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
