BQ #7: Unit V: Derivatives and the Area Problem

How is the Difference Quotient derived ?

Throughout this entire year, we have practiced and memorized the difference quotient through song, and rhythm. In order to understand it, one must know that it is derived from the slope of a secant line, that touches two points on the function.  

http://cis.stvincent.edu/carlsond/ma109/DifferenceQuotient_images/IMG0413.JPG

A secant line is when it touches two points of the functions graphed. touching two points we know that the slope is not steep, but is negative or positive, depending on where it points. 

http://upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Secant_line_and_curve_basic.svg/450px-Secant_line_and_curve_basic.svg.png

A tangent line is when it touches the graph only ONCE! This shows that the graph is steep, and or the slope is equal to 0, that means the line is horizontal. 

http://mathinsight.org/media/image/image/neuron_firing_rate_kink_tangent_line.png

BQ #6: Unit U

1. What is Continuity ?

A continuous function is predictable. It has no holes, no breaks, and no jumps. It can be drawn with a single, unbroken pencil stroke. A continuous graph has to have the SAME limit and value; meaning the limit can only be reached with a continuous graph.

What is Discontinuity? 

A discontinuity is any type of break, hole, or jump from a graph. The limit and the value of a discontinuous graph is always different from the value. There are two families of discontinuity: Removable and Non- Removable. Removable discontinuities (also known as holes), the limit does exist, but the value id undefined. Non- Removable discontinuities, the limit Does Not Exist, but the value can.

http://www.zweigmedia.com/RealWorld/calctopic1/pics3/pic1.gif

2. What is a limit? 

 A limit is the intended height of a graph. the limit can be a numerical, zero, undefined, and indeterminate. The limit DNE in Non-Removable discontinuities: jumps (comparing left and right behavior), oscillating, and infinite (unbounded behavior).

When does a limit exist?

 A limit exist when a height is being shown with the exceptions of : two opened circles, and any open circle in general.

http://i45.tinypic.com/152hn36.jpg

When does a limit not exist? 

A limit does not exist when we have opened circles, and wherever there may be vertical asymptotes. If the answer to a limit is 0/a number, the limit DNE. having different points on an X-point (jump) the limit DNE because of different Left and Right. Oscillating behavior, the limit DNE.

What is the difference between a limit and a value?

A limit is the INTENDED height of a graph, does not mean the value is the same.
A value is the ACTUAL height of a graph, it can be determined as undefined at times.

3. How do we evaluate graphs numerically, graphically, and algebraically? 

Numerically is evaluated with a table where we get VERY, VERY, VERY, VERY, VERY, close to the point, but never reach it.

https://finitemathematics.wikispaces.hcpss.org/file/view/limit_table.PNG/239144381/575x194/limit_table.PNG

We evaluate Graphically by using two fingers and following our graph to see if they meet. The identification of the graph and its discontinuities, and continuities.

http://www.coolmath.com/precalculus-review-calculus-intro/precalculus-algebra/images/21-graphing-04.gif

We evaluate Algebraically by using the direct substitution, dividing out/ factoring method, and rationalizing/ conjugate method.

http://www.analyzemath.com/calculus/limits/find_limit_3_2.gif

BQ #4: Unit T Concept 1-3: Why is a “normal” tangent graph uphill, but a “normal” cotangent graph downhill?

Why is a “normal” tangent graph uphill, but a “normal” cotangent graph downhill?

For a Tangent graph we know that it is positive only in quadrant I and III based off the Unit Circle. A tangent graph has the ratio of sine/cosine, and for an asymptote to appear the denominator must be zero; meaning that the value of cosine must equal 0.To complete a full period, an asymptote must go from a negative to a positive, creating an uphill graph.

A cotangent graph is bit similar to that of a tangent graph, but a few things differ. For instance the ratio is cosine/sine; meaning that the sine must equal 0 to give it an asymptote. so in order for the cotangent to complete a full period with an asymptote, the graph has to start from a positive and move to a negative, creating a downhill graph.

BQ #3: Unit T Concepts 1-3: How do the graphs of sine and cosine relate to each of the others?

Tangent's ratio is, as we all know, Cosine/Sine, which relates to the reason to why it has asymptotes. Sine has the possibility of having the value of 0, make the ratio undefined, creating an asymptote in the graph. According to the Unit Circle, Sine is positive in the I and II quadrants, Cosine is positive in the I and IV quadrants and Tangent is positive in the I and III quadrants. Cotangent is but the reciprocal of Tangent, meaning that the ratio is Sine/Cosine. Cotangent is also another that includes an asymptote in it's graph because of the undefined ratio solution. 

Secant is the reciprocal of Cosine, meaning the ratio is R/X giving it an asymptote if the x (cosine) is equal to 0. 

Cosecant is the reciprocal of Sine, meaning that the ratio is R/Y giving it an asymptote if the y (sine) is equal to 0 

According to this graph, there are no asymptotes present, but it does give a small interpretation of how the Unit Circle unfolds onto a graph. the loop going up is divided in half and the first half is the first quadrant and the second half is the second quadrant, so on and so on. these quadrants represent the value of the sign that the trig functions have in the graph. 

BQ #5: Unit T Concept 1-3: Why do cosine and sine not have asympototes, but the other four trig functions do?

According to the Unit Circle, Sine and Cosine were the only trig functions that were not able to equal zero with their ratio. Sine being y/r and Cosine being x/r, the r representing 1, NEVER can the ratios equal zero. An asympototes only appear when the ratio of a trig function is equal to zero, which is why every other trig function ( CSC, SEC, TAN, COT) do have asymptotes.

(http://www.sosmath.com/trig/Trig2/trig2/img6.gif)

BQ #2: Unit T Concept Introduction

How do the trig functions relate to the Unit Circle?

The trig functions relate to the unit circle based on the signs they hold in the Unit Circle that distinguish how the graph will look on the x and y graph. The x and y graph is the unfolded Unit Circle. Sin has a pattern of positive, positive, negative, negative. Cos has a pattern of positive, negative, negative, positive. Both sin and cos go through ONE cycle while covering 2pie units. The trig function of tan is different in the way that tan's pattern is positive, negative, positive, and negative and tan and cot go through ONE cycle while covering pie units.

How does the fact that sine and cosine have amplitude of one relate to what we know about the Unit Circle?

Sine and Cosine have amplitudes because of the ratio they share. Amplitudes are only found when the ratio is undefined, meaning that the denominator is 0. Sine and Cosine (CSC and SEC) have the ratio of R/ X or R/Y and with that the answer cannot equal zero, whereas Tan and Cot have the possibility of equaling and being left with a ratio of undefined, leaving it to have an Amplitude.

Reflection #1: Unit Q concepts 1-5: Identify Trig Functions

1. To verify a trig function means to break down an equation of trig functions to the smallest, most simplest trig function possible. that means to put all our knowledge of previous math to work. All the information we learned from the Unit Circle are being tested in this unit. All equations must break down to one trig function. That is your goal! 

2. The tips and tricks that i have found helpful to me is the Unit Circle and the identities that we have had to memorize. The Unit Circle's points and degrees are a part of Units 1-5. The trig identities are what come to finding our answers much faster than any other method. 

3. To begin breaking down a trig function, i first look to see if it is possible to split or substitute anything into the equation (You look for the fastest way to break down your equation). Next we see if any identities are found or can be made with what we have broken down. If not, we continue with what we have, either multiply or divide functions. The final thing i look for is the possibility if the function can be substituted, distributed, or identified (usually will be one of these).