AP Calc BC Review

1

sin²x + cos²x \=

csc²x

1 + cot²x \=

sec²x

tan²x + 1 \=

(1-cos2x)/2

(power reducing) sin²x \=

(1+cos2x)/2

(power reducing) cos²x \=

x \= (-b ± √(b² - 4ac))/2a

Quadratic Formula

A \= 1/2(b₁+b₂)h

Area of a Trapezoid

A \= 1/2 bh

Area of a Triangle

A \= πr²

Area of a Circle

m \= (y₂ - y₁) / (x₂ - x₁)

Slope of a Line

y - y₁ \= m(x - x₁)

Point-Slope Form - Equation of a Line

0

ln(1)

x

e^lnx

odd function

sin x, odd or even?

even function

cos x, odd or even?

d \= √[( x₂ - x₁)² + (y₂ - y₁)²]

Distance Formula

1

lim (x→0) (sinx)/x

0

lim (x→0) (1-cosx)/x

lim(h→0) [f(x + h) - f(x)] /h

Limit Definition of Derivative, as h→0

lim (x→a) [f(x) - f(a)] / [x - a]

Limit Definition of Derivative, as x→a

0

lim (x→∞) (sinx)/x

e

lim (n→∞) [1+(1/n)]ⁿ

cu'

d/dx [cu]

u' ± v'

d/dx [u ± v]

uv' + vu'

d/dx [uv] (product rule)

\[vu' - uv'] /v²

d/dx [u/v] (quotient rule)

0

d/dx [c]

nuⁿ⁻¹ × u'

d/dx [uⁿ] (chain rule)

u'/u

d/dx [ln u]

u'e^u

d/dx [e^u]

u'/(ln a)u

d/dx [log_a(u)]

u'(ln a)a^u

d/dx [a^u]

(cos u)u'

d/dx [sin u]

-(sin u)u'

d/dx [cos u]

(sec²u)u'

d/dx [tan u]

-(csc²u)u'

d/dx [cot u]

(sec u tan u)u'

d/dx [sec u]\=

-(csc u cot u)u'

d/dx [csc u]\=

1/[f'(f⁻¹(x))]

(f⁻¹(x))' or d/dx [f⁻¹(x)]

u'/√(1-u²)

d/dx [arcsin u]

-u'/√(1-u²)

d/dx [arccos u]

u'/(1+ u²)

d/dx [arctan u]

-u'/(1+ u²)

d/dx [arccot u]

u'/[ |u| √(u² -1) ]

d/dx [arcsec u]

-u'/[ |u| √(u² -1) ]

d/dx [arccsc u]

∫f(u) du ± ∫g(u) du

∫[f(u) ± g(u)] du

\[uⁿ⁺¹]/ [n+1] + C, n≠1

∫uⁿ du

sinu + C

∫cosu du

-cosu + C

∫sinu du

tanu + C

∫sec² u du

secu + C

∫secu tanu du

-cotu + C

∫csc² u du

-cscu + C

∫cscu cotu du

e^u + C

∫e^u du

(a^u/ ln a) + C

∫a^u du

-ln |cosu| + C

∫tanu du

ln |sinu| + C

∫cotu du

ln |secu + tanu| + C

∫secu du

-ln |cscu + cotu| + C

∫cscu du

ln |u| + C

∫1/u du

ulnu - u + C

∫lnu du

arcsin(u/a) + C

∫du/ √(a² - u²)

(1/a)arctan(u/a) + C

∫du/ (a² + u²)

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

∫du/ [u√(u² - a²)]

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

Average Rate of Change of f(x) on [a,b]

If f is continuous on [a,b] and differentiable on (a,b) then there exists a number c such that f'(c) \= (f(b) - f(a)) / (b-a).

Mean Value Theorem (MVT)

If f is continuous on [a,b] and k is any number between f(a) and f(b), then there exists at least one number c such that f(c) \= k

Intermediate Value Theorem (IVT)

f is continuous at c iff

f(c) is defined

lim (as x approaches c) f(x) exists

lim (as x approaches c) f(x) = f(c)

Definition of Continuity

Let f be defined at c. If f'(c) \= 0 or if f' is undefined at c, then c is a critical number of f.

Critical Number

Let c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows:


1. If f'(x) changes from negative to positive at c, then f(c) is a relative minimum of f.
2. If f'(x) changes from positive to negative at c, then f(c) is a relative maximum of f.

First Derivative Test

Let f be a function such that f'(c) = 0 and the second derivative of f exists on the open interval containing c.


1. If f"(c)>0, then f(c) is a relative minimum.
2. If f"(c)

Second Derivative Test

Let f be differentiable on an open interval I. The graph of f is concave upward on I if f' is increasing on the interval and concave downward on I if f' is decreasing on the interval.

Concavity

Let f be a function whose second derivative exists on an open interval I.


1. If f"(c) > 0 for all x in I, then the graph of f is concave upward in I.
2. f f"(c) < 0 for all x in I, then the graph of f is concave downward in I.

Test for Concavity

A function f has an inflection point (c, f(c)) if


1. f"(c) = 0 or does not exist
2. f"(c) changes sign at x = c or if f' changes from increasing to decreasing or vice versa at x = c

Inflection Point

∫ (from a to b) f(x) \= F(b)-F(a)

First Fundamental Theorem of Calculus

d/dx ∫ (from a to g(x)) f(t)dt\= f(g(x))*g'(x)

Second Fundamental Theorem of Calculus

1/(b - a) ∫(from a to b) f(x)dx

Average Value of f(x) on [a,b]

V\= π∫(from a to b) [r(x)]² dx

Volume around a horizontal axis by discs

V\= π∫(from a to b) ([R(x)]²-[r(x)]²)dx

Volume around a horizontal axis by washers

V\= ∫(from a to b) A(x)dx

Volume by cross sections perpendicular to the x-axis

v(t)\= s'(t) [where s(t) is position function]

Velocity

a(t) \= v'(t) \= s"(t)

Acceleration

∫(from a to b) v(t)dt

Displacement

∫(from a to b) ||v(t)|| dt

Total Distance (from a to b)

∫u dv \= uv - v du

Integration by Parts

s \= ∫(from a to b) √1+[f'(x)]² dx

Length of Arc for functions

dP/dt \= kP(L-P)

Logistic Growth Formula

P\= L/[1+Ce^(-Lkt)]

Solution of Logistics Differential Equation

|R(n)|≤ | [M/(n+1)!]* (b-c)^(n+1) |

LaGrange Formula for Error Bound

dy/dx = (dy/dt)/(dx/dt)

d²y/dx² = \[d/dt (dy/dx)\] /(dx/dt)

Parametrics

(x(t), y(t))

Position Vector

(x'(t), y'(t))

Velocity Vector

(x"(t), y"(t))

Acceleration Vector

||v(t)||\= √[(dx/dt)^2 + (dy/dt)^2]

Speed (Magnitude of Velocity Vector)

s\= ∫(from a to b) √[(dx/dt)²+ (dy/dt)²] dt

Distance traveled from t\=a to t\=b (arc length)

x= rcosθ, y= rsinθ, x² + y² = r², tanθ= y/x

Relationship of Polar and Rectangular

dy/dx\= (rcosθ+ r'sinθ)/ (-rsinθ + r'cosθ) or

Slope of a Polar Curve

A\= (1/2) ∫ (from alpha to beta) r² dθ

Area of a Polar Curve

lim (n→∞) of a{n} ≠ 0, then the series diverges

nth term test for divergence

Σ(n \= 0 to ∞) arⁿ \= a/(1-r) provided |r|

Geometric Series