AP Calc BC 1st Semester Exam Review- HARDCORE

everything u and ur mom need to know so you don't bomb the final, bitch. study tf up

Center of mass- equation of mass

integral f(x)-g(x) dx

Center of mass- My

integral x\[f(x)-g(x)\] dx

Center of mass- Mx

integral 1/2\[f(x)²-g(x)²\] dx

Center of mass- x center of mass

My/m

Center of mass- y center of mass

Mx/m

What is M?

Sum of individual moments of x or y axes

Arc length formula

integral sq rt\[1+f’(x)²\] dx

Surface area x-axis

integral 2pi f(x) \[arc length for f’(x)\]

Surface area y-axis

integral 2pi f(y) \[arc length for f’(y)\]

First 3 steps to approaching an integral?

Recognize it, algebra, u-substitution

5 basic approaches

U-substitution, partial fractions, long division, completing square, multiplying by 1

sin2x=

2sinxcosx

cos2x=

cos²x-sin²x, 1-2sin²x, 2cos²x-1

Integration by parts formula

integral udv = uv- (integral vdu)

Integral of lnx

xln - x + C

Power reducing formula cos²x

(1+cos2x)/2

Power reducing formula tan²x

(1-cos2x)/(1+cos2x)

Power reducing formula sin²x

(1-cos2x)/2

What is an improper integral?

Integral where one or both of the bounds is infinity

If the limit of an improper integral exists?

Converges

If the limit of an improper integral doesn’t exist?

Diverges

Trig substitution a²-x²

x = asin(theta)

Trig substitution x²-a²

x = asec(theta)

Trig substitution a²+x²

x = atan(theta)

How to rationalize substitutions?

u = sq rt of an expression, square both sides and find derivative, substitute for x and dx and use long division/partial fractions to solve

Weierstrass substitution dx=?

2/(1+t²)

sinx and cosx trig identity

sin²x + cos²x = 1

secx and tanx trig identity

sec²x - tan²x = 1

cscx and cotx trig identity

csc²x - cot²x = 1

Formula for geometric series

\[a(1-r^n)\]/(1-r)

Formula for infinite geometric series

a/(1-r)

Geometric series converges:

|r| < 1

Geometric series diverges:

|r| > 1 or = 1

Divergence test diverges:

lim ak is not 0

Divergence test is inconclusive:

lim ak = 0

Integral test converges:

integral ak < infinity

Integral test diverges:

integral ak = infinity

P-series form

1/k^p

P-series converges:

p > 1

P-series diverges:

p < 1 or = 1

Ratio test converges:

lim a(k+1)/ak < 1

Ratio test diverges:

lim a(k+1)/ak > 1

Root test converges:

lim (ak)^1/k < 1

Root test diverges:

lim (ak)^1/k > 1

Root test and ratio test are inconclusive if lim=?

1

Limit comparison test converges:

0

Limit comparison test diverges:

lim ak/bk > 0 and bk DIVERGES

Alternating series form

(-1)^k(ak)

Alternating series converges:

lim ak = 0

Alternating series diverges:

lim ak is not 0

Alternating series converges absolutely:

|ak| converges

Alternating series converges conditionally:

|ak| diverges and ak converges

Alternating series diverges absolutely:

ak diverges (by divergence test)

d/dx cotx

\-csc²x

d/dx secx

secxtanx

d/dx cscx

\-cscxcotx

integral tanx

ln|secx| + C

integral secx

ln|secx + tanx| + C

weierstrass substitution t=?

t=tan1/x