AP Calc AB Exam Flashcards

d/dx (xⁿ)

nx^n-1

d/dx (sin x)

cos x

d/dx (cos x)

-sin x

d/dx (tan x)

sec² x

d/dx (cot x)

-csc² x

d/dx (sec x)

sec x tan x

d/dx (csc x)

-csc x cot x

d/dx (ln u)

1/u du/dx

d/dx (e^u)

e^u du/dx

If f has an inverse function g then:

g’(x) = 1/f’(g(x)) because derivatives are reciprocal slopes

Average Rate of Change

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

Instantaneous Rate of Change

f’(x)_h-0 = lim f(x+h)-f(x)/h

Local Minimum dy/dx goes from

Negative to Positive

Local Minimum d²y/dx² Derivative

Greater than 0

Local Maximum dy/dx goes from

Positive to Negative

Local Maximum d²y/dx² Derivative

Less than 0

Point of Inflection

Concavity Changes

f’(x)>0

function is increasing

f’(x)<0

function is decreasing

f’’(x)>0

concave up

f’’(x)<0

concave down

Relative Maximum 2nd Derivative

f’’(x)<0

Relative Minimum 2nd Derivative

f’’(x)>0

write the equation of a tangent line

y-y₁= m(x-x₁)

If acceleration and velocity have the same sign then

the speed is increasing

If acceleration and velocity have different signs then

the speed is decreasing

The particle is moving right when velocity is

positive

The particle is moving left when velocity is

negative

Displacement

Integral v(t)dt

Total Distance

Integral final time over initial time (Absolute value of v(t))dt

Average Velocity

final position-initial position/total time

Accumulation

x(0)+Integral v(t)dt

ln N = p

e^p =N

ln e

1

ln 1

0

ln(MN)

ln M + ln N

ln (M/N)

ln M - ln N

p times ln M

ln M^p

1st Fundamental Theorem of Calculus

Int b over a f(x)dx =F(b)-F(a)

Average Value of Integral

1/b-a Integral b over a f(x)dx

sin (0)

0

cos (0)

1

tan (0)

0

sin (pi/6)

1/2

cos (pi/6)

root 3/2

tan (pi/6)

root 3/3

sin (pi/4)

root 2/2

cos (pi/4)

root 2/2

tan (pi/4)

1

sin (pi/3)

root 3/2

cos (pi/3)

1/2

tan (pi/3)

root 3

sin (pi/2)

1

cos (pi/2)

0

tan (pi/2)

infinite

sin (pi)

0

cos (pi)

-1

tan (pi)

0

Trapezoidal Sum

1/2h(b₁ + b₂)

sin² x+cos² x

1

1+tan² x

sec² x

cot² x +1

csc² x

tan x

sin x/cos x

cot x

cos x/sin x

csc x

1/sin x

sec x

1/cos x

Integral du

u+C

Integral e^u du

e^u+C

Integral sin (u) du

-cos u + C

Integral cos (u) du

sin u + C

Integral tan (u) du

-1/cos u (absolute value) + C

Integral cot (u) du

1/sin u (absolute value) + C

Integral sec (u) du

1/sec u + tan u (absolute value) + C

Integral csc (u) du

-1/csc u + cot u (absolute value) + C

Integral sec² (u) du

tan u + C

Integral csc² (u) du

-cot u + C

Integral sec (u) tan (u) du

sec u + C

Integral csc (u) cot (u) du

-csc u + C

Square Cross Section

Integral (base)² dx

Equilateral Triangle Cross Section

root ¾ Integral (base)² dx

Isosceles Right Triangle Cross Section

¼ Integral (base)² dx

Rectangle Cross Section

Integral (base times height) dx

Semi-Circle Cross Section

pi/2 Integral (radius)² dx