
Volume of a Solid with Cross Section
V=
Volume of a Solid with Cross Section
 V=ʃ_{a}^{b }A(x)dx
 (cross section is perp. to height of solid)

Euler's Method when y'=F(x,y)
x_{1}=
y_{1}=
_{}_{}Euler's Method when y'=F(x,y)
 x_{1}= x_{0}+h y_{1}= y_{0}+hF(x_{0},y_{0})
 h is step size
 (smaller step size, better approx.)

Disc Method about the line h
V=
Disc Method about the line h
 V= (pi) ʃ_{a}^{b}( f(x)  h )^{2 }dx,
 where  f(x)h =radius
 (if xaxis, h=0)

Washer Method between f(x) and g(x)
V=
Washer Method between f(x) and g(x)
 V=(pi) ʃ_{a}^{b }[ (f(x))^{2 } (g(x))^{2 }]dx
 Outer radius  smaller radius

What is the Squeeze Therom?
if g(x)< f(x) < h(x) and Lim_{x>a}[g(x)= h(x)]=L, then so does that same limit of f(x)

What is the Intermediate Value Therom?
If a function f is continuous on a closed interval [a,b] and k is a number with f(a) < k < f(b), there exists a number inside the interval where f(c)=k

What is...
d/dx (a^{u}) =
d/dx (a^{u}) = a^{u} ln(a) u'

What is...
d/dx log(u)=
d/dx log(u)= 1/(u lna) * u'

What is Rolle's Therom?
 Rolle's Therom: if f is continuous [a,b] and differentiable (a,b), and f(a)=f(b)=0
 then there is a number c that exists in (a,b) such that f'(c)=0
 *rel. mins and maxs

What is the Mean Value Therom?
 Mean Value Therom: if f is continuous [a,b] and differentiable (a,b) thn there exists a number c in (a,b) that f'(c)=(f(b)f(a))/(ba)
 *secant line slope passing through points a and b are equal to tangent slope of c

General Procedure for Related Rates Problems!!
 1.Read problem, draw diagram
 2.Variables and Math symbols for given info
 3. Write and equation involving the rate of change to be determined(reduced to one variable)
 4.Differentiate with respect to time
 5.Plug in values and solve for desired rate
 6.Answer with units of measure

General Procedure for Solving Applied Max and Min Problems!
 1. Write Primary Equation and Secondary Equation
 2. Solve so Primary Equation is in terms of one variable
 3. Find feasable domain
 4. Find critical values (f' = 0) of the function on the feasable domain
 5. Test critical numbers and endpoints
 6. Answer in context of the problem

What is the Average Value of a Function? from (a,b)
Average Value of a Function: 1/(ba) ʃ_{a}^{b }f(x)dx

Area of an equilateral triangle
A=
A=(s^{2 }3^{1/2}) / 4

Position Function
Instantaneous Velocity
Acceleration
Instantaneous Speed
 Position Function = s(t)
 Instantaneous Velocity = s'(t) = v(t)
 Acceleration = s''(t) = v'(t) = a(t)
 Instantaneous Speed = v(t)

First Fundamental Theorem of Calculus
ʃ_{a}^{b }f(x)dx = F(b)  F(a)

Second Fundamental Theorem of Calculus
F(x)= ʃ_{a}^{b} f(t)= f(x) , F'(x)=f(x) at every point of [a,b]

Area between two curves f(x) and g(x)
 ʃ_{a}^{b} [f(x)g(x)]dx
 upper curve  lower curve

Distance Trveled Problems!!
Displacement from t_{1} to t_{2}
 Displacement from t_{1 }to t_{2 }= ʃ_{t1}^{t2} v(t)
 Total distance traveled is the same thing but the absolute value of v(t)

pseries
 is a series of the form, 1+1/2^{p}=1/3^{p}=1/4^{p}+...+1/n^{p}+...= sum( 1/n^{p}, n, 1, infinity)
 Series converges when p>1
 Series diverges when 0<p<1

harmonic series
special p series, where p=1, so the series diverges

geometric series
 sum(ar^{(n)},n,0,infinity).
 series converges when r<1
 sum of the series is a/(1r)

integral test
if a_{n}=f(n) where f is continuous, positive, decreasing funciton on [c,infinity)

