AP Calculus AB: Stuff You Must Have Memorized

log a + log b =

log(ab)

log a - log b =

log(a/b)

c * log a =

log(a^c)

log 0 =

undefined

log 1 =

0

ln e =

1

sin 0 =

0

sin π/6 =

1/2

sin π/4 =

√(2)/2

sin π/3 =

√(3)/2

sin π/2 =

1

sin π =

0

sin 3π/2

-1

cos 0 =

1

cos π/6 =

√(3)/2

cos π/4 =

√(2)/2

cos π/3 =

1/2

cos π/2 =

0

cos π =

-1

cos 3π/2 =

0

f is increasing when...

f' is positive

f is decreasing when...

f' is negative

f has a local maximum when...

f' changes from positive to negative

f has a local minimum when...

f' changes from negative to positive

f is concave up when...

f'' is positive (or f' is increasing)

f is concave down when...

f'' is negative (or f' is decreasing)

f has a point of inflection when...

f'' changes sign (or f' has a local max or min)

d/dx k =

0

d/dx (mx+b) =

m

d/dx x^n =

n*x^(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 e^x =

e^x

d/dx b^x =

b^x / ln b

d/dx ln x =

1/x

d/dx log_b(x)

1/(x ln b)

d/dx Arcsin x =

1 / √(1 - x²)

d/dx Arccos x =

-1 / √(1 - x²)

d/dx Arctan x =

1 / (1 + x²)

d/dx Arccot x =

-1 / (1 + x²)

d/dx Arcsec x =

1 / (|x|√(x² - 1))

d/dx Arccsc x =

-1 / (|x|√(x² - 1))

∫ k dx =

kx + C

∫ x^n dx =

x^(n+1) / (n+1) + C

∫ sin x dx =

-cos x + C

∫ cos x dx =

sin x + C

∫ sec² x dx =

tan x + C

∫ csc² x dx =

-cot x + C

∫ sec x tan x dx =

sec x + C

∫ csc x cot x dx =

-csc x + C

∫ 1/√(1 - x²) dx =

Arcsin x + C

∫ 1/(1 + x²) dx =

Arctan x + C

∫ 1/[x√(x² - 1)] dx =

Arcsec x + C

∫ e^x dx =

e^x + C

∫ b^x dx =

b^x * ln b + C

∫ 1/x dx =

ln |x| + C

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

f'(x) g(x) + f(x) g'(x)

(Product Rule)

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

[g(x) f'(x) - f(x) g'(x)] / g(x)²

(Quotient Rule)

"Lo-D-Hi minus Hi-D-Lo,

Square the bottom and off you go!"

d/dx f(g(x)) =

f'(g(x))g'(x)

∫ₐᵃ f(x) dx =

0

∫ᵦᵃ f(x) dx =

-∫ₐᵇ f(x) dx

(The sign flips)

d/dx ∫ₐˣ f(t) dt =

f(x)

(Fundamental Theorem of Calculus, part II)

Average value of a function on [a,b]

(∫ₐᵇ f(x) dx) / (b-a)

Arrange these in order of slowest to fastest growth:

Exponential, Logarithmic, Polynomial

Logarithmic, Polynomial, Exponential

A particle is moving to the right when...

Its velocity is positive. (v(t) > 0)

A particle is moving to the left when...

Its velocity is negative. (v(t) < 0)

A particle is at rest when...

Its velocity is zero. (v(t) = 0)

A particle is speeding up when...

Its velocity and acceleration have the same sign.

A particle is slowing down when...

Its velocity and acceleration have different signs.

The integral of a rate is...

An amount of change.

What are two meanings of the derivative?

* The slope of the tangent line to a curve

* The instantaneous rate of change of a function

What are two meanings of the integral?

* The signed area between a curve and the x-axis

* The accumulation of a rate of change over time

What is the formula for average rate of change?

[f(b) - f(a)] / (b-a)

What is the derivative of position?

Velocity

What is the derivative of velocity?

Acceleration

What is the integral of acceleration?

Change in velocity

What is the integral of velocity?

Change in position (displacement)

What is the relationship between speed and velocity?

speed = |velocity|

How do you calculate the DISTANCE traveled by a particle?

Take the integral of the SPEED.

∫ₐᵇ |v(t)| dt

What do you do on your calculator to calculate a derivative at a point?

Hit the [MATH] button, scroll down to 8:nDeriv(

What do you do on your calculator to calculate a definite integral?

Hit the [MATH] button, scroll down to 9:fnInt(

Where does a function f have critical points?

Where f' is zero or undefined

What are the two limit formulas for the derivative?

Derivative at a point:

lim[x→a] [f(x) - f(a)] / (x-a)

Derivative as a function:

lim[h→0] [f(x+h) - f(x)] / h

What is the area of a circle?

A = πr²

What is the circumference of a circle?

C = 2πr or πd

What is the area of a triangle?

A = 1/2 b h

What is the Quadratic Formula?

x = [-b ± √(b² - 4ac)] / (2a)