Sem 2 Stewart

Question 1

method for 1st order differential equation

Answer 1

separating the variables

integrating factor if variables cannot be seperated

Question 2

method for second order homogeneous differential equations

Answer 2

auxiliary equation and then general solution depending on roots of auxiliary equation

Question 3

method for second order non-homogeneous differential equations

Answer 3

complementary function and particular integral in the same form as f(x) or multiply by x or x^2 if in same form as CF

Question 4

form of general solution for real unequal roots

Answer 4

y=Ae^ax+Be^bx

Question 5

form of general solution for real equal roots

Answer 5

y=(Ax+B)e^ax

Question 6

form of general solution for complex roots

m=a+/-bi

Answer 6

y=e^ax(Acosbx+Bsinax)

Question 7

steps for Riemann sums

Answer 7

1. divide domain into n strips of width Δx=b-a/n
2. find grid points
3. evaluate riemann sum

(could be left, right, upper or lower)

Question 8

midpoint rule

Answer 8

Δx[f(x1bar)+…+f(xnbar)]

where Δx=(b-a)/n and x bar is midpoint of interval

Question 9

trapezium rule

Answer 9

the average of the L and R Riemann sums

1/2(Ln+Rn)
=Δx/2[f(x0)+2f(x1)+…+2f(xn-1)+f(xn)]

Question 10

simpson’s rule

Answer 10

use parabolae instead of straight lines
n must be even

consider consecutive pairs of intervals and approximate the integral over each pair

Δx/3(y0+4y1+2y2+4y3+2y4+…+2yn-2+4yn-1+yn)

Question 11

general steps for approximate integration

Answer 11

1. find Δx
2. find x0,x1,…,xn
3. (midpoint) x1bar, x2bar…
4. f(x1), f(x2),… etc
5. apply formula

Question 12

function is continuous on an interval if…

Answer 12

continuous at every number in interval

Question 13

function is differentiable at a if..

Answer 13

f’(a) exists

differentiable on an open interval if differentiable at every point

Question 14

F is an anti derivative of f if

Answer 14

F’(x)=f(x) for all x

Question 15

what form are integrals in for integration by substitution?

Answer 15

∫f(g(x))g’(x)

u=g(x) is a differentiable function, continuous on I

∫f(g(x))g’(x) dx = ∫f(u) du

Question 16

limits in integration by substitution

Answer 16

change limits using expression for u and sub in current limits

limits are g(a) and g(b)

Question 17

trigonometric integrals

Answer 17

manipulate integrand using trig identities

can use substitution in reverse by setting x=g(f) and obtain inverse substitution

Question 18

integral on closed interval [-a,a], if f is even (i.e. f(-x)=f(x))

Answer 18

∫a/-a f(x)dx = 2∫a/0 f(x) dx

Question 19

integral on closed interval [-a,a], if f is odd (i.e. f(-x)=-f(x))

Answer 19

∫a/-a f(x)dx = 0

Question 20

integration by parts formula

Answer 20

∫uv’=uv-∫vu’

Question 21

what is needed to split into partial fractions?

Answer 21

proper function
degree numerator < degree denominator

if not, denominator into numerator

Question 22

4 different forms of partial fraction

Answer 22

distinct linear factors
repeated linear factors
irreducible quadratic factors
repeated irreducible quadratic factors

Question 23

distinct linear factors

Answer 23

A/ax+b + B/cx+d

Question 24

repeated linear factors

Answer 24

A/ax+b + B/(ax+b)^2

Question 25

irreducible quadratic factors

Answer 25

Ax+b/ax^2+bx+c

Question 26

repeated irreducible quadratic factors

Answer 26

Ax+b/ax^2+bx+c + Cx+d/(ax^2+bx+c)^2

Question 27

area between two curves y=f(x), y=g(x) and lines x=a, x=b

f(x)>/= g(x) for all x in [a,b]

Answer 27

A=∫a/b [f(x)-g(x)]dx

Question 28

area between two curves y=f(x), y=g(x) and lines x=a, x=b

f(x) < g(x)

Answer 28

A=∫a/b |f(x)-g(x)| dx

Question 29

volume: s is solid, lies between x=a and x=b.

if cross-sectional area of S in plane Px, through x and perpendicular to x=axis then V=

Answer 29

lim as n approaches infinity

∫ b/a A(x) dx

Question 30

solids of revolution

Answer 30

obtained by revolving a region about a lines

v=∫ b/a A(x) dx

where A is area of cross-section

Question 31

volume of revolution if cross -section is a disk

Answer 31

v=∫ b/a A(x) dx

A=pi(radius)^2

Question 32

volume of revolution if cross-section is a washer

Answer 32

v=∫ b/a A(x) dx

A=pi(router)^2-pi(rinner)^2

Question 33

volumes by cylindrical shells of revolution

rotating about y-axis, region under curve y=f(x), from a to b

Answer 33

v=∫b/a 2pix f(x) dx

=circumference.height.thickness

Question 34

volumes by cylindrical shells of revolution

rotating about x-axis, region under curve x=f(y), from c to d

Answer 34

v=∫b/a 2piy f(y) dy

Question 35

if f’ continuous on [a,b], length of curve is

Answer 35

L=v=∫b/a root(1+f’(x)^2 dx

(or root 1+ (dy/dx)^2 dx)

Question 36

differential of arc length

Answer 36

ds=root 1+(dy/dx)^2 dx

Question 37

arc length function

Answer 37

smooth curve equation y=f(x), s(x) is distance along curve from initial point P to Q

s(x) = ∫x/a root 1+ (f’t)^2 dt

Question 38

surface area of revolution

Answer 38

f +ve, continuous derivative, surface area by rotating y=f(x) about x-axis

s= ∫b/a 2piy ds

where ds= root 1+ (dy/dx)^2 dx

Question 39

what are parametric equations used for?

Answer 39

to describe curves not in the form y=f(x) i.e. fails vertical line test

Question 40

parametric equations

dy/dx=

Answer 40

dy/dt / dx/dt

Question 41

parametric area under y=F(x) from a to b, x=f(t), y=g(t)

t between alpha and beta

Answer 41

A=∫b/a F(x) = ∫beta/alpha g(t)f’(t) dt

Question 42

parametric arc length

Answer 42

L=∫beta/alpha root (dx/dt)^2+ (dy/dt)^2 dt

Question 43

parametric surface area of revolution

Answer 43

s=∫beta/alpha 2piy(t) root (dx/dt)^2+ (dy/dt)^2 dt

Question 44

limits of vector functions

Answer 44

take each component separately

make sure to sub point into original equation to get in terms of t

Question 45

d/dt [u(t)xv(t)] =

Answer 45

u’(t)xv(t) + u(t)xv’(t)

Question 46

unit tangent vector to space curve r(t) is

Answer 46

T(t) = r’(t)/|r’(t)|

Question 47

unit normal vector to space curve r(t) is

Answer 47

N(t)=T’(t)/|T’(t)|

Question 48

improper integral

Answer 48

interval is infinite or f has an infinite discontinuty.

Question 49

Type 1 improper integral

Answer 49

if ∫t/a f(x) dx exists for every t>/= a then

∫ infinity/a f(x) dx = lim as t appraoches infinity ∫t/a f(x) dx

same for negative infinity

Question 50

convergent

Answer 50

limit exists

Question 51

divergent

Answer 51

limit does not exist

Question 52

type 2 improper integrals

Answer 52

if f continuous on [a,b) and discontinuous at b

∫b/a f(x)dx=lim t –> b- ∫t/a f(x) dx

if f continuous on (a,b] and discontinuous at a

∫b/a f(x)dx=lim t –> a+ ∫b/t f(x) dx

Question 53

comparison theorem

Answer 53

if f(x) and g(x) continuous with f(x) >/= g(x)>/=0 for x>/=a

if ∫ infinity/a f(x) is convergent, ∫ infinity/a g(x) convergent

if ∫ infinity/a g(x) is divergent, ∫ infinity/a f(x) divergent