Ap Calculus Formula Test

FOR THE FELLOW AP CALC PEEPS

d/dx (x^n)= ____________

nx^n-1

d/dx (fg)= ____________

f’(g(x))g’(x)

d/dx (f/g)= ____________

f(g^-1)

d/dx f(u)= ____________

f‘(u)u’

d/dx (e^u)= ____________

e^u

d/dx (a^u)= ____________

a^ulna

d/dx (ln u)= ____________

u’/u

d/dx (log _*a*_ u)= ____________

1/xlna

d/dx (sin u)= ____________

cosu

d/dx (cos u)= ____________

\-sinu

d/dx (tan u)= ____________

sec^2u

d/dx (cot u)= ____________

\-csc^2u

d/dx (sec u)= ____________

sec u tan u

d/dx (csc u)= ____________

\-csc x cot x

d/dx (arcsin x)= ____________

1/√1-x^2

d/dx (arctan x)= ____________

1/1+x^2

∫ a dx\= \____________

a∫dx → a(x+c_1) → ax + ac_1 → ax + c

∫ x^n dx= ____________

x^n+1/n+1 + c

∫ dx\= \____________

x + c

∫ u'/u dx= ____________

ln|u| + c

∫ e^x dx= ____________

e^x + c

∫ a^x dx= ____________

a^x/lna + c

∫ sin x dx\= \____________

\-cos x + c

∫ cos x dx\= \____________

sin x + c

∫ tan x dx\= \____________

\-ln |cosx| + c

∫ cot x dx\= \____________

ln|sinx| + c

∫ sec^2 x dx= ____________

tan x + c

∫ sec x tan x dx\= \____________

sec x + c

∫ csc^2 x dx= ____________

\-cot x + c

∫ csc x cot x dx\= \____________

\-csc x + c

∫ u'/a^2+u^2 dx= ____________

1/a arctan u/a + c

∫ u'/ √a^2-u^2 dx= ____________

arcsin u/a + c

Fundamental Theorem of Calculus:

∫ab f'(x)dx=_________

f(b) -f(a)

Second Fundamental Theorem of Calculus:

d/dx ∫uv f(t)dt=_________

f(v)v’ - f(u)u’

Area Between 2 Curves

For functions of x:

**A=**

________ curve - _______ curve

A= ∫a/b {f(x) - g(x)} dx

top - bottom

Area Between 2 Curves

For functions of y:

**A=**

________ curve - _______ curve

A= ∫a/b \[f(y) - g(y)\]dy

right curve - left curve

Volumes of Revolution:

Disks-

V=

v= 𝝅∫a/b r^2 dx (or dy)

\

Volumes of Revolution:

Washers-

V=

v=𝝅∫a/b(R^2 - r^2) dx (or dy)

\

Volumes of known Cross Sections

V=

V= ∫a/b Adx or dy where A represents the area of a representative cross section

Position-Velocity-Acceleration

d/dt s(t)=

v(t)

Position-Velocity-Acceleration

d/dt v(t)=

a(t)

Position-Velocity-Acceleration

∫ a(t)dt=

v(t) + c

Position-Velocity-Acceleration

∫ v(t)dt=

s(t) + c

Position-Velocity-Acceleration

Speed = ___________

|v(t)|

Position-Velocity-Acceleration

Displacement = ___________

∫a/b v(t) dt

Position-Velocity-Acceleration

Total Distance Traveled

∫a/b |v(t)|dt

Tangent Line: \________________________

y-y_1 = m(x-x_1)

First Derivative Test:

f is increasing when: _______

f’(x) > 0

First Derivative Test:

f is decreasing when: _______

f’(x) < 0

Points of Inflection:

f is concave upward when: __________

f’’(x) > 0

Points of Inflection:

f is concave downward when: ___________

f’’(x) < 0

Average Value of a Function: favg\= \_____________

favg = ∫a/b f(x) dx / b-a

Average Rate of Change: AROC\= \_____________

m= f(b) - f(a) / b-a

Instantaneous Rate of Change: IROC\= \_____________

Use the derivative (slope at one point)