AP Calculus AB

dy/dx

derivative of y with respect to x

d/dx

takes the derivative of f(x) with respect to x

line tangent

y + “y value” = slope(x - (“x value”))

continuity 

drawing a graph without lifting your pencil

d²y / dx²

take the second derivative — f’’(x)

a function is not continuous (discontinuous) if you

1. divide by zero

2. take a square root of a negative number

3. take ln of zero or a negative number

4. piecewise function (if heights at the boundary don’t match

to find continuity for dividing by zero (a rational fraction)…

• does not need a sign chart

• numerator is irrevelant

• make denominator = 0

will be continuous everywhere but the discontinuity point

to find continuity of a piecewise function…

• check if heights match at boundaries

  • if not, boundary is discontinuity point (if do, then continuous everywhere

• does not need sign chart either

non removable discontinuity

jump. gap that cannot be fixed by a single dot with your pencil, limit will not exist

cannot cancel out

0/0

hole

what would give away if it was nonremovable discontinuity?

numerator = 0 and a denominator

rational function

only discontinuity that has possibility of being a removable discontinuity

removable discontinuity

infinity small, limit will exist

something on the top and something on the bottom can cancel out

there is a non removable discontinuity (jump discontinuity) if…

| 0 | / 0 ≈ | 0 |

there is a non removable discontinuity (vertical asymptote - infinite discontinuity) if…

#/0

intermediate value theorem

pass through every value between low and high IF CONTINUOUS

when finding one sided limits use…

the same rules as limits

if a problem asks for if it is continuous at a given point

look for any of the continuity problems:

1. plug in given number and look for any of the continuity problems (divide by 0, square root a negative, taking the ln of a negative or a number)

2. justify using claim and evidence

vertical asymptote

setting denominator of a fraction = 0

fraction rules - faster/slower

equals to infinity or DNE

fraction rules - slower/faster

0

fraction rules - same/same

coefficients

order of functions from slowest to fastest growing

y = #, “trig”, constant and waves, ln and sqrt over x, linear (y = mx + b), polynomials (xⁿ), e^x or exponential (2^x, 5^x)

fraction rules work no matter where the…

function is

• #/infinity → 0

• #/0 → infinity

to find a horizantal asymptote…

use limit as x approaches infinity and limit as x approaches (-) infinity

slope formula

m = y₂ - y₁ / x₂ - x₁

derivative

equation that gives the instant rate of change at any given x-value or the equation that gives the slope at any x value

to find a derivative:

f(x) - f(#) / x - #

other notations of derivative

dy / dx, y’, f’(x)

f(#) means…

height of f at a #

what is considered not differentiable…

not continuous, sharp turn, vertical

d/dx(#) = 

0

d/dx(x) = 

1

d/dx(x^#) = 

a number times x^{# - 1}

d/dx[f(x) + g(x)] =

f’(x) + g’(x) (do each individually)

d/dx(# times f(x)) = 

x times f(x)

d/dx(sin x) =

cos(x)

d/dx(cos x) =

-sin(x)

d/dx(tan x) =

sec²(x) —> (sec(x))²

product rule

f x g, f’g + g’f

quotient rule

f/g, f’g - g’f / (g)²

quotient rule is not necessary for…

equation / # — 1/# times f(x)

\frac{\differentialD y}{\differentialD x}\vert x=\# is the same as…

f’(#), plugging things in

if you ever have parenthesis (layers) (imaginary or not) you must use the…

chain rule

formula for chain rule

\frac{d}{\differentialD x}=\left\lbrack f\left(g\left(x\right)\right)\right\rbrack\longrightarrow{}f^{\prime}\left(g\left(x\right)\right)\longrightarrow{}g^{\prime}\left(x\right)

formula for natural log differentiation

\frac{d}{\differentialD x}\left\lbrack\ln u\right\rbrack=\frac{1}{u}\cdot u^{^{\prime}}

why never do the chain rule for ln?

it is built in

3.8 exponential differentiation - formula for exponential differentiation

\frac{d}{\differentialD x}\left\lbrack e^{x}\right\rbrack=e^{x}

3.8 exponential differentiation - formula for exponential differentiation with a # as an exponent

\frac{d}{\differentialD x}\left(e^{\#}\right)=e^{\#}\cdot\#^{^{\prime}}

3.8 exponential differentiation - ln of an exponential…

x\ldots\ln e^{\#}=\#

3.8 exponential differentiation - formula for a # raised to the power of x

\frac{d}{\differentialD x}\left\lbrack\#^{x}\right\rbrack=\#^{x}\cdot\ln\left(x\right)

\frac{d}{\differentialD x}\left\lbrack\arcsin x\right\rbrack= to what… (3.9 inverse trig differentiations)

\frac{1}{\sqrt{1-x^2}}

\frac{d}{\differentialD x}\left(\arctan x\left(u\right)\right)= to what… (3.9 inverse trig differentiations)

\frac{1}{1+x^2}

first derivative forms

y’, f’, g’, u’, f’(x), dy/dx

second derivative forms

\frac{d^2y}{\differentialD x^2},y^{\doubleprime},f^{\doubleprime} … etc

third derivative forms

\frac{d^3y}{\differentialD x^3},y^{\prime\prime\prime},f^{\prime\prime\prime}\left(x\right) … etc

fourth derivative forms

\frac{d^4y}{\differentialD x^4},f^4\left(x\right),g^4\left(x\right)

3.10 higher order, calculator, piecewise differentiation - to take the derivative of a piecewise function..

1. take the derivative of each piece

2. change any ≤ or ≥ to < or >

3.11 implicit differentiation - when an equation is not just __ equals something never solve for y, instead… (2 pieces of information to plug in)

y, use implicit differentiation

3.11 implicit differentiation - how to implicit differentiate?

1. take the derivative of each piece

2. put dy/dx when you take the derivative of a y (chain rule with y)

3. get all the dy/dx stuff on one side and get the other stuff on the other side

4. divide out/pull out the dy/dx (if there is more than one)

5. divide the stuff attached to the dy/dx to the other side

4.1 lhopitals rule - steps to limits

1. plug it in

2. if num = 0 & deno = 0, then lhopitals rule

4.1 l’hopitals rule - if numer = 0 & deno = 0 then…

\lim_{x\to a}=\frac{f^{\prime}\left(a\right)}{g^{\prime}\left(a\right)}

4.2 rates of change - instantaneous rate of change

the derivative = slope at a point

4.2 rates of change - average rates of change

slope over interval

equation: y₂-y_{1 /} x₂-x₁

4.2 rates of change - blanket statement for rates:

at t = #, the words of the equation is increasing/decreasing at a rate of answer units

4.2 rates of change - to approximate a derivative on a table…

use average rate of change aka slope

4.3 critical numbers - what are critical numbers?

x value where the derivative = 0 OR derivative DNE

4.4 increasing and decreasing - a function is increasing if…

f^{\prime}\left(x\right)>0

4.4 increasing and decreasing - a function is decreasing if…

f^{\prime}\left(x\right)<0

4.6 concavity and points of inflection - a function has a POI if…

f’’(x) changes signs

4.6 concavity and points of inflection - a function is concave up if…

f’’(x) > 0

4.6 concavity and points of inflection - a function is concave down if…

f’’(x) < 0

4.7 the 2nd derivative test - the second derivative test is another method for finding…

extrema

4.10 IVT MCT EVT & Rolles theorem - constraints necessary for IVT

continuous on a ≤ x ≤ b and a ≤ c ≤ b

4.10 IVT MCT EVT & Rolles theorem - what does the IVT theorem say?

f\left(a\right)\le f\left(c\right)\le f\left(b\right)

OR

f\left(b\right)\le f\left(c\right)\le f\left(a\right)

4.10 IVT MCT EVT & Rolles theorem - constraints necessary for MVT?

1. continuous on a ≤ x ≤ b

2. a ≤ c ≤ b

differentiable on a ≤ x ≤ b

f’(x) is continuous on a ≤ x ≤ b

_{4.11 equation of secant and tangent lines - equation of a tangent line}

same as any other line

y = y₁ + m(x-x₁)

4.11 equation of secant and tangent lines - slope of tangent line?

m_tan = f’(x₁) —> derivative of x = x₁

4.12 tangent line approximation - to approximate a value using tangent line…

1. find equation of tangent line

2. plug the nearby x into x value

5.2 related rates and basic shapes - to solve a related rate use…

1. sketch

2. find the rates (given)

3. make an equation involving only the rates from step 2 (probably given)

4. take the derivative

5. plug in the when (x=,b=)

6. write down units (not used)

5.2 related rates and basic shapes - formula for area of a square

A=s^2

5.2 related rates and basic shapes - formula for area of a circle

A=\pi r^2

5.2 related rates and basic shapes - formula for circumference of a circle

C=2\pi r

5.2 related rates and basic shapes - formula for perimeter of a square

P=4\cdot s

5.2 related rates and basic shapes - formula for SA of a cube

A=6s^2

5.2 related rates and basic shapes - formula for SA of a sphere

A=4\pi r^2

5.2 related rates and basic shapes - formula for volume of a cube

V=s^3

5.2 related rates and basic shapes - formula for volume of a sphere

V=\frac43\pi r^3