ap calc bc

Definition of e

e = lim_{n→∞}(1 + 1/n)^n

Absolute value

|x| = x if x ≥ 0; |x| = -x if x < 0

Definition of the derivative

f'(x) = lim_{h→0} [f(x+h) - f(x)] / h

Alternate derivative form

f'(a) = lim_{x→a} [f(x) - f(a)] / (x - a)

Continuity

f is continuous at c if f(c) exists, lim f(x) exists, and both are equal

Average rate of change

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

Average value of a function

(1 / (b - a)) ∫_a^b f(x) dx

Rolle's Theorem

If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then f'(c)=0

Mean Value Theorem

f'(c) = (f(b) - f(a)) / (b - a)

Intermediate Value Theorem

If f is continuous on [a,b], it hits every value between f(a) and f(b)

sin(2x)

2 sin x cos x

cos(2x)

cos²x - sin²x

cos(2x) alternate

1 - 2 sin²x

cos(2x) alternate

2 cos²x - 1

sin²x

(1 - cos(2x)) / 2

cos²x

(1 + cos(2x)) / 2

Derivative power rule

d/dx[x^n] = n x^{n-1}

Product rule

(fg)' = f'g + fg'

Quotient rule

(u/v)' = (v u' - u v') / v²

Derivative of sin u

( sin u )' = cos u · u'

Derivative of cos u

( cos u )' = -sin u · u'

Derivative of tan u

( tan u )' = sec²u · u'

Derivative of cot u

( cot u )' = -csc²u · u'

Derivative of sec u

( sec u )' = sec u tan u · u'

Derivative of csc u

( csc u )' = -csc u cot u · u'

Derivative of ln u

(ln u)' = u'/u

Derivative of a^u

(a^u)' = a^u ln(a) · u'

Derivative of e^u

(e^u)' = e^u u'

Derivative of arcsin u

u' / √(1 - u²)

Derivative of arccos u

• u' / √(1 - u²)

Derivative of arctan u

u' / (1 + u²)

Derivative of arccot u

• u' / (1 + u²)

Derivative of arcsec u

u' / (|u| √(u² - 1))

Derivative of arccsc u

• u' / (|u| √(u² - 1))

Inverse function derivative

(f^{-1})'(a) = 1 / f'(f^{-1}(a))

∫ sin u du

• cos u + C

∫ cos u du

sin u + C

∫ sec²u du

tan u + C

∫ csc²u du

• cot u + C

∫ sec u tan u du

sec u + C

∫ csc u cot u du

• csc u + C

∫ du/u

ln|u| + C

∫ tan u du

• ln|cos u| + C

∫ cot u du

ln|sin u| + C

∫ sec u du

ln|sec u + tan u| + C

∫ csc u du

• ln|csc u + cot u| + C

∫ e^{au} du

(1/a) e^{au} + C

∫ a^u du

a^u / ln(a) + C

∫ du / √(a² - u²)

arcsin(u/a) + C

∫ du / (a² + u²)

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

∫ du / (u √(u² - a²))

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

Critical number

c is critical if f'(c)=0 or undefined

First derivative test

−→+ gives min, +→− gives max

Second derivative test

f''(c)>0 min; f''(c)<0 max

Concavity (definition)

Concave up if f' increasing; concave down if f' decreasing

Concavity test

f''>0 concave up; f''<0 concave down

Inflection point

Where f'' changes sign

FTC Part 1

∫_a^b f(x) dx = F(b) - F(a)

FTC Part 2

d/dx ∫_a^x f(t) dt = f(x)

FTC chain rule

d/dx ∫_a^{g(x)} f(t) dt = f(g(x)) g'(x)

Volume (disks)

V = π ∫ [r(x)]² dx

Volume (washers)

V = π ∫ ([R(x)]² - [r(x)]²) dx

Volume (shells)

V = 2π ∫ r(y) p(y) dy

Volume (cross-sections)

V = ∫ A(x) dx

Velocity

v(t) = s'(t)

Acceleration

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

Displacement

∫ v(t) dt

Total distance

∫ |v(t)| dt