Pre-midterm

Question 1

Method of exhaustion

Answer 1

Using super tiny polynomials to approximate curved stuff

Question 2

Riemann Sum

Answer 2

Lim (n-> inf) (sum)(i=1)(n)(f(xi))Δx = S(b,a)f(x)dx

Question 3

Trapezoid rule

Answer 3

Riemann sum average (f(xi-1) + f(xi))/2

And n->N

Question 4

Properties of the definite integral

Answer 4

• Linearity
• can break it up
• from a to a = 0
• a to b = -b to a

Question 5

Mean value theorem for integrals

Answer 5

If f is continuous on [a,b], then there exists a point x* in [a,b] such that the integral from a to b of f(x)dx = f(x*)(b-a)

Question 6

Proof for MVT for integrals

Answer 6

1) by EVT, has max and min
2) area within rectangles made with Max and min
3) by IVT f takes all values between it, so f is avg area

Question 7

Fundamental Theorem of Calculus

Answer 7

Given a function f which is cts on an interval I, containing the point a, then
1) Let the function F be defined on II by
F(x) = S(a,x)f(t)dt
F(x) is differentiable on II, and F’(x) = f(x). So F(x) is the anti-derivative of f on II
d/dx[S(a,x)f(t)dt]=f(x)
2) if G(x) is any anti-derivative of f on II, so that G’(x) = f(x) on II, then for any point b in II, we have
S(a,b)f(x)dx = G(b)-G(a)

Question 8

FTC proof

Answer 8

1) from def of derivative
Plug in F
Use MVT to replace x with c, c->x as h->0
2) G’(x) = f(x)
F(x) = G(x) + C on II for some constant C
Show integral
Let x = a, x = b

Question 9

Indefinite integral

Answer 9

Integral +C

No limits

Question 10

Integral of x^n

Answer 10

x^n+1/n+1, n NOT 1

Question 11

Integral of e^x

Answer 11

e^x

Question 12

Integral of 1/x

Answer 12

ln(|x|)

Question 13

Integral of b^x

Answer 13

b^x/lnb

Question 14

Integral of sin(x)

Answer 14

-cos(x)

Question 15

Integral of cos(x)

Answer 15

Sin(x)

Question 16

Integral of sec^2(x)

Answer 16

Tan(x)

Question 17

Integral of 1/x^2+1

Answer 17

Arctan(x)

Question 18

Integral of 1/sqrt(1-x^2)

Answer 18

Arcsin(x)

Question 19

Integral of sinh(x)

Answer 19

Cosh(x)

Question 20

Integral of cosh(x)

Answer 20

Sinh(x)

Question 21

Substitution

Answer 21

1) identify substitution (derivative should appear in numerator!)
2) change infinitesimal
3) change limits

Question 22

Integration by parts

Answer 22

integral of f’g = fg-integral(fg’)

Question 23

Cos power reduction

Answer 23

Cos^2(x) = 1/2(1+cos(2x))

Question 24

Sin power reduction

Answer 24

sin^2(x) = 1/2(1-cos(2x))

Question 25

When you have a bunch of cyclic stuff...

Answer 25

Integrate by parts several times! Integrand comes back

Question 26

When you're stuck?

Answer 26

Try multiplying by 1 and IBP

Question 27

Trig sub sqrt(a^2-x^2)

Answer 27

Sub x=asin (angle between -+ Pi/2)

Question 28

Trig sub sqrt(a^2+x^2)

Answer 28

Sub x=atan (angle between +-Pi/2)

Question 29

Trig sub sqrt(x^2-a^2)

Answer 29

Sub x=asec (angle between 0 and Pi/2)

Question 30

Partial fraction - OP(x) >_ OQ(x)

Answer 30

ZIT-FR it!!!

Question 31

Partial fraction - repeated root in denom

Answer 31

Include repeated root (has n different times now)

Question 32

Partial fraction - irreducible quadratic factor

Answer 32

Has Ax+B over quad factor You'll always get arctan and ln factors

Question 33

Improper integrals

Answer 33

Improper if the integrand exhibits an asymptote over domain of integration Approach asymptote via limits (if lim exists)

Question 34

Comparisons you should know

Answer 34

Integral from 0-inf of e^-x converges | Integral from 1-inf of 1/x^p converges iff p>1

Question 35

Euler-Lagrange equation

Answer 35

δF/δy - d/dx(δF/δy') = 0 If a function minimizes another function, it satisfies this

Question 36

Beltrami's identity

Answer 36

When there's no explicit x dependence F - y'(δF/δy') = constant

Question 37

Differential Equation (def)

Answer 37

A differential equation is an algebraic equation involving the derivative (or multiple, or higher order) of a function

Question 38

Most common way to write differential equations

Answer 38

Derivative (thing) = (all effects that increase) - (all effects that decrease) Sources and sinks

Question 39

Separable 1st order ODEs

Answer 39

Separable if we can write dy/dx = A(x)B(y) So then rearrange and integrate both sides Integral(1/B(y))dy = integral(A(x))dx

Question 40

Evaluating sin^mxcos^nx - cos is odd

Answer 40

Save a cos, express rest in sin, u=sin

Question 41

Evaluating sin^mxcos^nx - sin is odd

Answer 41

Save a sin, express rest as cos, u=cos

Question 42

Evaluating sin^mxcos^nx - sin and cos both odd

Answer 42

Can do the same method for either sin or cos being odd, doesn't matter

Question 43

Evaluating sin^mxcos^nx - both sin and cos are even

Answer 43

Use power reduction formulas (aka half angle IDs) Also maybe use sinxcosx = 1/2sin(2x)

Question 44

Evaluating tan^mx•sec^nx - sec is even

Answer 44

Save a factor of sec^2, express rest as tan | u = tan

Question 45

Evaluating tan^mx•sec^nx - tan is odd

Answer 45

Save a factor of sectan, express rest as sec, u=sec

Question 46

Evaluating sin(mx)cos(nx)

Answer 46

Use sin(A)cos(B) = 1/2[sin(A-B) + sin(A+B)]

Question 47

Evaluating sin(mx)sin(nx)

Answer 47

Use sin(A)sin(B) = 1/2[cos(A-B) - cos(A+B)]

Question 48

Evaluating cos(mx)cos(nx)

Answer 48

Use cos(A)cos(B) = 1/2[cos(A-B) + cos(A+B)]

Question 49

Comparison theorem - integrals

Answer 49

Suppose that f(x) and g(x) are continuous functions with f>_g>_0 for x>_a If S(inf,0)f(x)dx is convergent, then S(inf,0)g(x)dx is convergent If S(inf,0)g(x)dx is divergent, then S(inf,0)f(x)dx is divergent

Question 50

What do the Euler-Lagrange equations tell us about the curve y

Answer 50

Give us the curve y that minimizes J=S(x2,x1)F(x,y(x),y'(x))dx, the action