BQ#7

you know there is a tangent line on the graph when the line only touches the graph once as you can see in the picture

on thispicture the line touches the graph twice which meants it is a secant line. this changes which the coordinates are (x+h, f(x+h) )

with this we can derive the quotient formula and by that we plug in the secant coordinates to our slope formula which is y2-y1/x2-x1 when we plug everything in all we need to do is cancel the x's and end up with the formula which is  (f(x+h)-f(x)) /h

BQ6 continuity

Continuity has no breaks no jumps no holes, you can draw it without lifting your pencil , it is predictable and when the limit and value are the same it is continuous.

discontinuity is broken down into two families- removable and non removable they are broken down into two because one has limits and the other doesn't . a discontinuity has limits. the non removable discontinuities re jump infinite and oscillating

A limit is an intended value it exists when it is a continuous function or if it is a hole a limit does not exist when there is a vertical asymptote because it has unbounded behavior or when it is oscillating and when it jumps because it is different from left to right. the difference of a limit and a value is that a limit is an intended height and a value is the actual height

We evaluate limits numerically by using a table, graphically we just go from left to right going toward the same point, algebraically nu using substitution which is plunging in the limit and if we get 0/0 we use different algebraic methods that can work

BQ#4

Tangent Graphs

Tangents graphs g uphill while cotangent graphs go downhill. the reason being is because of their asymptotes. these you can get by your unit circle. the ratio for tangent is y/x and when x=0 we get an asymptote and when that happens is when it is at 90 and 270 degrees which convert to pi/2 and 3pi/3 so knowing this we know that the graph cant touch it but what also has the affect s the quadrants in a unit circle we know that tangent is positive in the 1 and in the 3rd and negative in the 2 and 4th so this shows you were the part of the graph is going to be above or below the x axis

The same goes to cotangent. its ratios are x/y so this time y has to = 0 and when it does it is at the 0 and 180 degree and the asymptotes for that is pi and 2pi even though that have the same four quadrants the asymptote are the ones that make this graph go downhill instead of uphill like tangent

BQ#3

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Tangent the picture graph on the top shows sine cosine and tangent. since tangent is y/x we know that x is cosine so we get asymptotes whenever the denominator = 0 so that means whenever cosine =0 there will be an asymptote. if one of sine or cosine is negative tangent will be negative
Cotangent is on the bottom of the picture. here we know that cotangent is the opposite tangent so the ratio is x/y so now whenever y=0 there will be an asymptote. but if sine and cos are positive the graph is going down hill if its negative it goes up.

secant is on the top picture as you can see wherever cosine =0 there you can find secant it looks like a parabola because there are asymptotes

 for cosecant It is the last picture as you can see it is the same as secant but you can find cosecant whever sine =0 and it too has asymptotes which give it its shape
 

BQ#5

BQ#5
 Sine and Cosine do not have asymptotes the reason being that sine and cosine will never have a denominator equal 0.when the denominator equals zero it means it is undefined and when it is undefined it will give u an asymptote. if we go back to the unit circle we know that sine and cos will always be over one so it can ever be over zero there for not having asymptotes.  but tangent and the others do because they can be over zero.

BQ #2 Unit T Intro

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BQ#2

There is a period for sine and cosine that is 2pi, it is 2pi because it is one revolution of the unit circle, it is how much times it takes to repeat its cycle. When we use A,S,T,C on out graph we know that Sine is positive n quadrant 1 and 2 and will be negative on 3,4 . now for cosine it is positive on quadrant 1 and 4 but will be negative on 2 and 3 . as you can see it takes 4 to repeat the pattern which is why it takes 2pi but with tangent it only takes pi because it is positive negative then positive and negative again so it repeats after the second one.


the fact that it has amplitudes it makes sine and cosine have restrictions. the restriction are 1 and -1 with this restrictions they cant go past a certain paint un like tangent. if we try any number it wont work because of the restriction so its the same like the unit circle.

Reflection #1 unit Q Trig identities

1.  To verify is to prove an identity. you have to use logical steps and identities to show how you can make one side equal the other.

2. the tricks I found help full was to feel my identities next to me and write everything down because it keeps track of what you are doing.
3. I usually just try to convert everything to sine and cos. I try to look at the problem as a whole and see if anything can cancel out or steps I can take to solve it. it takes me a while but after I know something isn't going to work and move on to try something else I can end up solving .