Review for final exam

Teng Zhang

University of Minnesota

December 12, 2012

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General considerations

FRIDAY, December 14, 2012 1:30 pm - 4:30 pm Moos Tower,Room 2-650

13 multiple choice questions (5 points each), and 8hand-graded questions (10-25 points each).

Formula sheet available at the final study guide online

Please refer to the final study guide for the material we willcover in final exam (past exams might contain material wewill not cover).

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Section 7.1: Integration by parts

Choose u in the order of logarithmic, inverse trigonometric,polynomial and exponential.

Choose exponential function for dv .

Examples:

ln u du,

ln2 u du (integration by parts twice),tan1 x dx ,

ex sin x dx (integration by parts twice)

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Section 7.2: Trigonometric integrals

sinm x cosn x dx (when m is odd, u = cos x , when n is odd,

u = sin x , and the other case)tanm x dx secn x (when m is odd, u = sec x when n is even,

u = tan x and the other case)sin(Ax) cos(Bx)dx ,

sin(Ax) sin(Bx)dx ,

cos(Ax) cos(Bx) dx (half-angle identity)

Examples:

sin3 x cos2 x dx ,

tan3 x sec4 x dx ,

tan x sec3 x dx

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Section 7.3: Trigonometric substitutions

If there is an additional linear term (e.g.,x2 + 2ax + b), use

substitution (e.g., u = x + a).

Example:

5 + 4x x2 dx , x31x2 dx

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Section 7.4: Integration of rational functions

Step 1: if P(x)/Q(x) is improper, decompose it to apolynomial plus a proper rational function R(x)/Q(x) (bylong division).

Step 2: Factorize Q(x) into linear factors (ax + b) andirreducible quadratic factors (ax2 + bx + c and b2 4ac < 0).Step 3: write R(x)/Q(x) as sum of partial fractions of theform

A

(ax + b)ior

Ax + B

(ax2 + bx + c)j

Step 4: solve for the constants A and B (maybe by pluggingin the root of linear factors)

Step 5: integrate each of of partial fractions

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Section 7.5: Strategy for integration

Simplify the function and use substitution if possible.

Examples: sin3(x)

xdx ,

ex

e2x+1dx ,

x31x2 dx

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Approximate Integration: definition (Section 7.7)

Left endpoint rule, Right endpoint rule, Midpoint rule, Trapezoidalrule (will be on formula sheet)

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Approximate Integration: Over/under estimates (Section7.7)

For an increasing/decreasing function f (x), find the order of

Ln, I = ba f (x) dx and Rn.

For an function concave down (f (x) < 0 for a x b),(exercise 47)

Tn 0, the function is increasing,when it is < 0 it is decreasing.

Q: What does this mean in terms of the system thats beingmodeled?A: increasing/decreasing in terms of population/amount ofsalt/

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Section 10.1: Curves Defined by Parametric Equations

Sketching a parametric curve (pay attention to thedomain/range).

Eliminating the parameter to find a Cartesian equation of thecurve.

Example: eliminate the parameter t and sketch the parametriccurve:

x = et 1, y = e2t .

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Section 10.2: Calculus with Parametric Curves

Find the slope of tangent, without eliminating the parameter.

Example: at point (1, 1), what is the slope of the tangent lineto the parametric curve y = t3 and x = t2?

Example: When does the parametric curve y = t3 1 andx = t + 1 have vertical/horizontal tangent lines?

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Section 10.3: Polar Coordinates

Connection between polar coordinates (r , ) and Cartesiancoordinates (x , y):

x = r cos , y = r sin , r2 = x2 + y2, tan =y

x.

Example: find a polar representation for the Cartesian pointx = 3, y = 4.There are infinitely many representations in polar coordinates,for example (r , + 2kpi) and (r , + (2k + 1)pi) are thesame points.Definition of polar curves: what is r = 2 or = 1?Conversion between the equation for polar curves and theequation for Cartersian curves. Example: r2 sin 2 = 1.Slope of the tangent line for the polar curve:

dy

dx=

drd sin + r cos drd cos r sin

,

and when is the slope horizontal/vertical? 21 / 35

Section 10.4: Areas and Lengths in Polar Coordinates

Area of a polar region bounded by r = f (), and rays = aand = b:

A =

ba

1

2r2 d.

Sometimes we need to find a, b by the intersection of twopolar curves r = f () and r = g(). The intersection isobtained by solving the equation f () = g().

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Section 11.1: Sequences

Definition of a limit of s sequence. Example: limn(1)n.Relationship to limit of a function: if limx f (x) = L andf (n) = an, then limn an = L. Example: limn ln nn+1(lHospitals rule).

If limn an = L and f is continuous at L, thenlimn f (an) = f (L). Example: limn e1/n.Squeeze Theorem: if an bn cn andlimn an = limn cn = L, then limn bn = LMonotonic Sequence Theorem: Every bounded, monotonicsequence is convergent.

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Section 11.2 Series

Definition:

n=1 an = limn sn, where sn =n

i=1 ai .

Geometric series:

a + ar + ar2 + + arn1 + = a1 r when |r | < 1,

and it diverges for |r | 1.Divergence test:

an is divergent if

1 limn an does not exists.2 limn an exists, but is nonzero.

Telescoping sums. Example:

n=11

n(n+1)

If both

n=1 an and

n=1 bn are convergent,n=1(an bn) =

n=1 an

n=1 bn.

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Sequence and series

Dont mix up sequence and series.

Example of sequence: limn 1n ;Examples of series:

n=1

1n .

For example, divergence test and root/ratio tests are forseries, not sequence.

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Section 11.3: Integral test

Integral test: if an = f (n), and f is continuous, positive,decreasing on [k ,). Then n=k an is convergent if and onlyifk f (x) dx is convergent.

The p-series

n=1

1

npis

{convergent if p > 1

divergent if p 1.

Example:

n=21

n ln n

Dont forgot to show that the function is decreasing,sometimes by differentiation of f (x).

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Section 11.4: The Comparison Tests

Comparison test for series: Assume two series

an and

bnwith bn an 0, then

1 if

bn is convergent, so is

an.2 if

an is divergent, so is

bn.

The limit comparison test: Assume two series

an and

bnwith an, bn 0, then if

limn

anbn

= c ,

where c is a finite number and c > 0, then either both seriesconverge or both diverge.

Usually apply p-series and geometric series to compare, andfind the series to compare by the dominant terms. Examples:

n=1nn3+1

.

Note the difference between comparison test and limitcomparison test. Example:

n=2

nn32 .

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Section 11.5: Alternating Series

Alternating series test: If the alternating series

n=1

(1)n1bn orn=1

(1)nbn

satisfies bn 0, limn bn = 0 and bn+1 bn for all n, thenthe series is convergent.

Sometimes we need to use derivative to show that bn is

decreasing. Examples: determine if

n=1(1)nn2n2+5

isconvergent or divergent.

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Section 11.6: Absolute/Conditional Convergence andRatio/Root Tests

an is absolutely convergent if

|an| is convergent.an is conditionally convergent if

|an| is divergent andan is convergent. Example:

n=1(1)n 1n .

Ratio/root test:

limn

{an+1an n|an|=

L < 1

L > 1 or 1

, then

an is

absolutely convergent

divergent

(no conclusion.)

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Section 11.8: Power Series

A power series is a series of the form

n=0

cnxn = c0 + c1x + c2x

2 + c3x3 + ,

There are only three possibilities for

n=0 cn(x a)n:1 It converges only when x = a.2 It converges for all x .3 There exists R such that it converges when |x a| < R and

diverges when |x a| > R. (R is called radius of convergence)We first find the radius of convergence R by ratio test or roottest, and then test the endpoints a R for convergenceseparately.

Example: find the interval of convergence for the seriesn=2

(3)nnn

(x + 1)n converge.

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Section 11.9: Power series as functions

Power series representation for 11x :

1

1 x = 1 + x + x2 + x3 + =

n=0

xn, for |x | < 1.

Domain of the power series = interval of convergence.

Basic manipulation. Example: find power seriesrepresentations for 2x

2

1+x3or x5x .

Inside the radius of convergence we can performintegration/differentiation term by term. Example: find powerseries representations for f (x) = 1

(1x)2 and f (x) = tan1(x).

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Section 11.10: Taylor series

f (x) = f (a)+f (a)

1!(xa)+ f

(a)2!

(xa)2+ f(a)3!

(xa)3+ is called the Taylor series of f at a.Exercise: what is f (2) for f (x) =

n=0

cnn! (x 2)n.

Tn(x) = f (a)+f (a)

1!(xa)+ f

(a)2!

(xa)2+ + f(n)(a)

n!(xa)n

is called nth-degree Taylor polynomial of f at a, andRn(x) = f (x) Tn(x) is called remainder.Taylors inequality: if |f (n+1)(x)| M for |x a| d , then

|Rn(x)| M(n + 1)!

|x a|n+1 for |x a| d .

Example: prove that ex = 1 + x + x2

2! +x3

3! + for all xBasic manipulation. Example: find Taylor expansion for sin xxat a = 0. 32 / 35

Section 12.1-12.2: Vector and 3D-coordinate system

The distance |P1P2| between points P1(x1, y1, z1) andP2(x2, y2, z2) is|P1P2| =

(x2 x1)2 + (y2 y1)2 + (z2 z1)2.

Definition: A vector is a quantity that have both a magnitudeand a direction.

Addition (Triangle/Parallelogram law), subtraction, scalarmultiplication of vectors.

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Section 12.3: Dot product

Dot product between vectors for a =< a1, a2, a3 > andb =< b1, b2, b3 >:

a b = a1b1 + a2b2 + a3b3.

Geometric interpretation: if is the angle between vectors aand b, then

a b = |a||b| cos .When two vectors are orthogonal, =pi/2, cos = 0.

When two vectors are parallel, =0 or pi, cos = 1.

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Section 12.4-12.5: Cross product

Definition: the cross product of a and b is

a b =< a2b3 a3b2, a3b1 a1b3, a1b2 a2b1 > .If is the angle between a and b (0 pi), then

|a b| = |a||b| sin .Example: Find the area of the triangle with verticesP(1,2, 0), Q(3, 1, 4) and R(0,1, 2).The plane through P0(x0, y0, z0) and with normal vectorn =< a, b, c > is described by

a(x x0) + b(y y0) + c(z z0) = 0.Example: find the plane that passed through P(1,2, 0),Q(3, 1, 4) and R(0,1, 2) (main point: find its normal vectorby cross product).

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