AP Calculus A/B- FINAL REVIEW

sin(7π/6)

-½

sin(5π/4)

-√2/2

tan(2π/3)

-√3

f^-1 represents what?

An inverse function

ƒ''(c)>0

Concave Up

When does a derivative not exist at 'x' (with a graph)?

Corner Cusp Vertical Tangent Discontinuity

limit as h approaches 0 of [f(x+h)-f(x)]/h

Formal definition of derivative

Absolute Extrema

The absolute highest and lowest points of a graph.

If f¹(x) changes from positive to negative, then

by the First Derivative Test, f has a maximum

no breaks gaps or holes

continuous function

derivative

rate

∫v(t)dt

s(t) position

Product rule?

uv'+vu'

y' = sec²(x)dx

y = tan(x), y' =

derivative of log a (u)

(1/( ln(a) * a^u)) * u'

integral of a^u ∫a^udu

(1/ln(a)) * a^u +c

Chain Rule

(F o G)' (x) = f'(g(x)) *g' (x)

rate of change of y is proportional to y

(dy/dt) = Ky

Implicit differentitation

(image)

Change of base Equation with log a (X)

(logb(x) / logb(a))

derivative of arcsin

(u' / sqrt(1-u^2))

Derivative of arcsec

(u'/ abs(u)* sqrt(u^2-1)))

Derivative of arctan

(u'/1+u^2)

Quotient rule?

(vu'-uv')/v^2

(x^2)/(y^2)

(x/y)^2 =

x^(a+b)

(x^a)(x^b) =

x^(a-b)

(x^a)/(x^b) =

x^(ab)

(x^a)^b =

Derivative of cosine inverse

- 1/sqrt(1-x^2)

Derivative of arccos

-(u' / sqrt(1-u^2))

Derivative of arccsc

-(u'/ abs(u)* sqrt(u^2-1)))

Derivative of arccot

-(u'/1+u^2)

sin(3π/2)

-1

antiderivative of sin

-cos +c

antiderivative of csc^2

-cot +c

antiderivative of csc*cot

-csc +c

antiderivative of tan

-ln |cos(x)|+c

antiderivative of csc

-ln|csc+cot| +c

sin(5π/3)

-√3/2

tan(5π/6)

-√3/3

sin(π)

0

cos(2π)

1

How do you find a Local Extrema?

1. Find the first derivative of f using the power rule. 2. Set the derivative equal to zero and solve for x. x = 0, -2, or These three x-values are the critical numbers of f.

How to find a vertical asymptote

1. Set the denominator equal to zero 2. Simplify the fraction 3. Cancel out like terms on the top and the bottom

4 guidelines on finding extrema on a closed interval

1. find critical #'s of f(x) in (a,b) 2. evaluate original f(x) at each critical number in (a,b) 3. evaluate f(x) at each endpoint on [a,b] 4. the least is the minimum, the greatest is the maximum

How do I find an equation of a line tangent to a curve

1.) Calculate the slope of a secant through P and a point Q nearby on the curve. 2.)Find the limiting value of a secant slope (if it exists) as Q approaches P along the curve. 3.)Define the slope of the curve at P to be this number and define the tangent to the curve at P to be the line through P with this slope.

∫du/(u*√u^2-a^2)

1/a * arcsec( abs(u)/a) +c

∫du/(a^2+u^2)

1/a * arctan(u/a) +c

Derivative of sine inverse

1/sqrt(1-x^2)

the derivative of the ln|x|=

1/x * x'

What are the 1st and 2nd derivatives of displacement?

1st derivative is velocity and the 2nd is acceleration. These are found by identifying the slope of displacement to find velocity, and slope of velocity to find acceleration

- shortest side is opposite 30° angle - medium side is opposite 60° angle - longest side is opposite 90° angle

30-60-90 Right Triangle

- 2 shorter sides are equal in length "n" - hypotenuse = (n)(√2)

45-45-90 Right Triangle

Where can you not draw a tangent line?

A Corner

Rules of Piecewise Functions

A Piecewise function is made up of sub-functions that apply to a certain interval of the main function's domain. First look at the conditions on the right to see where x is. Then just plug that number into the equation.

Limit

A limit is the value that a function or sequence "approaches" as the input or index approaches some value.

Critical Number

A number in the domain of a function where the derivative is 0 or DNE

What is an inflection point?

A point of a curve at which a change in the direction of curvature occurs.

What is a secant line? *Used in Mean Value Theorem

A secant line is a straight line joining two points on a function. It is also equivalent to the average rate of change, or simply the slope between two points. The average rate of change of a function between two points and the slope between two points are the same thing.

What does a tangent line look like?

A straight line that hits a curve at exactly one point

What graph comes as a result of finding the derivative of a speed graph?

Acceleration Graph

limit as x approaches a of [f(x)-f(a)]/(x-a)

Alternate definition of derivative

Slope of secant line between two points, use to estimate instantanous rate of change at a point.

Average Rate of Change

What does a Corner look like?

Corner at (17, 600) and (18, 530)

What will override a critical number being so?

Discontinuity

Rate of Change

Expressed as a ratio between a change in one variable relative to a corresponding change in another. When the function y=F(x) is concave up, the graph of its derivative y=f'(x) is increasing. When the function y=F(x) is concave down, the graph of its derivative y=f'(x) is decreasing.

Mean Value Theorem

F'(c)= f(b)-f(a)/b-a

True or false: All critical numbers are relative extrema

False

How do you find the absolute extrema of a function? How can you find the absolute extrema of a function on an interval with end points?

Find critical points by funding where the first derivative is 0 or undefined, then plug in end points to f(x) and critical points to find extrema.

How do you find the local extrema of a function?

Find the first derivative and set it equal to zero

Quotient Rule

Function (f/g) Derivative

Sum Rule

Function - f + g Derivative - f' + g'

Difference Rule

Function - f - g Derivative - f' − g'

Product Rule

Function - fg Derivative - f g' + f' g

Power Rule

Function - x^n Derivative - 〖nx〗^(n-1)

Reciprocal Rule

Function 1/f Derivative −f'/f2

Chain Rule (Using ' )

Function f(g(x)) Derivative f'(g(x))g'(x)

∫ f(x) dx on interval a to b = F(b) - F(a)

Fundamental Theorem of Calculus

g + ∫ (F (t) − E (t)) dt from 0→m

Given a water tank with g gallons initially being filled at the rate of F(t) gallons/min and emptied at the rate of (t) gallons/min on [0,b], find the amount of water in the tank at m minutes

(d/dt) ∫ (F (t) − E (t)) dt from 0→m == F(m) - E(m)

Given a water tank with g gallons initially being filled at the rate of F(t) gallons/min and emptied at the rate of (t) gallons/min on [0,b], find the rate the water amount is changing at m

Solve F (t) − E (t) = 0 to find candidates, evaluate candidates and endpoints as x = a in "g + ∫ (F (t) − E (t)) dt from 0→a" choose the minimum value

Given a water tank with g gallons initially being filled at the rate of F(t) gallons/min and emptied at the rate of (t) gallons/min on [0,b], find the time when the water is at a minimum

Extreme Value Theorem

If a f(x) is continuous, it is guaranteed to have a max/min

f(x) has a relative maximum

If f '(x) = 0 and f"(x) < 0,

f(x) has a relative minimum

If f '(x) = 0 and f"(x) > 0,

How do you find the derivative of an inverse function?

If f and g are inverse functions, then f'(x)=1/(g'(f(x))

Intermediate Value Theorem (IVT)

If f is continuous on a closed interval [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c) = k.

Extreme Value theorem

If f is continuous over a closed interval, then f has maximum an minimum values over that interval.

f'(x) = g(x)dx

If f(x) = ∫ g(t) dt on interval 2 to x, then f'(x) =

Power Rule

If function f is f(x) = x^n, where n is any integer, then f' (x) = n·x^n-1.

Slope of tangent line at a point, value of derivative at a point

Instantenous Rate of Change

U-Substitutions

Integration technique that is used to integrate a product, quotient and/or composition

If f(1)=-4 and f(6)=9, then there must be a x-value between 1 and 6 where f crosses the x-axis.

Intermediate Value Theorem

critical points

Is where there is a point in the domain of a function f at which f'=0 or f' does not exist is a critical point of f. *critical points are not always maximum and minimum values.

What graph comes as a result of finding the derivative of an acceleration graph?

Jerk Graph

When are limits nonexistent?

Jump Discontinuities: both one-sided limits exist, but have different values. Infinite Discontinuities: both one-sided limits are infinite. Endpoint Discontinuities: only one of the one-sided limits exists. Mixed: at least one of the one-sided limits does not exist.

if f(a)/g(a)=0/0 or infinity/infinity, then lim as x→a of f(x)/g(x)= lim as x→a of f'(x)/g'(x) *continue to take the derivative until a number (not 0/0) is reached*

L'Hopital's Rule

If f is decreasing (f¹ < 0), then

LRAM is an over approximation and RRAM is an under approximation.

If f is increasing (f¹>0), then

LRAM is an under approximation and RRAM is an over approximation.

What are discontinuities? When are limits nonexistent?

Limits dont exist when the values from the left and righ are3 no equal

How do you move a term from the denominator to the numerator?

Make the power of the denominator negative than multiply the denominator by the numerator

How do we handle negative exponents?

Negative exponents are moved to the bottom of a fraction to make the exponent positive. When finding derivatives, it's easier to solve when you put a factor from the denominator of the fraction to the top with a negative exponent and use the power rule.

f''(c)=0 f''(c-)<0 f''(c+)>0

POI

f''(c)=0 f''(c-)>0 f''(c+)<0

POI

velocity is negative

Particle is moving to the left or down

velocity is positive

Particle is moving to the right or up

who found out about the relationship between relative extrema and critical numbers

Pierre de fermat

Optimization Problems

Problems where you are trying maximize or minimize "something"

RAM

Rectangle Approximation Method of equal widths

Reimann Sum

Rectangle Approximation Method of unequal widths

Types of discontinuity

Removable Discontinuity: when a point on the graph is undefined or does not fit the rest of the graph (there is a hole) Jump Discontinuity: when two one-sided limits exist, but they have different values Infinite Discontinuity:

When can removable discontinuities be fixed?

Removable discontinuities can be "fixed" by re-defining the function.

How do you find a local maxima on a graph?

Set derivative equal to zero and solve for "x" to find critical points. Critical points are where the slope of the function is zero or undefined.

Unit Circle

Since C = 2πr, the circumference of a unit circle is 2π. A unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system.

CHANGE TO x=ysomething, y bounds

Solids of revolution: stacking disks like a coin stack things to remember

How to find derivative?

Solve a problem for (dy/dx). Solutions may involve power, product, chain or quotient rule.

What graph comes as a result of finding the derivative of an position/displacement graph in absolute valuation?

Speed Graph

What is the derivative of a position function? How do you find where the function is decreasing?

Speed/Velocity. The function is decreasing when y' is negative (below the x-axis)

How do you find the limit of a piece-wise function?

Step 1 Evaluate the one-sided limits for each function. Step 2 If the one-sided limits are the same, the limit exists. If the one-sided limits are different, the limit doesn't exist.

Second Derivative

Take first derivative. Then, find the derivative of the first derivative. f'(x), then f''(x).

Find the derivative of the square root of f(x)

The derivative of the square root of a function is equal to the derivative of the radical divided by the double of the root.

Extrema

The relative maximum and relative minimum of the graph in the interval

If the appropriate conditions are satisfied, what does the Mean Value Theorem guarantee?

There is at least one point c in the interval (a, b) at which f'(c) = [f(b) - f(a)] / [b - a]

plug (x,y) coordinates into differential equation, draw short segments representing slope at each point

To draw a slope field,

critical points and endpoints

To find absolute maximum on closed interval [a, b], you must consider...

pair appropriate variables, integrate + C, use initial condition to find C, solve for y

To find particular solution to differential equation, dy/dx = x/y

Recognizing Implicit Differentiation

Used when an equation contains a variable besides x For example: 3x + 4y = 12 derivative of y= dy/dx

Related Rates

Variables that are changing over time (rate at which "something" is changing with respect to time)

What graph comes as a result of finding the derivative of a displacement graph?

Velocity Graph

How do you interpret a velocity graph to determine speed?

Velocity is the first derivative of position. In order to graph speed from velocity then you need to find the derivative of velocity from the graph. In order to do that you need to reflect the negative terms across the x-axis making them positive.

GCF

WHEN YA GET SUCK think to

Chain Rule

We use chain rule to find the derivative of the composition of two functions. formula : dy/dx f(g(x)) = f'(g(x))*g'(x)

What does a cusp look like?

When a function becomes vertical and then virtually doubles back on itself. Such pattern signals the presence of what is known as a vertical cusp.

mark which function you are using

When doing an value analysis

relative maximum

When f '(x) changes fro positive to negative, f(x) has a

point of inflection

When f '(x) changes from increasing to decreasing or decreasing to increasing, f(x) has a

relative minimum

When f '(x) changes from negative to positive, f(x) has a

concave down

When f '(x) is decreasing, f(x) is

concave up

When f '(x) is increasing, f(x) is

decreasing

When f '(x) is negative, f(x) is

increasing

When f '(x) is positive, f(x) is

corner, vertical tangent, discontinuity (no defined derivative)

When is a function not differentiable

Finding the vertical asymptote

When the denominator of the function equals 0.

When is a function decreasing?

When the first derivative/ slope is negative

When is a function increasing?

When the first derivative/ slope is positive

When is the second derivative of a function negative?

When the graph of the function is concave down

When is the second derivative of a function positive?

When the graph of the function is concave up

Increasing Functions

Where the graph of the first derivative shows the original function being continuous, differentiable and increasing.

Power rule

X^2 --> 2X^2-1 ---> 2X

Why can't you draw a tangent line on a corner?

You can't draw a tangent line because the tangent line from the left and the right will be going different directions.

area of trapezoid

[(h1 - h2)/2]*base

Monotonic Function

a function that never changes direction monotone increasing - always increasing monotone decreasing - always decreasing

(a^x)(lna)

a is a positive real number

If a function is concave down, then

a linearization will be an over approximation and a secant line will be an under approximation

If a function is concave up, then

a linearization will be an under approximation and a secant line will be an over approximation

If f is concave up (f¹¹ > 0 and f¹ is increasing), then

a trapezoidal sum will be an over approximation.

If f is concave down (f¹¹ < 0 and f¹ is decreasing), then

a trapezoidal sum will be an under approximation.

derivative of a^u

a^u * ln(a) * u'

speed

absolute value of velocity

If v(t) is the velocity function, then v¹(t) is

acceleration

Critical Point

an x where f¹(x) = 0 or is undefined

∫ du/(√a^2-u^2)

arcsin(u/a) +c

positive

area above x-axis is

negative (negative x-values are just absolutes!)

area below x-axis is

∫ f(x) - g(x) over interval a to b, where f(x) is top function and g(x) is bottom function

area between two curves

h ∫(top)-(bottom) dx

area of a solid built with rectangles

.5π ∫((top-bottom)/2)^2 dx

area of a solid built with semi-circles

∫(top-bottom)^2 dx

area of a solid built with squares

Ln(x)=

atiderivative from 1 to x of (1/t) dt

= 1/(b-a) ∫ f(x) dx on interval a to b

average value of f(x)

If f¹(x) changes from negative to positive, then

by the First Derivative Test, f has a minimum

If f¹(x) = 0 and f¹¹(x) < 0, then

by the Second Derivative Test, f has a maximum

If f¹(x) = 0 and f¹¹(x) > 0, then

by the Second Derivative Test, f has a minimum

c

c is a real number

f¹(g(x)) × g¹(x)

chain rule

Displacement over [a,b]

change in position s(b) - s(a)

Average Velocity over [a,b]

change in position over change in time

f''(c) <0

concave down

when f' is decreasing, f is

concave down

when f' is increasing, f is

concave up

(√below) / 2

cos(θ) hand rule

cos(x)/sin(x)

cot(x)

propagated error

created error from problem

1/sin(x)

csc(x)

Domain and range of arcsin

d: [-1,1] r: [-pi/2, pi/2]

Domain and range of arccos

d: [-1,1] r: [0,pi]

Domain and range of arccsc

d: [-inf, -1) U (1,inf] r: [-pi/2, 0) U (0, pi/2]

Domain and range of arcsec

d: [-inf, -1) U (1,inf] r: [0, pi/2) U (pi/2, pi]

Domain and range of arctan

d: [all reals] r: [-pi/2, pi/2]

Domain and range of arccot

d:all reals r:[0,pi]

Population derivative

dN/dt = K (population- N)

Definition of a limit

definition of a limit

you le integrate duh

distribute the negative when

dy

dx*f'(x)

Newton's law of cooling

dy/dt= K (y-medium)

Derivative of y

dy/dx

0

dy/dx of a #

Derivative of e^x

e^x * derivative of argument

∞

e^x *lim x →∞ *

0

e^x *lim x →∞− *

(x-h)²+(y-k)²=r²

equation of a circle

graph is symmetrical with respect to the y-axis; f(x) = f(-x) *cos, sec*

even function

point of inflection (POI)

exchange of concavities

If f¹¹(x) changes signs, then

f has an inflection point

If f¹(x) < 0, then

f is decreasing

If f¹(x) < 0 and f¹¹(x) > 0, then

f is decreasing at a decreasing rate

If f¹(x) < 0 and f¹¹(x) < 0, then

f is decreasing at an increasing rate

f'(c) < 0

f is decreasing at c

If f¹(x) > 0, then

f is increasing

If f¹(x) > 0 and f¹¹(x) < 0, then

f is increasing at a decreasing rate

If f¹(x) > 0 and f¹¹(x) > 0, then

f is increasing at an increasing rate

f'(c) > 0

f is increasing at c

what determines concavity

f''

stationary point

f'=0

singular point

f'=DNE

delta y

f(deltax+x)-f(x)

linearize, approximate, etc

f(deltax+x)= f(x)+( f'(x)*dx)

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

f(g(x)) has the derivative of

∫f¹(x)dx

f(x)

What conditions must be to satisfied for the Mean Value Theorem to be valid?

f(x) is continuous in the interval [a, b] and differentiable in the interval (a, b)

antiderivative power rule

f(x)= (x^n+1)/n+1

f''(c)=0

further investigation needed

f'(c)=0 f''(c)=0

further investigation needed

∫f¹¹(x)dx

f¹(x)

If a function is decreasing...

f¹(x) < 0 LRAM is an over approximation RRAM is an under approximation

If a function is increasing...

f¹(x) > 0 LRAM is an under approximation RRAM is an over approximation

If f¹¹(x) < 0, then

f¹(x) is decreasing and f is concave down

If f¹¹(x) > 0, then

f¹(x) is increasing and f is concave up

If a function is concave down...

f¹¹(x) < 0, f¹(x) is decreasing Trapezoidal approximation is an under approximation

If a function is concave up...

f¹¹(x) > 0, f¹(x) is increasing Trapezoidal approximation is an over approximation

If f¹(x) is changing directions, then

f¹¹(x) changes signs making f change concavity

∫ abs[v(t)] over interval a to b

given v(t) find total distance travelled

f'(c) = 0

horizontal tangent at c

Mean value theorem for derivatives

if f(x) is continuous over [a,b] and differentiable over (a,b), then at some point c is between a and b.

f is strictly monotonic when slope is either

increasing or decreasing Examples: cubic, linear, log,

no limits, find antiderivative + C, use inital value to find C

indefinite integral

antiderivative is the same as

indefinitinte integral differential equation general equation particular solution

get the sum of what is being rated

integrate a rate

A relation formed by reversing x and y in a function.

inverse function

Half life constant

k= -ln(2) / #of yrs for populations that triple every x years use k= -ln(3) / #of yrs

1

lim as x→0 of sin(x)/x

When is the Vertical asymptotes

lim(x→0)⁡ (1/x)

When is the horizontal asymptotes

lim(x→∞) (1/x)

What must be true for a limit to exist?

limit from the left = limit from the right

plug in: if over 0 DNE, if 0/0 rationalize radicals or simplify complex fractions or factor/reduce

limits

antiderivative of cot

ln | sin | +c

antiderivative of sec

ln |sec +tan| +c

anti derivative of 1/u

ln |x| +c

0

ln(1)

1

ln(e)

ln(m/n)

ln(m) - ln(n)

ln(mn)

ln(m) - ln(n)

∞

ln(x) *lim x →∞ *

-∞

ln(x) *lim x →∞− *

B^E=N

log(base B)N = E

(1/2)log(x)

log(x)^(1/2)

log(x) - log(y)

log(x/y)

log(x) + log(y)

log(xy)

Extrema

maximums and minimums (occur at critical points and/or endpoints if on a closed domain)

if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b) f '(c) = [f(b) - f(a)]/(b - a)

mean value theorem

Relative Error

mistake/ perfect

nx^(n-1)

n is a real number

function is decreasing when slope is

negative

when f' is concave down, f'' is

negative

ln(mⁿ)

nln(m)

graph is symmetrical with respect to the origin; f(-x)=-f(x) *sin, tan, cot, csc*

odd function

one y-value for each x-value

one to one function

2 equations, solve for one variable, plug into the other equation, take derivative, set equal to 0, find variable 1, plug in to find variable 2

optimization

If v(t) = 0 and undergoes a sign change, then

particle changes direction

The derivative of the function f at the point x=a is the limit... if the limit exists.

picture on email

If v(t) < 0, then

position is decreasing (left or down)

If v(t) > 0, then

position is increasing (right or up)

function is increasing when slope is

positive

when f' is concave up, f'' is

positive

f¹(x)g(x) + g¹(x)f(x)

product rule

(f¹(x)g(x) - g¹(x)f(x))/(g(x))²

quotient rule

be careful about product and chain rules when taking the derivative

rate of change

draw a picture always

related rates

absolute extrema can be one of two points

relative extrema or endpoints

f'(c)=0 f''(c)<0

relative maximum at c

f'(c)=0 f'(c-)>0 f'(c+)<0

relative maximum at c

f'(c)=0 f''(c)>0

relative minimum at c

f'(c)=0 f'(c-)<0 f'(c+)>0

relative minimum at c

percent error

relative* 100

*subtract a* from f(y) if to the *left* of the y-axis (which is essentially just adding it), *subtract f(y)* from a if it is to the *right*

rotate around the line x=a (this is penny-stacking rotation)

*subtract a* from f(x) if rotating *below* the area, *subtract f(x)* from a if rotating *above* the area

rotate around the line y=a (this is side-ways rotation)

antiderivative of sec*tan

sec +c

1/cos(x)

sec(x)

How do you find POI's

set f''(x) to zero and undefined and solve for points

if the derivative is undefined, this will produce what type of graph

sharp turns

show that *lim x →a− * f(x) = *lim x →a+ * f(x); exists and are equal

show that f (x) *lim x→a* exists

antiderivative of cos

sin +c

What are the derivatives of trig functions?

sin(x) = cos (x); cos (x) = -sin(x); tan(x) = sec^2(x)

(√above) / 2

sin(θ) hand rule

zero

slope of horizontal line

undefined

slope of vertical line

if the derivative is 0, this will produce what type of graph

soft hills and valleys

be in terms of X (dx)

solids build PERPENDICULAR to the x-axis must

If v(t) < 0 and a(t ) > 0, then

speed is decreasing (slowing down)

If v(t) > 0 and a(t ) < 0, then

speed is decreasing (slowing down)

If v(t) < 0 and a(t ) < 0, then

speed is increasing (speeding up)

If v(t) > 0 and a(t ) > 0, then

speed is increasing (speeding up)

How do you determine the end behavior model of a polynomial function going to positive or negative infinity?

take the variable with the largest exponent and substitute the variable with the limit

antiderivative of sec^2

tan(x)

sin(x)/cos(x)

tan(x)

√(above/below)

tan(θ) hand rule

Linearization

tangent line equation that is used to approximate a function's value near the point of tangency

use trapezoids to evaluate integrals (estimate area)

trapezoidal rule

True or false: All relative extrema are critical numbers.

true

Product Rule

u*v' + v*u'

tan(π/2)

undefined

a circle whose center is at the origin and has a radius of one, cos is x-value, sin is y-value, tan is y/x

unit circle

arcsin graph

up on the right

when the chain rule would be necessary

use u substitution to integrate when

∫a(t)dt

v(t) velocity

shell method

v= 2 pi (integral from a to b) radius*height

If s(t) is the position function, then s¹(t) is

velocity

If a(t) < 0, then

velocity is decreasing

If a(t) > 0, then

velocity is increasing

What does a Vertical Tangent look like?

vertical tangent image

(1/3)πr²h

volume of a cone

s³

volume of a cube

πr²h

volume of a cylinder

L x W x H

volume of a prism

(1/3)Bh

volume of a pyramid

(4/3)πr³

volume of a sphere

π ∫ r² dx over interval a to b, where r = distance from curve to axis of revolution

volume of solid of revolution - disk method

π ∫ R² - r² dx over interval a to b, where R = distance from outside curve to axis of revolution, r = distance from inside curve to axis of revolution

volume of solid of revolution - washer

U-substitution

when integrating a trig function, use

add C

when integrating without bounds always

newtons method

x1= x0- f(x)/f'(x)

1

x^0

Euler's Method

x₁= x₀ + ∆x y₁= y₀ + (m* ∆x) where m= dy/dx

y' = a^(x) ln(a)dx

y = a^x, y' =

y' = -sin(x)dx

y = cos(x), y' =

chain rule

y = cos²(3x), state rule used to find derivative

y' = -1/√(1 - x²)dx

y = cos⁻¹(x), y' =

y' = -csc²(x)dx

y = cot(x), y' =

y' = -1/(1 + x²)dx

y = cot⁻¹(x), y' =

y' = -csc(x)cot(x)dx

y = csc(x), y' =

y' = e^(x)dx

y = e^x, y' =

y' = 1/(x)dx

y = ln(x), y' =

quotient rule

y = ln(x)/x², state rule used to find derivative

y' = 1/(x) 1/ln(a)dx

y = log (base a) x, y' =

y' = sec(x)tan(x)dx

y = sec(x), y' =

y' = cos(x)dx

y = sin(x), y' =

y' = 1/√(1 - x²)dx

y = sin⁻¹(x), y' =

y' = 1/(1 + x²)dx

y = tan⁻¹(x), y' =

product rule

y = x cos(x), state rule used to find derivative

rate of change for growth and decay in "y=" form

y= ±Ce^(kt) If the point is positive, use plus if the point is negative, use negative k= Positive, then its growth k=negative, then its decay

ln(y)=xln(x), 1/y(dy/dx)=ln(x)+1, y'=y(ln(x)+1)

y=x^x then y'=

indeterminate form (more mathematical work needs to be done in order to evaluate) You can try L'Hopital's Rule.

±∞/±∞ OR 0/0

cos(π/3)

½

0

√(0)

cos(π/4)

√2/2

tan(π/3)

√3

cos(11π/6)

√3/2

tan(π/6)

√3/3