Calc BC Cram Session!!!

Average value of a function f(x)

Average value of a function f(x)

1/(b-a) * ∫ f(x) dx from a to b

Average rate of change of f(x) from a to b

f(b) - f(a) / b-a

Polar to rectangular coordinates: x=?

rcosθ

Polar to rectangular coordinates: y=?

rsinθ

Rectangular to polar coordinates: r=?

v(x²+y²)

Rectangular to polar coordinates: θ=?

arctan(y/x)

Area of a sector of a circle

½θr²

Area of a region defined by a polar function

½∫r²dθ from θ1 to θ2

Area between two polar curves

½ ∫r_{outer curve}²-r_{inner curve}²dθ from θ1 to θ2

Displacement of f(x)

∫f’(x)dx from a to b

Distance of f(x)

∫|f’(x)|dx from a to b

Speed of f(x)

|f’(x)|

Arclength of f(x)

∫v(1+[f’(x)]²)dx from a to b

Displacement of ⟨x(t), y(t)⟩

Distance of ⟨x(t), y(t)⟩

Speed of ⟨x(t), y(t)⟩

v((x’(t))²+(y’(t))²)

Arc length of ⟨x(t), y(t)⟩

∫x^a dx

[x^(a+1)] / (a+1) + c

∫e^x dx

e^x + c

∫lnx dx

xlnx - x + c

∫cosx dx

sinx + c

∫secxtanx dx

secx + c

∫csc²x dx

-cotx + c

∫secx dx

ln|secx + tanx| + c

∫cscx dx

ln|cscx - cotx| + c

∫1/(a²+x²) dx

1/a arctan(x/a) + c

∫1/x dx

ln|x| + c

∫a^x dx

a^x/lna + c

∫sinx dx

-cos(x) + c

∫sec²x dx

tanx + c

∫cscxcotx dx

-cscx + c

∫tanx dx

ln|secx| + c

∫cotx dx

ln|sinx| + c

∫1/v(a²-x²) dx

arcsin(x/a) + c

∫1/xv(x²-a²) dx

1/a * arcsec(x/a) + c

In the exponential model y=Ce^(Kx), what is y?

Final amount

In the exponential model y=Ce^(Kx), what is C?

Initial amount

In the exponential model y=Ce^(Kx), what is K?

Growth/decay constant

In the exponential model y=Ce^(kx), what is x?

Time

In the logistic model y = a/(1+be^(-aKt)), what is a?

Carrying capacity

What does a geometric series look like?

When does a geometric series diverge?

When |r| ≥ 1

How do you find the sum that a geometric series converges to?

Using the N-th term test, when does a series diverge?

If the limit as n approaches infinity of a_n does not equal 0

What are the criteria for f(x) in the integral test for convergence?

f(x) must be positive, continuous, and decreasing

When does a series converge according to the p series test?

When p>1

When does a series diverge according to the p series test

When 0<p≤1

What is it called when p=1 in a p series?

Harmonic series

What does a p series look like?

What is the first condition for an alternating series to converge?

The limit as n approaches infinity of a_n = 0

What is the second condition for an alternation series to converge?

a_n+1 ≤ a_n (a_n are non-increasing)

How is the error for an estimate of an alternating series found?

error ≤ |a_n+1|

sinx’ = ?

cosx

cosx’ = ?

-sinx

tanx’ = ?

sec²x

secx’ = ?

secxtanx

cotx’ = ?

-csc²x

cscx’ = ?

-cscxcotx

arcsinx’ = ?

1/v(1-x²)

arccosx’ = ?

-1/v(1-x²)

arctan’ = ?

1/(1+x²)

arccotx’ = ?

-1/(1+x²)

arcsecx’ = ?

1/(|x|v(x²-1))

arccscx’

-1/(|x|v(x²-1))

lnx’ = ?

1/x

log_ax’ = ?

1/xlna

e^x’ = ?

e^x

a^x’ = ?

a^xlna