Calculus 1 & 2 Review
what is the range of inverse tangent
(-π/2, π/2) (NOT inclusive)
how to integrate dx / (x^2 + a^2)
(1/a) * inverse tan (x/a) + c
trig identities for (cos(theta))^2 and (sin(theta))^2 that are helpful when solving equation for polar area
(cos(theta))^2 = (1 + cos(2*theta)) / 2 (sin(theta))^2 = (1 - cos(2*theta)) / 2
what are the three equivalent expressions to cos2x
(cosx)^2 - (sinx)^2 2*(cosx)^2 - 1 1 - 2(sinx)^2
sum of i^3
(n(n+1)/2)^2
derivative of tanx
(secx)^2
what is the integral of (sinx)^5 * cosx
(sinx)^6 /6
derivative of cotx
-(cscx)^2
derivative of cscx
-cscxcotx
integral of cscx
-ln|cscx + cotx|
derivative of cosx
-sinx
lim x --> 0 of (1-cosx)/x
0
(sinx)^2 + (cosx)^2 =
1
what is secx equivalent to
1/cosx
what is cscx equivalent to
1/sinx
define riemann right and left hand sums
LHR: say that you are doing the approximation [1,4] with intervals of 1, would be using f(1) through f(3) RHR: say that you are doing the approximation [1,4] with intervals of 1, would be using f(2) through f(4)
how many times is the r = c circle drawn in 2pi radians
ONCE
what is the range of inverse sine
[-π/2, π/2] (inclusive)
what is the formula for total distance traveled versus total displacement
distance = integral from t1 to t2 of |v(t)|dt displacement = integral from t1 to t2 of v(t)
what is the formula for the derivative of an inverse
h'(x) = 1/ (g'(h(x))
what is Rolle's theorem (it's a specific version of the MVT)
if f(a) = f(b) then there must exist f'(c) = 0 where c is between a and b
formula to find the area between two curves
integral from a to b of (f(x) - g(x))
what is the formula for the trapezoid rule
integral from a to b of f(x) is approximately equal to (b-a)/n * (Y0 + 2Y1 + 2Y2 + ... + 2Yn-1 + Yn)
how to find parametric arc length
integral from t1 to t2 of the square root of (x')^2 + (y')^2
formula for square cross sections
integral of (f(x)-g(x))^2 where f(x) is the function above g(x)
what are the two versions of the parametric distance formula aka formula for arc length
integral of root ((x')^2 + (y')^2) integral of root (1 + (dy/dx)^2)
what is the formula for integration by parts
integral of udv = uv - integral of vdu
what is the integral and derivative of 10^(5x)
integral: 10^(5x) / (ln10 * 5) derivative: 1/5 * ln(10) * 10^(5x)
how to find the average value
interval across the interval divided by the interval width
what is the integral of -1 / (a^2 - x^2)^(1/2)
inverse cosine of x/a
formula to model an integral with a summation
lim n--> infinity of the sum from i=1 to n of f(xi) delta(x) where delta(x) = (b-a)/n and xi = a + i*delta(x) (a and b are the values that you are integrating between)
what is the definition of e
lim n-->infinity of (1 + 1/n)^n
what is l'hopital's rule and when can it be used
lim x-->? f/g = lim x-->? f'/g' BUT this can only be used where lim x-->? = 0/0 or infinity/infinity
what is the integral of secx
ln|secx + tanx|
what is the integral of tanx
ln|secx| + c
integral of cotx
ln|sinx|
how do you use the washer method to rotate around the y axis
make the equations in terms of y, and make it so that the two numbers that the integral is between are numbers on the y axis rather than x axis (y coordinate of the same points) and then use pi * integral from y1 to y2 of (f(y) - g(y)) where f(y) is farther from the y axis ends up like pi * integral from Y1 to Y2 of ((f(y)^2 - g(y)^2))dy
sum of i
n(n+1) / 2
sum of i^2
n(n+1)(2n+1)/6
formula to find speed parametric
root of (x')^2 + (y')^2
formula for equilateral triangle cross sections
root3 / 4 * integral of (f(x)-g(x))^2 where f(x) is the function above g(x)
what is the shell method for
rotating around the y axis
derivative of secx
secxtanx
lim x-->infinity of sinx / x
1
what are the three requirements for continuity at a point x = c
1. f(c) exists 2. lim x-->c f(x) exists 3. f(c) = lim x-->c f(x)
what are the three steps for finding the derivative of the inverse
1. find the y-value when x = what is given 2. find the derivative of the original function and plug in x 3. find the reciprocal
what are the four steps for solving differential equations
1. separate (put y on one side and x on the other) 2. evaluate (find the integral of each side) 3. solve for c 4. solve for y (plug c back into the equation and solve for y so it's y on one side with everything else on the other side)
what is the derivative of inverse tan (tan^(-1)x)
1/(1+x^2)
formula for isosceles triangle cross sections
1/2 * integral of (f(x)-g(x))^2 where f(x) is the function above g(x)
what is the formula for the shell method
2*pi* integral of x*f(x)
shell method if you are rotating the area between TWO lines around the y axis
2*pi*integral of x*(f(x)-g(x))
what is the integral of tan(x/2)
2ln|sec(x/2)|
if n is even in a petaled flower, how many petals will there be
2n
trig identity: sin2x =
2sinxcosx
in 2pi radians, how many times is the r = sin(theta) circle drawn
2x
formula for rectangular cross sections where one side of the rectangle is 3x the length of the other
3* integral of (f(x)-g(x))^2 where f(x) is the function above g(x)
if y = 8^x, what is the derivative of y
8^x * ln8
formula to find area of a polar curve when there are two curves
A = (1/2) * integral from theta1 to theta2 of (r1^2 - r2^2) dtheta
formula to find the area of a polar curve
A = (1/2) * integral from theta1 to theta2 of r^2 dtheta
what is the formula for continuous interest
A = Pe^(rt)
what is the traditional algebraic distance formula
D = square root of (X2-X1)^2 + (Y2-Y1)^2)
what is the population formula and what do each of the variables mean?
P(t) = cc / (1 + Ae^kt) cc is carrying capacity, k is constant, t is time
how to get the original function from an integral
SUB AND CHAIN plug the two numbers into g(x) where g(x) is the function inside the integral and do g(top)*g'(top) - g(bottom)*g'(bottom) top is the top number of the integral and bottom is the bottom
what is the range of inverse cosine
[0, π] (inclusive)
what should you never forget to add when you solve an integral and ln is in the answer
absolute value of whatever is going inside the ln
where is growth on the logistic curve fastest?
at the inflection point, which is also cc/2
why do you swap the order of the functions in the washer method when you are rotating them around a line that is above both of them
because relative to a line that is above both functions, the function that is below the other one is now further from what it is being rotated around than the upper function
what shape does r = cos(theta) make and where
circle that is centered around y = 0 and x = r/2 it is tangent to the y axis and the x axis passes through the middle
what shape does r = sin(theta) make and where
circle that is centered around y = r/2 and x = 0 like it is tangent to the x axis and the y axis passes through the middle
what is the shape of r = c where c is a constant and where
circle, centered around the origin with radius r
how to remember that r = cos(theta) goes on the x axis
cos(0) = 1 so the diameter of the circle must lay on the x axis because this is where r is greatest
how are iimacons/cardioids oriented when the equation is sine vs cosine
cosine: oriented horizontally (like the heart would be sideways) sine: oriented vertically (like heart right side up)
derivative of sinx
cosx
what is the logistic growth formula and what do each of the variables in it mean
dP/dt = kP(1 - P/cc) k is the constant, P is the population, cc is the carrying capacity
what is the integral of 1 / (a^2 - x^2)^(1/2)
inverse sine of x/a
describe what r = sin(4*theta) looks like graphically
it has eight petals, two in each quadrant, and none of them are on the x or y axis
what is the mean value theorem
it is about derivatives. say you have two points a and b that are located on a function. if you calculate the slope between a and b, there must be a point between a and b on the curve where the derivative is equal to the slope between a and b
what is the intermediate value theorem
it is about y values. if a function is said to be continuous from (1,1) to (10,10) (for example), it must have passed through 1,2,3,4...
how to find a polar derivative (find the slope at a point on the polar curve) this is effectively asking how to find dy/dx of theta
it is dy/dx where y = rsin(theta) and x = rcos(theta)
what is the formula to find the difference quotient (aka the algebraic first derivative)
lim h-->0 (f(x+h) - f(x))/h = f'(x)
what is the derivative of log7(x^2-4) (log base 7) and what is the process for getting there
log7(x^2-4) = (ln(x^2-4) / ln(7)) (change of base formula) y' = 1/ln7 * 2x / (x^2 - 4)
what is crucial when doing integration by parts
need to make sure that you can integrate the part of the original equation that you classify as dv because you will need to find v
if you have sine in an equation for the petaled flower, what do you know about where the petals will be located
petal will never lay on the x axis because sin0 = 0 so when theta = 0, r=0.
formula for washer method to find the volume of a space between two lines rotated around y = c (which is above both of them)
pi * integral of ((c-g(x))^2 - (c-f(x))^2) where f(x) is the function above g(x)
formula for washer method to find the volume of a space with only one line f(x) rotated around y = c (which is below the function)
pi * integral of ((f(x) - c)^2
formula for washer method to find the volume of a space between two lines rotated around y = c (which is below both of them)
pi * integral of ((f(x)-c)^2 - (g(x)-c)^2) where f(x) is the function above g(x)
formula for washer method to find the volume of a space between two lines rotated around the x axis
pi * integral of (f(x)^2) - (g(x)^2)) where f(x) is the function that is above g(x)
formula for semicircle cross sections
pi/8 * integral of (f(x)-g(x))^2 where f(x) is the function above g(x)
how do you find position, velocity, and acceleration in parametric
position: <x,y> velocity: <x',y'> acceleration: <x", y">
how to use the shell method to rotate around the x axis
pretty sure that... put the numbers that the integral is going between as the y coordinate of the two points (rather than the x coordinate) and put the function in terms of y and then do 2*pi* integral between y coordinates of f(y)
what is the general form of the equation to create a petaled flower
r = a f(n * theta) where f is either sine or cosine
what is the form of the polar equation that gives you limacons and cardioids
r = b + a*f(theta) where f is cosine or sine
how to go from rectangular coordinates to polar coordinates
tan(theta) = y/x and x^2 + y^2 = r^2
if you have cosine in an equation for the petaled flower, what do you know about where the petals will be located
the first petal will always lay on the x axis because cos0 = 1 so r is maximized when theta = 0
describe what r = sin(5*theta) looks like graphically
there are 5 petals, they are all evenly spaced and there is a petal whose max is on the positive y axis (like pointing directly upward)
what is the second fundamental theorem of calculus
to get the original function, take the derivative of the integral
in 2pi radians, how many times is the r = cos(theta) circle drawn
twice
when do convex limacons occur and draw one
when 2a <= b it is like a heart but without the punched in thing at the top. more like a weird shaped rectangle
when do inner loop limacons occur, and draw one
when b < a they are like a heart, but with a little loop where the inner point typically would be
when do cardioids occur, and draw one
when b = a it is a perfect heart
normally, when doing cross sections, they are done perpendicular to the x axis. in this case, everything is in terms of ___. however, if you are doing things perpendicular to the y axis, everything should be switch such that it is in terms of ___.
x y (i.e.y = 2x become x = y/2
how to go from polar coordinates to rectangular coordinates
x = r*cos(theta) y = r*sin(theta)
product rule
y = f*g y' = fg' + gf'
quotient rule
y = f/g y' = (gf' - fg') / g^2
