final exam Flashcards
Slicing method
Integral of area
disk method
pi*integal of radius squared
washer method
pi* integral of outer radius square minus inner radius squared
shell method
integral of 2pi(r)*height
arc length formula
L=integral of sqrrt(1+(f’(x))^2)
surface area
integral of 2pi(r)*ArcL
integration by parts
uv-Integral(vdu)
Int. of cosx
sinx
Int. of sinx
-cosx
Int. of sec^2(x)
tanx
Int. of secx
ln|secx+tanx|
Int. of csc^2(x)
-cotx
Int. of secx tanx
secx
Int. of cscx cotx
-cscx
Int. of cscx
-ln|cscx+cotx|
Int. of tanx
ln|secx|
Int. of cotx
ln|sinx|
sin^2(x)+cos^2(x)
1
1+cot^2(x)
csc^2(x)
tan^2(x) +1
sec^2(x)
sin^2(x)
(1-cos(2x))/2
cos^2(x)
(1+cos(2x))/2
sin(2x)
2sinxcosx
Trig substitution 1st step
draw a triangle
fraction decomposition
linear factor: A + B
Repeated factor: A/x + B/x-a + C/(x-a)^2
Quadratic Factor: A + (Bx + C)
improper integrals
use a limit
Geometric Series
a[(1-r^k)/(1-r)]
i. |r|>= 1 diverges
ii. |r| < converges
Divergence Test
lim(ak) not 0 then diverges
Integral test
f(x) is postive, continuous, decreasing
ak behaves the same as integral
p series (1/n^p)
p>1 converges
p<=1 diverges
comparison test
bk > ak behave the same
direct comparison test
if ak < bk and bk converges, ak converges
if ak>bk and bk diverges, ak diverges
sequence convergence test
USE LIMIT
limit comparison test
if lim (an/bn) not = 0 then behave the same
if lim (an/bn) = 0 and bn converge, then an converges
if lim (an/bn) = infinity and bn diverges, then an diverges
Alternating series test
i. divergence test, lim = 0
ii. a(k+1) < ak
converges
Absolute convergence test
if sum |ak| converges, ak converges absolutely
Ratio test
r = lim |a(k+1)/ak|
i. r<1 converges
ii. r>1 diverges
iii. r=1 inconclusive
root test
p = lim krt(|ak|)
i. p<1 converges
ii. p>1 diverges
iii. p=1 inconclusive
convergence of power series
ratio test, |r| <1, solve for x
check endpoints
plug in endpoints for x then do a test (alternating series or p test)
finding Taylor series
p(x) = c0+c1(x-a)+c2(x-a)^2+c3(x-a)^3+…
c0 = f(a)
c1 = f’(a)/1!
c2 = f’‘(a)/2!
c3 = f’’‘(a)/3!
Maclaurin series
taylor series where a=0
binomial expansion
sum of (r n) x^n = (r 0)x^0 + (r 1)x + (r 2)x^2 + (r 3)x^3 +… = 1 + rx/1! + r(r-1)x^2 /2! + r(r-1)(r-2)x^3 /3! +…
parametric equations
make table with x, y, t and the draw the graph and mark direction
eliminate parameter
solve for x or y and substitute
finding tangent line
find dy/dx and then plug in point for tangent line y =mx +b
critical points
dy/dx = 0 or UND
polar coordinate formulas
x^2 + y^2 = r^2
x=rcosO
y=rsinO
tanO = y/x
Area of cardiod
1/2 Int. r^2 dO
Use symmetry
completing the square
ax^2+bx+c+(b/2)^2-(b/2)^2
