Sequence And Series + Parametrics

Question 1

For a sequence whose terms increase at a constant rate by addition, what is the formula for an?

Answer 1

an= a1 + d( n-1 )

Question 2

What is the test for divergence and when can we use it

Answer 2

WE use it to test the DIVERGENCE of a SERIES not a sequence or anything else! If the lim as n approaches infinity is not equal to zero then the series diverges. If it equals zero the test is inconclusive!

Question 3

What do we do to test the convergence or divergence of a sequence ?

Answer 3

Take the limit

If it’s a finite number it converges

If it DNE\ is equal to infinity it converges

Question 4

What is ln (a) + ln(b) = ?

Answer 4

Ln(a*b)

Question 5

If I have to find the convergence of ln(2n) / ln(4n) , how do I start ?

Answer 5

Notice that ln(4n) = ln(2) + ln(2n)

So the new formula is

1/ (original bottom/ original top) + 1

1/ (ln(2)/ln(2n)) + 1

Question 6

If we have a problem that is in the form ROOT n of ( 3^4+3n) what do we do to find convergence

Answer 6

Notice that this is the same as

(3^4+3n)^(1/n) which is the same as

3^((4/n) + (3n/n) )

Because we simply can multiply by the exponents by the outer n exponent

Question 7

What equals 1 in terms of the special limit sine rule?

Answer 7

Lim x-> ZERO Sin(x) / x

Question 8

What are the basic parametric forms when we are looking at a circle

Answer 8

X= rcos(t) + h

Y= rsin(t) + k

Where (h,k) is the center

Question 9

How do we change the direction of rotation in a parametric particle if we need to manipulate an equation ?

Answer 9

Sub in NEGATIVE t

Question 10

Sin (-t) =

Answer 10

-sin(t)

Question 11

Cos (-t) =

Answer 11

Cos (t)

Question 12

Formula for a circle

Answer 12

(X-h)^2 + (y-k)^2 = r^2

Question 13

If you have the parametric equations for a particle that goes around a circle, ONCE, in a specific direction and are asked to find equations for the particle when it goes around 4 times in that same specific direction, what must you do?

Answer 13

Simply multiply the end parameter of t by the number of times you go around and keep all the equations the same

Question 14

Cos ( A MINUS B)

Answer 14

cosAcosB + sinAsinB

Question 15

Cos ( A PLUS B)

Answer 15

CosAcosB - sinAsinB

Question 16

Sin( A+ B) =

Answer 16

sinAcosB + cosAsinB

Question 17

Sin ( A - B ) =

Answer 17

sinAcosB - cosAsinB

Question 18

For a shift in parameterization what are the rules

Answer 18

(t + shift) CCW

(t-shift) Clockwise

Question 19

What are collision points

Answer 19

When x1=x2 and y1=y2 FOR THE SAME T

Question 20

What is the formula for finding the tangent of a PARAMETRIC curve

Answer 20

dy / dx= (dy/dt) / (dx/dt)

Think y ‘ (t) / x ‘ ( t)

Question 21

ln(? ) = 0

Answer 21

1

Question 22

How do we find the tangent by eliminating the parameter ?

Answer 22

1) eliminate t by choosing a formula
2) plug t into the other formula
3) differentiate that formula
4) use the given point to plug in the variable of the equation
5) the answer is equal to the slope
6) use point slope formula

Question 23

How do we find the tangent of a parametric equation without eliminating the parameter?

Answer 23

1) find dy/ dx. This gives us an answer in terms of t
2) find t by setting the x or y equation or its respective point value and solve for t
3) sub t into the answer to dy/dx to find the slope
4) point slope formula

Question 24

What do we do when we want to find the Cartesian coordinate or even the slope of the tangent of parametric TRIG equations.

Answer 24

Use a Pythagorean identity and sub in x and y appropriately

If we want to find the slope just take the derivative of this equation, isolate dy/dx, and use the given point to solve for m

Question 25

What is the formula for d^2y/ dx^2

Answer 25

d/ dt ( dy/dx ) / dx/dt Which is essentially, (dy/dx) PRIME / x PRIME

Question 26

How to we find the points of a horizontal tangent to a parametric curve ?

Answer 26

1) find dy/dt 2) solve for t 3) plug t in to x and y formulas for points

Question 27

How to we find the points of a vertical tangent to a parametric curve ?

Answer 27

1) find dx/dt 2) solve for t 3) plug t in to x and y formulas for points

Question 28

How can we tell if a geometric series converges or diverges?

Answer 28

If |r| < 1 it converges If |r| >= 1 it diverges

Question 29

Form of a geometric series and form of its sum

Answer 29

E a(r)^n Or E a(r)^n-1 Sum = a1 / 1-r Where r is the constant ratio that is being multipled or divided in each instance

Question 30

When there is a series where every other term has the same ratio what do we do to find the sum?

Answer 30

Break it up into two geometric series Determine convergence by finding r Find each individual sum and add them together

Question 31

Parametric equation for ellipse

Answer 31

X=acos(theta) Y= bsin(theta)

Question 32

Area for parametric equations formula and what does x and y equal.

Answer 32

Y= gtheta. X= f(theta) Integral from theta 1 to theta 2 Y dx = g(theta) * f ‘ (theta) dtheta So it’s y * x prime

Question 33

What is the bounds if we want to find the area of one arch under a trachoid?

Answer 33

0 -> 2pi

Question 34

Describe how s1 and s2 and so on work

Answer 34

S1= an S2 = S1 + an+1 S3 = s2 + an+2 And so on

Question 35

ln( infinity) =

Answer 35

Infinity

Question 36

What is the formula for the surface area by rotating a curve about the x axis ?

Answer 36

Integral from a to b 2🥧r * ROOT ( 1 + (dy/dx) ^2 ) dtheta

Question 37

What is the arc length formula of a parametric curve

Answer 37

Integral from a to b ROOT ( x ‘ ^2 + y ‘ ^ 2 ) dt

Question 38

What is the formula for the surface area by rotating a curve about the Y axis ?

Answer 38

2🥧r* ROOT ( 1 + ( dy/dx) ^2 ) dtheta

Question 39

What is the form of polar coordinates in (:,:)

Answer 39

( r, theta )

Question 40

What is the x formula and y formula for polar coordinates ?

Answer 40

X = rcos(theta) Y= rsin(theta)

Question 41

Where is cos = 1/2 on the unit circle ( near which axis)

Answer 41

The verticle ones Think the half foot men stand straight up tall and proud

Question 42

What is the polar coordinate value r equal to in terms of x and y

Answer 42

R = ROOT ( x^2 + y^2 )

Question 43

How do we get theta from Cartesian coordinates ?

Answer 43

Arctan ( y/x)

Question 44

Describe the process of finding theta for a Cartesian coordinate and the general formula

Answer 44

Theta = calculated theta + pi*n - find theta if its negative pull the negative out in front of - figure out where the point in the quadrant is which is how you decide which value of pi we use

Question 45

Describe r> 0 or <0 process for polar coordinates

Answer 45

For r> 0 just use the positive r value and theta as found by (theta = calculated + pi*n) For r< 0 change r to a negative sign and add or subtract 180 degrees/ pi to the angle

Question 46

When we are finding other polar coordinates what are the rules for adding pi vs 2pi

Answer 46

When r changes sign add PI when r is the same sign add TWO PI

Question 47

Cos theta in terms of polar coordinates

Answer 47

X/r

Question 48

Sin theta in terms of polar coordinates

Answer 48

Y/r

Question 49

How do we find a Cartesian equation given a polar equation!

Answer 49

Replace r and trig forms with their terms of x,r, and y. Set it all to zero Complete the square Determine if it’s a circle, parabola, limacon, hyperbola, or ellipse

Question 50

Circle formula

Answer 50

(x - h)² + (y - k)² = r²

Question 51

Ellipse formula . What is a^2

Answer 51

x²/a² + y²/b² = ONE!!! THE LARGER NUMBER IS A

Question 52

Hyperbola formula WHERE IS A

Answer 52

is (x-h)²/a² - (y-k)²/b² = 1 A CAN BE UNDER X OR Y BUT IS ALWAYS UNDER THE POSITIVE FRACTION

Question 53

formula for asymptote of a hyperbola . What must be the case for the slope.

Answer 53

Y-k= +- b/a (x-h) It can be b/a or a/b because its rise over run

Question 54

Parametric vs polar curves

Answer 54

parametric curves define coordinates (x, y) in terms of a third variable (parameter, often 't'), while polar curves define coordinates in terms of distance from a point (pole) and an angle. Parametric equations use two equations for x and y, each depending on the same parameter, while polar equations use a single equation for the distance (r) from the pole, which is a function of the angle.

Question 55

Formula for length of a curve of a POLAR equation

Answer 55

Integral from a to b ROOT ( r^2 + (dr/dtheta)^2 )

Question 56

Limacon formula

Answer 56

r = a ± b sin(θ) or r = a ± b cos(θ)

Question 57

How to graph limacon from r= “ formula

Answer 57

Make r vs theta chart and get points Graph Cartesian Then graph the r, theta version

Question 58

Area for polar coordinates formula

Answer 58

A= integral from a>b 1/2 r^2 dtheta

Question 59

What is the length of a polar curve

Answer 59

L= integral from a to b ROOT ( r^2 + (dr/dtheta) ^2 ) dtheta

Question 60

Ellipse: What is the major axis equal to ? The minor axis ?

Answer 60

Major: 2a Minor: 2b

Question 61

Ellipse: Do the foci lie on the major or minor axis and what is its formula

Answer 61

Major : c^2= a^2 - b^2

Question 62

Hyperbola: if a^2 is under the y fraction how does it look? Under the x fraction?

Answer 62

Y: open up and down on y axis X: opens left and right in x axis

Question 63

Where are the foci in a hyperbola and what is their formula ?

Answer 63

Cupped by the curves ( on outside ) C=a^2 + b^2

Question 64

Parabolas: Formula for a parabola opening right or left and what’s the key difference

Answer 64

-p = left +p= right (Y-k)^2 = 4p(x-h)

Question 65

Parabolas: Formula for opening up and down and what is difference

Answer 65

-p= down +p= up (X-h)^2 = 4p(y-k)