AP Calc AB/BC Final Identity Quizlet

lim x->0 sinx/x =

1

Implicit differentiation

When taking d/d__ of a function which has y in it, when you do that you take the ____ but then multiply the ____ (for that __ variable) with ____

x, derivative, derivative, y, dy/dx

log(a/b) =

(same with ln)

loga - logb

Polar area = ___∫(theta2, theta1) ___d(___)

0.5, r², theta

S(1/(x+b) - 1/x) =

S(1/(x+b) + 1/x) =

ln((x+b)/x)

ln(x² + bx)

Slope fields

If only in terms of x, all _____ columns are same

If only in terms of y, all _____ columns are same

vertical, horizontal

Eulers method

Line formula

x, y: x, y value at ____ ___

xapprox, yapprox: x, y value ____ _____

yapprox=(____ (value calculated at __)) * (__−____​) + ____​

When going backwards, the ________ will be ______ which works that out

current point, being approximated dy/dx, x, xapprox - x, y, xapprox - x, negative

area of a sector =

0.5r² * theta

speed vector = sqrt((__(t)))² + (__(t))²)

x’, y’

Partial Fractional Decomposition

S(f(x)/g(x))dx = S(_/_____)dx + S(_/_____)dx +…

= (__/__)(ln|____| + (__/__)(ln|____| +…

A, ax + b, B, cx + d, A/a, ax + b, B/c, cx+d

Integration by parts: ∫f___ =

f: function thats easier to ______

g: function thats easier to ______

Choosing f: use LIATE

L- _______

I- ________ _______

A- ________

T- ________

E- ________

g’, fg - Sgf’, differentiate, integrate, logarithmic, inverse trigonometric, algebraic, trigonometric, exponential

for polar equations, dy/dx = (_____)__(_____)

dy/dtheta, /, dx/dtheta

for parametric/polar equations, d²y/dx² = ____(____) = ((_____)(_____))/(_____). This is the same for ___ equations and taking the ___ derivative

d/dx, dy/dx, d/dt, dy/dx, dx/dt, polar, second

For parametrics, vertical tangent lines may occur where ____ = 0 while horizontal tangent lines may occur where _____ = 0

These tangents don’t occur at the non-______ parts of a not ______ _______

dx/dt, dy/dt, smooth, smooth curve

Arc length of a ______ curve: (using _____ equations) from a to b

S(lower term ___, upper term ___) sqrt((____)² + (____)²)dt

smooth, parametric, a, b, dx/dt, dy/dt

∫speed(t)dt =

distance traveled

∫(t2, t1) v(t)dt =

displacement

d/dx h(f(g(x)))

h’(f(g(x)))f’(g(x))g’(x)

a^x = b^((___)log(base __ of __))

(same with ln)

x, b, a

lim x->0 (cosx-1/x) =

0

rcos(theta) =

x

rsin(theta) =

y

a^(x+y) =

a^x * a^y

a^(x-y) =

a^x/a^y

(e^x)^y =

e^xy

Quotient rule

f'(x)g(x)-g'(x)f(x)/g(x)^2

d/dx a^x =

a^x * ln(a)

∫a^x =

a^x/ln(a)

d/dx loga(x) =

dx/xlna

∫logax =

(x/lna)(lnx - 1) + c

tan =

sin/cos

csc =

1/sin

sec =

1/cos

cot =

cos/sin

d/dx tan =

sec^2

d/dx cot =

-csc^2

d/dx csc =

-csccot

d/dx sec

sectan

logb(a^r) =

(same with ln)

rlogb(a)

logb(ac) =

(same with ln)

logb(a) + logb(c)

∫(lower base a, upper base b) = -∫(lower base ___, upper base __)

b, a

a < b < c

∫(lower base a, upper base c) = ∫(lower base a, upper base __) + ∫(lower base ___, upper base ___)

b, b, c

ln(a(x))^b =

bln(a(x))

For riemann sums, xi = ______________________ and deltax = ____________________

xo + i(deltax), (b-a)/n

f avg for an integral bounded by [a, b] = ___________∫(lower bound __, upper bound __)__________

1/(b-a), a, b, f(x)dx

If g = f^-1, then g' =

1/f'(g)

∫lnx =

xlnx - x + C

d/dx arcsin x =

1/sqrt(1-x^2)

d/dx arccos x =

-1/sqrt(1-x^2)

d/dx arctan x =

1/(1 + x^2)

d/dx arccot x =

-1/(1 + x^2)

d/dx arcsec x =

1/(|x|sqrt(x^2 - 1))

d/dx arccsc x =

-1/(|x|sqrt(x^2 - 1))

For an even function, f(-x) = _________, and for an odd function, f(-x) = ____________

f(x), -f(x)

sin(a + b) =

sinacosb + cosasinb

sin(a - b) =

sinacosb - cosasinb

cos(a + b) =

cosacosb - sinasinb

cos(a - b) =

cosacosb + sinasinb

sin(2x) =

2sinxcosx

cos(2x) =

cos^2 x - sin^2 x

∫1/x

ln|x|

∫1/x^2

-1/x

∫-1/x^2

1/x

When f''x is greater then 0, critical points are ___________ ___________, and when f''x is less then 0, critical points are ___________ ___________

relative minimums, relative maximums

Linear approximation formula: l(x) =

a = value near x

f = function

x = value for l(x) to be found at

f(a) + f'(a)(x - a)

Σ(i =1, n) i =

(n(n+1))/2

Σ(i =1, n) i^2 =

(n(n+1)(2n+1))/6

Σ(i =1, n) i^3 =

(n^2 (n+1)^2 )/4

∫du/(a^2 + u^2) =

1/a arctan (u/a) + c

∫du/sqrt(a^2 - u^2) =

1/a arcsin (u/|a|) + c

∫du/(u * sqrt(u^2 - a^2)) =

1/|a| arcsec (|u|/a) + c

Area between curves f(x) and g(x) (thru [a, b], f(x) > g(x)) formula = ∫(lower bound __, upper bound __)______________

a, b, f(x) - g(x)dx

Volume of cylinder =

pir^2h

Volume of cone =

1/3 pir^2h

Volume of pyramid =

1/2 lwh

If arrows in a slow field poitn away from the solution, the solution is asymptotically __________, but if they point towards the solution, the solution is asymptotiaclly ____________

unstable, stable

Exponential decay formula =

Exponential growth formula =

k(like what does it represent) =

y = yo e^-kt, y = yo e^kt, growth constant

Doubling time/Half life =

ln2/k

Compounded interest Balance formula: Pf =

r =

p =

Pe^rt, compounded interest rate, initial balance

Newtons law of cooling formula: T =

Ta =

(To - Ta)*e^-kt + Ta, outside temp

Disk method formula

V =

pi * ∫(lower bound a, upper bound b) (f(x))^2 dx

Washer method formula (f(x) above g(x))

V =

pi * ∫(lower bound a, upper bound b) (f(x)^2 - g(x)^2)dx

lnx = ∫(lowe bound __, upper bound __) ________dt

1, x, 1/t

a^x = (in terms of e and ln)

(same with log)

e^(xlna)

Usub

∫(lower bound a, upper bound b) f(g(x))g'(x)dx =

∫(lower bound ___, upper bound ___)__________

g(a), g(b), f(u)du

Trapezoidal sum formula:

((b-a)/2n)(f(xo) + 2f(x1) + 2f(x2)... + f(xn))

The Intermediate value theorem is used for ________, and states that if f is ________ over [a,b] and k is a number _______________ _______ and _______, there is at least 1 _________ c such that _________ = _____

lines, continuous, between f(a) and f(b), number, f(c)=k

The Mean value theorem is used for ______, and states that if f(x) is __________ and ___________ over [a, b], the slope of the _______ line (which is evaluated thru the formula ___________/__________) equals the slope of the _________ line at least once in the interval. ___________'s theorem is the same as this exxcept it applies when the average __________ _______ _______ over the interval is _____

derivatives, continues, differentiable, tangent, f(b) - f(a), b-a, secant, rolles, rate of change, zero

The mean value theorem part 2 for __________ states that if a function is _________ over interval [a, b], then it must cross over its ____________ calue at least ________ at some __-value in the interval

integrals, continuous, average, once, x

The extreme value theorem states that for a ____________ function on a bounded interval [a, b], there must be a _________ and ______________ value along this interval

continuous, maximum, minimum

For FTC part 1, the bounds of the integral F(x) = ∫(__, __)f(t)dt are

0, x

volume of triangular prism

(1/2bh)h

volume of sphere

4/3 * pir³

surface area of sphere

4pir²

surface area of cylinder

2pir² + 2pirh

If numerators degree < denominators degree, horizontal asymptote: y= __

If numerators degree = denominators degree, y = _______ ___ _________ _____________

If numerators degree > denominators degree, y = _________

If numerators degree is one greater than denominators degree, there is a ________ _________

0, ratio of leading coefficients, n/a, slant asymptote

d/dx cos^2 (x)

-2sinxcosx

d/dx sin^2 (x)

2sin xcos x

volume of revolution formula for

-semicircles: ____ x S ____

-rectangles: ____ x S____

-equilateral triangles: ___ x S____

pi/8, s^2, k, s^2, sqrt(3)/4, s^2

cos^2 (x) =

(1 + cos 2x) / 2