
35
Fundamental Theorem of Calculus 2
FTC2 is a consequence of FTC1, reduces computing definite integrals to that of computing antiderivatives
- find the definite integral of any function (especially those that are geometrically complicated) by finding the indefinite integral/antiderivative and evaluate at the end points
Proof: see doc
S f(x)dx=F(x)+C= indefinite integral of f is the general antiderivative of f
Proof: see doc
36
Area of a circle using riemann sum
Total Sum=sum of basesheight2=circumferenceradius2=2rr2
37
Summation Notation
E Summation operator defines sums of numbers that follow a pattern
i,j,k,l,m,n,p,q Index, starting number
n Ending number
f(x) Rule Pattern
38
Area Function
Area f(x) changes as limits of integration change x, not dependent on variable of integration
A(x)=xSa f(t)dt
39
Geometric Interpretation of Integral
Difference of Areas b/w f(x) & x-axis
Definite Integral is a number
Area is always a positive number or 0
40
Probability, Sample Space, Event, Value
* **Probability**: a number b/w 0 (0% never) & 1 (100% certain) interpreted as the chance of an experiment returning a particular value, 12 equal chances of happening and not happening
* **Sample Space**: all possible outcomes (ex. {tails, heads})
* Experiment: coin flip → **Event**: coin flipped and landed heads up → **Value** {heads}
41
Random Variable (Discrete vs Continuous)
**Random Variable (RV)**: number determined from a random event
**Discrete Random Variable Pr(R=#):** finite sample space/outcomes, measuring sum probability of finite outcomes, ex.rolling dice
**Continuous Random Variable Pr(R=a)**: infinite sample space/outcomes on a continuum, measuring probability of an interval on this continuum, ex.height of population
* Each outcome has probability 0 as there are infinite possible outcomes, thus probability becomes the likelihood of an outcome falling in a certain range = proportion of areas
42
Probability Density Function
**PDF**: defines the relationship between a random variable and its probability/likelihood
If aSb f(x)dx=Pr(a<=X<=b), random variable X has a probability density function f(x)
**Must Satisfy:**
1. All possible events probability: -infSinf f(x)dx=P(-inf
43
Gaussian PDF/Normal PDF/Bell Curve
f(x)=1/sqrt(pi) x e^(-x2) , even function so positive values are equally as likely as negative values
* Hard to prove as no simple antiderivative exists for error function=erf(x)=2/sqrt(pi) 0Sxe^(-t2)dx
44
Given a series which test to use?
1. Key Features
* Alternating Series (-1)^n or cos(npi), not (1)^2n+1--> Alternating Test
* Simple quotient a(1/n^p) --> Integral/P-Test
* Geometric Series (r)^n --> Geometric Rule
* Power Series/Factorials/Exponents --> Ratio Test
2. Second Resort:
* Comparison Test
* Limit Comparison Test
3. Check: divergence test
45
Which series test can give the exact value of the sum?
* Geometric Series
* Taylor polynomials of classical functions
46
Expectation (Discrete vs Continuous)
Mean of random variable X: specific number denoting the long-term average of all possible values weighted according to probability,
Ex. Heads=1, Tails=-1, Expectation=0
Discrete Random Variable Expectation: 𝔼(x)=SUMx6omega(x*1/6)Pr(X=x)
Continuous Random Variable Expectation: 𝔼(x)=-infSINTEGRALif (x*f(x))dx
Integral/Sum Bounds=sample space
x=values of random variable
f(x)=weighted probability
47
Expected Value of Bell Curve and if X+2
1) Method 1: By symmetry
𝔼(x)=0
Method 2: 𝔼(x)=-infSINTEGRALinf (x1/sqrt(pi)e^(-x^2)dx=0
2) Method 1:
𝔼(Y)=0+2=2, because (𝔼(X)=0) expectation of X is 0
Method 2 Graph transformation, find new mean
Method 3: 𝔼(x)=-infSinf((x+2)f(x))dx=-infSinf(xf(x))dx+-infSinf (2f(x))dx
=0+2-infSinf(f(x))dx=0+2(1)=2
48
General formula for X transformations affect on Expected Value
Let X be random variable with PDF f(x) and g(X) be a random variable representing values of x transformed by g
𝔼(g(X))=-infSINTEGRALinf(g(x)f(x))dx
49
Standard Deviation
Standard Deviation (X): measures amount of deviation/dispersion around mean as two different random variables with same expectation can behave/be dispersed very differently (ex.Y=50, X={0,1,… 100})
50
Variance, Conventional & Alternative
Variance of X: Var(X)=omega(X)^2 sqrt(omega(X))=Var(X)
* Conventional Variance: Var(X)=𝔼((X-𝔼(X))2)=-infSinf(x-𝔼(X))2f(x)dx
Explanation: (X-𝔼(X))^2=difference from mean,𝔼((X-𝔼(X))^2)=average difference from mean
(x𝔼(X))^2=transformation/deviation from mean, f(x)=weighted average
* Alternative Variance: Var(X)=𝔼(X2)-(𝔼(X))20
Proof: see doc
51
Variance of Smaller Interval vs Larger Interval
Larger the interval, the larger the variance, no matter the PDF
52
Algebraic vs Differential Equation
**Algebraic Equation**: unknown is a number
**Differential Equation**: unknown is a function solved using the derivative of the function
* Indefinite integral is a simple differential equation y'(x)=f(x) y(x)=SINTEGRALf(x)dx
* Finding a solution can be hard, there can be multiple
* Checking if function satisfies differential equation is easy
53
Autonomous Differential Equation
Independent variable doesn’t appear in the equation
54
Phase Line Diagram + What It Tells You
(dy/dx=(y+4)(y-3)
Differential Equation Diagram: used to study first-order autonomous differential equations.
-4 stable critical point
3 unstable critical point
When in doubt find limits to finities and find roots.
What it tells you: given an initial condition, where the differential equation will end. Tells you the limit as it approaches infinity.
55
General vs Particular Solution
**General Solution:** involves constant, multiple possible
**Particular Solution:** satisfies initial condition y(0)=#, sub x=0 & isolate for constant
56
Separable Differential Equation + How to Solve
dydx=f(x)g(y), f & g single variable functions
1. Y & Xs on opposite sides & integrate 1g(y)dy=f(x)dt
1. Combine C1+C2=C3
1. Isolate for y(t), get rid of
1. If given initial condition y(0)=y0 isolate for C3
1. Consider constant solutions (ex. y(t)=0)
57
Make a differential equation for: Bacterial population of initial size 10 grows at a rate equal to double the current population
dy/dt=2y, y(0)=10
58
Logistic Equation/Population Model Solution: Derivation + Phase Line Diagram+Sketch on x y axes
General Solution: dy/dt=y(1-y),y(t)=1/1+Ae^(-t)
Deriving Solution: see doc
Initial value Problem Solution:
y(t)=1/1+A((1-y0)/y0)e^(-t)
Deriving solution y(0)=y_0>0: see doc
59
Sequence vs Function
{a_n}={a_n}n=1=a1,a2,a3... : an infinite list of definite ordered numbers, can be defined by a formula
* Both ways of associating outputs to given inputs, function differs as it is defined for all real numbers (xR), sequence is defined only for integers (n=1,2,3,4...)
60
What does it mean when a Sequence Converges/Diverge
**Converges** to a number L if a_n values get closer and closer to L as n increases: limn>inf a_n=L, L is the limit of the sequence {a_n}
**Diverges to +- infinity** if an values increase/decrease without bound as n increases: limn>inf a_n=+-inf, is the limit of the sequence {an}
**Diverges to DNE** in any other case, oscillates, random sequence or doesn’t approach any number: limn>inf a_n=DNE, limit of {a_n} can be anything, exist or not
61
Nth Partial Sum
S_N=NSUMn=1 a_n=a_1+a_2+a_3+...+a_N: sum of a finite number of sequence terms
62
Partial Sum Sequence + What it means to converge/diverge
{S_N}=S_1,S_2,S_3... S_inf
* If {SN} converges to a number S, it is denoted as n=1an=S
* If {SN} diverges to a number S, it is denoted as n=1an diverges
63
Series + Connectin with Integrals
S_N=infSUMn=1 a_n=limN>inf S_N=limN>inf infSUMn=1 a_n=a_1+a_2+a_3+...+a_inf: sum of infinite amount of terms {a_n}/limit when n→ infinity of the sequence of partial sums
**Conceptual connection b/w improper integrals & series**: sum of infinite number of terms computed as limits that can converge or diverge, but differ as one is defined by integrating functions of real variables, the other defined by summing terms of a sequence
64
a_n=1/n--> sequence vs Partial Sum Sequence vs Series
See doc table pg46
65
Divergence Test + When Inconclusive + Explanation + Proofs + When to Use
**Series infSUMn=1 a_n Diverges:**If sequence {a_n} doesn’t converge to 0 (limn>inf a_n=/0)
**Inconclusive:** a_n converges to 0
**Explanation:** Keep summing numbers that aren’t getting smaller/not converging to 0, series has no chance of settling on a number
**Proofs**:
1) Improper Integral Proof: if improper integral doesn’t converge to 0 it diverges just as series diverges if sequence doesn’t converge to 0
* Diverges to : area under curve contains an infinite area diverges
* Converges to L: area under curve contains rectangle of base + and height L diverges
2) Counter Positive Proof: a series that converges, it’s sequence converges to 0, (series that not diverge, its sequence not not converge to 0)
SN+1=SN+aN+1
aN+1=SN+1-SN
If n=1an=L, Then SN=SN+1=L
aN+1=L-L=0 converges
See doc pg46 for clearer math
**When To Use:** if limn>inf can be taken or DNE and doesn't converge to 0. A quick check at the start or end.
66
Integral Test/P-Test + Conditions + Proof + When To Use + Inconclusive
**Conditions**:
1. an=f(x)
1. f(x) is continuous
1. f(x)>0 for all x1 (f is positive)
1. f(x) is decreasing for all x>1 or any +ve number if 1-3 are met cuz only tail matters
**Converges:** if 1Sinf f(x)dx converges
**Diverges:** if 1Sinf f(x)dx diverges
**Inconclusive:** conditions aren't met, akward large x behavior, f hard to define
**Proof Using P-Test+Comparison Test:**series becomes a riemann sum + comparison test smaller than a convergent will converge
Conditions:
an=1np f(x)=1xp
f(x) is continuous
f(x)0 for all x1 (f is positive)
f(x) is decreasing for all x1
If p>1 converges, p<1 diverges
**When To Use:** when conditions above are met, quotients of polynomials, look for 1/n^p
67
Geometric Series+Common Ratio+Test+Proof+When to Use+Inconclusive
S_N=NSUMn=0 ar^n=a_1-r^(N+1)/1-r, r=/1
Is also a power series where c=0, An=1
**Common Ratio abso(r):**next term is previous term multiplied by a constant, previous term is next term divided by a constant abso(a_n+1/a_n or a_n/a_n-1)
**Converges to 1/1-r** if abs(r)<1, Math: n=0rn=1-r+11-r=11-r convergent
**Diverges to inf** If r>1, n=0rn=1-r+11-r= inf divergent
**Proof Using Integral Test**:
infSUMn=0r^n=1+infSUMn=11r^n
Conditions Met:
* f(x)=rx=(elog(r))x=exlog(r)
* f(x) is continuous
* f(x)>0 for all x>1 (f is positive)
* f(x) is decreasing for all x>1
limb>inf e^xlog(r)=converges to -r/log(r) see doc for math pg47
By integral test, if 1Sinf f(x)dx=converges, series infSUMn=1 a_n=converges
**Proof** pg49:
SN=n=0Nrn=1+r+r2+r3+...+rN
r SN=rn=0Nrn=r+r2+r3+r4...+rN+rN+1
SN-rSN=1+rN+1(1-r)SN=1-rN+1SN=1-rN+11-r
**When To Use**: geometric series, look for (r)^n, a^n/b^n, (a/b)^n
**Inconclusive**: when it isn't a geometric series, if it is alternating
68
Give example of series that divergent but divergence test doesn’t detect this.
Harmonic Series 1/n
69
Comparison Test + When To Use + Inconclusive
b_n=K SERIESf(x) with known behavior
**Converges** if aKb for interval and b diverges, bigger than a divergent
**When To Use:** if comparison is easy and b_n has known behavior (p-series, geometric, alt.)
**Inconclusive:** if a>b and b converges tells you nothing, if a
70
Limit Comparison Test + Conditions + Proof + When To Use + Inconclusive+Formatting
L=limn>inf a_n/b_n
**Same behavior as SERIES b_n** if 0b, b div.)
**When To Use:** quotient of polynomials are best for simpler b_n focusing on main terms, a_n is similar to simpler b_n focusing on main terms
**Inconclusive:** alternating, not easy to compare to simpler series
See doc pg51
71
Ratio Test + Speed + When To Use + Inconclusive
L=limn>inf abso(a_n+1/a_n)
**Converges Absolutely** if L<1
**Diverges** if L>1 or +inf
**Speed:** Super Fast Conv. L=0 <> Super Slow Conv. L=1
**When To Use:** combinations of geometric series, factorials, power series
**Inconclusive:** L=1 or DNE, quotient of polynomials
72
Alternating Test + Conditions + Proof + When To Use + Inconclusive
Alt Series = infSERIESn=1 (-1)^nAn= (-1)^n+1An = cos(npi)^nAn (NOTE: not (1)^2n+1)
**Converges if conditions met:**
* Terms are positive and decreasing eventually
* lim n>inf An=0
**Absolutely Converges** if series and abso(series) converge
ex. (-1)^n1/n^2
**Conditionally Converges** if series converges but abso(series) diverges
ex. (-1)^n1/n
**Proof:** SERIES (-1)^n 1/n still converges no matter the minus sign, alt. helps it converge faster by cancelling out terms, draw drawing
**When To Use:** alt series (-1)^n or (-1)^n-1 with 0
73
Power Series + Power Series Representation + Graph
Infinite degree polynomials used to approximate functions, compute limits and solve differential equations. It also is a Taylor series of the function f(x) expanded around x=c.
a(1/1-x)=infSERIESn=0 ax^n for x<1, a=first term
or
infSERIESn=1 An(x-c)^n=A0+A1(x-c)^1+A2(x-c)^2+A3(x-c)^3+...
c=center, An=sequence of coefficients, A0=constant term=A0(x-c)0
See doc
74
When can operations be done to series or power/taylor series? What happens to Radius of Convergence
**Operations Term by Term on Power Series within Radius of convergence**: adding/subtracting together power series, multiplying/factorig out by a constant or a power of xk, differentiating, integrating a power series
**Resulting/New R>=R of Original Series**
75
Radius of Convergence + Process to Solve
Distance from the center in both directions where power series converges, allows us to find the integral of convergence
**Process:**
1. **Rewrite** in terms in n=1An(x-c)n, identify what is An,c
1. **Ratio test**: nAn+1(x-c)n+1An(x-c)n=(x-c)nAn+1An
- If nAN+1AN=exists, findnAN+1AN=A, then x-c<1A, R=1A, Interval of convergence=x(c-R,c+R)
- (Note: must factor out a from ax-c>1A and divide for R)
- If nAN+1AN=DNE=AN=oscillates, random, blows up, use comparison test, find bigger An substitute (smaller than conv→conv.) and repeat entire process
1. **Endpoints** must be individually tested to check if it converges or diverges
**Possibilities**
1. **R=+inf, A=0, xER**
2. **R=0, A=+inf, x=c**
3. R=#, A=nonzero exists, XE(c-R,c+R) check endpoints, abso(x-c)
76
Taylor Series + Use + Maclaurin Series
**Taylor Series**: power series representation for f(x) near/centered at infSERIESn=0 f^(n)(x)/n!(x-c)n=f(c)+f'(c)(x-c)+f''(c)2!(x-c)2+f'''(c)3!(x-c)3...
**When To Use:** tailor a power series representation to functions (infinitely differentiable) by
1) If easily differentiable, matching all their derivatives
2) If hard to differentiate, find relation to known T(x) functions and apply relation to its power series term by term or all at once (ex.log, arctan)
* If 2 functions take same values on small interval containing c, they will have same Taylor series around x=c
**Maclaurin Series c=0** infSERIESn=0f(n)(x)n!xn=f(0)+f'(0)(x)+f''(0)2!(x)2...
See doc for more clarity pg58
77
Error Formula in Taylor Approximations + Max error + How to prove no error in approximation
**Error Formula:** En(x)=f(x)-TN(x) how far partial sum (Taylor Series) is from the f(x)
**Max Error:**
EN(x)=f(x)-TN(x)M(N+1)!x-cN+1, where M=max{f(N+1)(y):y[c,x]or [x,c]}
**No Error Proof:**
limN>inf E_n(x)=0
78
Remember T(x) of 1/1-x and e^x
See doc pg61
79
Derive T(x) of log(1-x),log(1+x),log(x)+arctan(x)
see doc pg61
80
Optional: T(x) of sin(x) and cos(x)
see doc pg61
81
Find p so that and limit so that limit n->0 of (6xsin(8x)+px^2)/(24-24cos(2x)-48x^2)
p=-48
lim=512/16
82
Find limit x-->infinity of (1+7/5x)^(9x)
e^(63/5)
83
Compute sinm(x)cosn(x) when one exponent is odd, when both are even, when both are odd
See paper notes
84
What concavity will each approximation method underestimate or overestimate?
Draw it bub
85
Intuitive argument to check if expected value is more/less/halfway point of the space.
More probability weight one side, the more leaning to that area the expectation will be
86
Halflife, Newton's Law of Cooling, Decay, Mixing Differential Equations + Key Parameters
Google + Solve
- Decay/Half Life: dy/dx=-ky, y(half life)=1/2, y(0)=1
- Newton's Law of Cooling: dy/dx=k(y-b), y=
87
How to solve expectation for piecewise function
Solve individual expectations for each piece and add them up to get total expectation.
88
Harmonic Series + infSERIESn=1 1/n^2
infSERIESn=1 (1/n) & pi^2/6
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