AP Calculus AB Flashcards

AB Content ONLY

Linear function

y = x

Quadratic Function

y = x^2

Cubic Function

y = x^3

Absolute Value Function

y = |x|

Absolute Value Function

y = sqrt(x^2)

Square Root Function

y = sqrt(x)

Reciprocal Function

y = 1/x

Sine Function

y = sin(x)

Cosine Function

y = cos(x)

Tangent Function

y = tan(x)

Natural Base Function

y = e^x

Natural Logarithm

y = ln(x)

Reciprocal Squared Function

y = 1 / x^2

Semicircle Function

y = sqrt(a^2 - x^2)

y = |x| / x

x = ±1

ln e

1

ln 1

0

ln(mn)

ln m + ln n

ln(m/n)

ln m - ln n

ln(m^n)

n ln m

e^m * e^n

e^(m+n)

(a + b)^2

a^2 + 2ab + b^2

(a - b)^2

a^2 - 2ab + b^2

(a + b)(a - b)

a^2 - b^2

csc(x)

1/sin(x)

sec(x)

1/cos(x)

cot(x)

cos(x)/sin(x)

tan(x)

sin(x)/cos(x)

cot(x)

1/tan(x)

sin^2(x) + cos^2(x)

1

1 + tan^2(x)

sec^2(x)

1+cot^2(x)

csc^2(x)

Even function

Symmetric about the y-axis

Odd function

Symmetric about the origin

Vertical Asymptote

lim (x -> c) f(x) = ±∞

Horizontal Asymptote

lim (x -> ±∞) f(x) = c

When we can use dominant terms to find a limit?

When x approaches positive or negative infinity

Conditions for Continuity

1. lim (x -> c) f(x) exists

2. f(c) is defined

3. lim (x -> c) f(x) = f(c)

Conditions for Intermediate Value Theorem

1. f(x) is continuous on [a, b]

2. f(a) ≠ f(b)

3. By IVT, there exists a value c where f(c) = k on the domain a ≤ c ≤ b and the range f(a) ≤ k ≤ f(b) since f(x) takes on every value in the interval

f’(x)

f'(a) = lim (h -> 0) [ f(a + h) - f(a) ] / h

Point-Slope Form

y - y1 = m(x - x1)

d/dx[x^n]

n*x^(n-1)

d/dx[u(x)v(x)]

u'(x)v(x) + u(x)v'(x)

d/dx[u(x)/v(x)]

(u'v - uv')/v^2

d/dx[sin(u)]

cos(u) * u’

d/dx[cos(u)]

-sin(u) * u’

d/dx[tan(u)]

sec^2(u) * u’

d/dx[cot(u)]

-csc^2(u) * u’

d/dx[sec(u)]

sec(u)tan(u) * u’

d/dx[csc(u)]

-csc(x)cot(x) * u’

d/dx[e^u]

e^u * u’

d/dx[ln(u)]

u’ / u

d/dx[a^u]

a^u * ln(a) * u’

d/dx[log_a(u)]

u’ / (u * ln(a))

d/dx[arcsin(u)]

u’ / sqrt(1 - x²)

d/dx[arccos(u)]

u’ / sqrt(1 - x²)

d/dx[arctan(u)]

u’ / (1 + u²)

d/dx[arccot(u)]

-u’ / (1 + u²)

d/dx[arcsec(x)]

u' / |u|sqrt(u² - 1)

d/dx[arccsc(u)]

-u' / |u|sqrt(u² - 1)

d/dx[sqrt(u)]

u' / (2sqrt(u))

d/dx[1/x^2]

-2/x^3

f’(x) > 0

Function is decreasing

f’(x) < 0

Function is increasing

f’(x) = 0 or DNE

Critical values of x

f’(x) = 0 or DNE and sign of f’(x) changes from + to -

Relative maximum

f’(x) = 0 or DNE and sign of f’(x) changes from - to +

Relative minimum

When do we need to check endpoints?

When finding absolute max or min

Position function

x(t)

Velocity function

v(t) = x'(t)

Acceleration function

a(t) = v’(t) = x''(t)

Speed

| velocity |

Particle is speeding up

when the velocity and acceleration have the same sign

Particle is slowing down

when the velocity and acceleration have different signs

Particle is moving rightward

when velocity is positive

Particle is moving leftward

when velocity is negative

Function is concave up

f’’(x) > 0

Function is concave down

f''(x) < 0

Conditions for Rolle’s Theorem

1. Continous on closed interval [a, b]

2. Differentiable on open interval (a, b)

3. f(a) = f(b)

   1. There is at least one number x = c in (a, b) such that f’(c) = 0

Right Riemann Sum (RRAM)

(b - a) / n (n is number of right outputs)

Left Riemann Sum (LRAM)

(b - a) / n (n is number of left outputs)

Midpoint Riemann Sum (MRAM)

(b - a) / n (n is number of midpoints of outputs)

Trapezoidal Rule for Area

(b - a) / n * (f(a) + f(b) + 2 * sum of f(xi)) for n subintervals.

∫ 5 dx

5x + C

∫ sin(x) dx

-cos(x) + C

∫ cos(x) dx

sin(x) + C

∫ sec^2(x)

tan(x) + C

∫ csc^2(x)

-cot(x) + C

∫ sec(x)tan(x) dx

sec(x) + C

∫ csc(x)cot(x)

-csc(x) + C

∫ 1/x dx

ln|x| + C

∫ x^-1 dx

ln|x| + C

∫ x^-2 dx

1/x + C

∫ e^x

e^x + C

∫ 3^x dx

(3^x / ln(3)) + C

∫ ln(u) du

u ln(u) - u + C

∫ sin(kx) dx

-1/k cos(kx) + C

∫ e^(kx) dx

(1/k)e^(kx) + C

∫ 1 / (3x + 4) dx

(1/3) ln|3x + 4| + C

∫ 1 / (1 + x²) dx

arctan(x) + C