Differentiation

A Derivative of a function is the instantaneous rate of change of that function at a point. It's the limit as the change in x approaches zero of the average slope of a graph. The operation of taking a derivative of a function can be written in different notation, like f"(x) or d(f(x))/dx . The rules for taking derivatives of some basic functions are outlined below:

if n is a constant and u and v v are functions of x
                                                 
when f(x)=                                             |                                                    f '(x)=
un                                                           |                                             u'(x)ᐧ (n)un-1 
nu                                                           |                                             u'(x)ᐧln(n)ᐧnu
n                                                             |                                                 0
uᐧv                                                          |                                          (u'(x)ᐧv)+(v'(x)·u)
u                                                             |                                        (u'(x)ᐧv)-(v'(x)·u)
v                                                             |                                                   v2 
u(v(x))                                                    |                                          u'(v(x))ᐧv'(x) (d(u(v(x)))/(v(x)) )(v'(x))
sin(x)                                                      |                                                   cos(x)
cos(x)                                                      |                                                   -sin(x)
tan(x)                                                     |                                                   (sec(x))2
cot(x)                                                     |                                                    -(csc(x))2
arcsin(x)                                                 |                                                 u'(x)ᐧ    1    
                                                               |                                                        √1-u2
arccos(x)                                                 |                                                     -u'(x)ᐧ    1    
                                                               |                                                               √1-u2
arctan(x)                                                 |                                                     u'(x)ᐧ    1    

                                                               |                                                               1+u2
ln(x)                                                       |                                                       1/x
log base a (x)                                         |                                                     1/x(ln(a))

to differentiate a function like xx,you need to use the natural log rule: ln(xu) =uᐧln(x).
When given a table of values you can estimate the derivative at a point at or in between input values given.

If a function is differentiable on an interval then the function is continuous on the interval but if the  function is continuous on the interval then the function is not necessarily differntiable ( i.e. functions can be non-differentiable and continuous ( with sharp cusps and the like ).

The MVT states that if a function is continuous and differentiable on an interval [a,b] then the is some point c where f '(c) is equal to the slope between a and b or the average rate of change over the interval. Rolles theorem is a special case of the mean value theorem when the average rate of change is 0.


Do differentiate parametric functions you take the derivative of y with respect to t and then divide id by the derivative of x with respect to t.

To differentiate polar functions you use the definitions y=rsinθ and x=rcosθ and then use parametric differentiation.

Two take second derivatives ( f "(x) ) you take the derivative of the derivative, to take third derivatives you take the derivative of the derivative of the derivative or the derivative of the second derivative and so on.

To take derivatives of inverse functions at a point, you take the reciprocal of the derivative of the corresponding point.

When a derivative of a function is undefined or equals zero you have got a critical point of that function. You can use derivatives to determine if a function is increasing( i.e. the derivative is positive ), or decreasing ( negative f '(x) ), concave up or concave down ( the second derivative is positive for concave up and negative for concave down. ). These determinations can be used to find maxima or minima of functions. They can also be used to find where the concavity of the graph changes ( a inflection point). There are multiple methods to do this including finding critical points and evaluating the second derivative at the points ( second derivative test, if it is positive it is a local minimum , if it is negative it is a local maximum ), or evaluating f ' before and after the critical point.
Finding local maxima and minima and points of inflection are all part of derivative applications, like useful thing you can use differential Calculus for. Approximating the value of a function using slope tangent line approximation is also part of Derivative applications. Basically you use the equation
y-y1= f '(x1)ᐧ(x-x1). Where you can approximate y values ( the y ) of x values ( the x ) near the  x1 value. The y1  is the y value at the x1.

Another application of derivatives is related rates. Which is when you have a object, and some measurement of it is changing at a given constant rate, while some other property or properties is/are changing at a non-constant rate. You use an equation that relates the two properties and then solve it for the rate of change of the non-constant one at some point in time.

One application of differential calculus has a bunch to do with Physics, namely the analyzation and study of the movement of a particle given its function of either , position, velocity, or acceleration with time. The derivative of position is velocity, the derivative of velocity is acceleration. Which means that the slope of a position versus time graph is the velocity and the slope of a velocity versus time graph is the acceleration. These definitions can help us analyze and determine when a particle is moving in a certain direction , when it is not moving, when it is slowing down or speeding up ( when the acceleration and velocity are in the same direction then the particle is speeding and when the velocity and acceleration are in different directions the particle  is slowing ), and how fast or how much a particle is accelerating at a point.

Differential calculus can also be used to solve optimization problems, in which you write a function for a thing to me maximized or minimized and then take the derivative(s) of that function to find the local an global maxima or minima.

Limits and Continuity

This is the first big idea in AP Calculus BC, and it lays the basis for most of Calculus. So Calculus is interested in finding the values of things that are infinitesimally small ( like the slope of a point or the area under a curve) and to find these values limits are used. Take for instance the area under the curve, to approximate it you can divide the curve into rectangles, but the area will only be that an approximation. The smaller the width and the more rectangles there were the better the approximation would be. So what calculus does is take the limit of the sum of the rectangles as the width of the rectangles approaches zero ( or the number of rectangles approaches infinity ) which is the exact area. I'll write more on finding areas under curves when I get to big idea 3 Integration.

So a limit of a function as the input approaches some value is really the value the output or the function approaches. So the function doesn't have to be defined at a point to have a limit there.
There are many ways to find limits of function, including using graphs and just plain algebra. Limits can come from two different directions , the negative direction ( or from the left side of the graph , i.e. the value the function approaches from input values less then the value you are taking the limit at ) and the Positive direction ( from the right side of the graph with values larger than the desired input value ). The notation for the two different types of limits looks like this:

lim f(x) = L                                                                            lim f(x) = L     
x→c -                                                                                      x→c +

From the negative direction                                         Positive Direction

So the → means approaches and the x→ c means as x approaches c . The Limits from each side do not have to be the same, and when they are not there is a weird jump in the function.

Some times to approximate limits algebraically you just have to plug in the c value for x in the function and it gives you the limit. But if this limit is undefined ( ∞/∞  or 0/0 ) then you are going to have to try some different methods. If the function has a quotient , then divide the top and bottom by the highest power of x , and then take the limit. There is also a way that you can factor a function out and then cancel expressions leaving a defined limit. A very useful rule for evaluating limits with indeterminate form  ( +∞/∞  or 0/0 ) is L'Hopitals rule which states that if you have a limit in indeterminate form , then you can take the derivative of the top and bottom ( separately  ) and then take the limit of it.

lim    f(x) =+∞/∞  or 0/0                then           lim    f(x) =lim  f '(x)           
x→c g(x)                                                           x→c g(x)  x→c g'(x)     

A function is continuous if ( all of the following conditions are met ):
            lim f(x) = lim f(x) = L                                                        
            x→c -         x→c +

( so the lim f(x)  is defined and equals L )                                                                             
             x→c 

lim f(x) = L =f(c)                                                                               
x→c 

( or in other words the function must be defined at x=c, and the limit of the function as x approaches c must be defined and equal the function's value at x=c )

A point that needs to be covered here is that if the limits of the function from the right hand and the left hand are not the same than the limit of the function at that point is not defined ( the limits from either side have to equal each other for the whole limit to be defined. )

Asymptotes  are places where either the limit as x approaches infinity equals that value ( for y value asymptotes or horizontal asymptotes ) or the limit as x approaches that value equals infinity ( for x values , or vertical asymptotes )

Continuity allows us to have some theorems, like the intermediate value theorem and the extreme value theorem ,and later on the Mean value and Rolles theorem. The Intermediate value theorem states that if a function is continuous on the interval [a,b] then there is some value c that if between the values f(a) and f(b) for the function. The extreme value theorem states that if a function is continuous on an interval it will hit a highest value on an interval. The Means value theorem ( MVT ) and Rolles theorem which is a special case of MVT have to do with derivatives.

There are different types of discontinuity , jump, removable and infinite discontinuities:

jump discontinuities happen when the right hand and the left hand limits don't equal each other ( the function " jumps " at the discontinuity )

removable discontinuous are when the limits equal each other but not the functions defined value.

Infinite discontinuities happen when functions approach positive or negative infinity at a point ( which actually means that their limit and value are undefined )

AP Calculus BC !

I took the AP Calculus BC exam last Tuesday and it was awesome!

I had studied for it for about 4 months before and It was quite fun to take it. I know that I could never have taken it without my Heavenly Father's help.
The free response questions ( released on college board here ) were harder than the past few FRQs that I have taken ( like the 2013, 2012, and 2016 ones).

For example the first calculator one ( or #1) did not ask for a estimation of a derivative ( as the earlier ones usually do) and asked for the rate of change of a volume that required an integral, which I had pondered about earlier while I was taking the course. Because I had learned about related rates for objects and shapes that had well defined formulas for area , volume and length ( circles, right triangles, cubes and the like), and I started learning how to take volumes of irregular shapes using integrals ( like volumes of rotation using discs and washers, and volumes with cross sections that had known areas ). So I wondered, If you knew the rate at with the upper bound of integration was increasing , could you solve for the rate at which the volume was changing ? I was not able to figure it out right then, and I still wondered it as a kept learning calculus.

But later on ( like 1-2 weeks before the exam ) it came to me! All I had to do was solve a basic differential equation, sub the solved value for the upper bound of integration as a function of time into the integral, and then use the chain rule and fundamental theorem to solve it !
It was a wonderful blessing from my Father in Heaven to allow me to figure this out, and have it on the exam!

The second FRQ was crazy!! it was a BC topic only that involved two polar functions and their graphs and had some really interesting parts. Part a was basic as it had you take the area of one of the polar graphs in the first quadrant, which since a calculator was allowed was not to complicated after you set up the integral.

Part b was cool because it asked you to write an equation that had a ray k that separated one of the graphs into two equal areas in the first quadrant. But parts c and d were where it got me really thinking !

Part c had two functions whose x and y coordinates were determined by functions ( which was basically a parametric form).The x coordinate of the first was determined by the a function of theta that equaled one of the polar functions, while the other was determined as a function of theta that equaled the other polar one. Both of the y coordinated were determined as theta. The question asked me to find an function in terms of theta that was the distance between the two graphs at any theta. The way I went about this problem was to use the vector formula for distance ( sqrt(x^2+y^2) ) and then minus the one with greater x coordinate from the one with lesser x coordinate.

Then the question asked for the average value ( over an interval ) for the function I had written , which made my have to plug in a whole bunch of notation into my calculator and then use it to integrate.  The last part of the question was where I had to take a derivative at a point of the function and determine if the function was increasing or decreasing. I remember urging my calculator on when it was taking to long to integrate the function

There were some hard parts on the exam that I did not really know how to do, but overall the exam was good. I had time left during Section I after finishing so I wrote James 1:5-6 from memory and my testimony.

The AP Calculus BC exam ( and AB ) was (were ) changed this school year to include a few more topics and with a different format. This exam I think made students think more and use multiple things they learned to answer questions.

Calculus was fun and I hope to get a 5 on the exam with Heavenly Father's help.
The course had 4 big ideas:

1. Limits

• definition,
•  evalutaion,
•  L'Hoplitals rule,
•  asymptoes , 
• continutity, 
• types of discontinuty
• theorems 

       2. Derivatives

• definition( slope of a point )
• rules ( polynomial, exponential, trigonometric, logarithmic, chain, product, quotient )
• as a limit
• approximation
• theorems ( mean value, Rolle's )
• differentiability and continuity
• tangent line approxiamtion
• parametic differation
• polar differentiation
• related rates
• higher degree derivatives 
• of inverse functions
• other applications ( maximum and minimum, points of inflection and concavity )
• movement of a particle ( distance, position, velocity, speed and acceleration)

       3. Integrals

• definition
• antiderivatives
• rules ( polynomial, exponential, trigonometric, logarithmic )
• fundamental theorem 
•  u-substitution
• by parts
• by partial fractions
• definite integral
• definite integral properties
• applications of integration
• areas under curves
• volumes of irregular solids
• approximation of using Riemann sums
• parametric integration
• polar integration
• movement of a particle ( distance, position, velocity, speed and acceleration)
• differential equations ( including logistic growth and euler's method )
• improper integrals

        4. Series 

• definition ( sum of a sequence )
• geometric
• infinite
• interval and radius of convergence
• p-series
• tests for convergence ( nth term, comparison, limit comparison, ratio, integral etc. ) 
• error bound ( for alternating and with Lagrange error bound )
• expressing functions as a series
• Power series ( Taylor and Macluraian )

It is pretty hard to wait for my scores until July 5th but I can't do much about it.

I will be posting about every week on AP Calulus BC topics ( Limits, Derivative, Integrals, Differential Equations, and Series ).

Calculus was awesome and fun to do ( once I learned the rules ) but it also has some practical and useful applications, since it allows you to find areas and slopes of things not defined in regular algebra and geometry. Graphs of thins like membership versus time , or profit versus time usually are not straight lines and perfect circles, and calculus enables you to find the instantaneous rate at with your membership ( or profit) is growing.  With differential equations you can also find the population at some time in the future if given a solvable differential equation for the rate of growth of the population, and an initial t=0 population.

AP Government , Art, Running and Precalculus

AP US Government
I have been taking AP US Government on shmoop. I also have been reading the complete Idiots Guide to US Government  to prepare for the AP Exam in may.
Art

I have been taking Art 1 at the high school and have been taking an online 3D animation course trough youth digital with Blender. 

Precalculus

I have been doing Precalculus on Khan Academy and I have completed~50% of it. 
Running

I have been running recetly and have been doing 2-7 miles every day except Sunday, cross workout ( swimming ) days, and snow/rain days. 

Some Math and Surtsey

In math I have recently been evaluating sequences in recursive form. 

$Here\; is\; problem\; that\; I\; copied\; off\; Khan$Academy ( and how to solve it ):

$f$ is a function that describes a sequence, and is therefore defined over the positive integers. This is the recursive definition of the function:

$\qquad f(n) = \begin{cases} -3 & \text{if } n=1 \\ -5 \cdot f(n\!-\!1) -7 & \text{if } n > 1.\end{cases}$

Generate the first four terms of the sequence.

$f(1)=$
$f(2)=$
$f(3)=$
$f(4)=$

so f(1)=-3 ( as the table tells us )

and then f(2)=-5*f(2-1)-7=-5*f(1)-7=-5*-3-7=15-7=8

and f(3)=-5*8-7=-40-7=-47

finally f(4)=-5*-47-7=228

I learned about a volcanic island today, named Surtsey, which is off the coast of southern Iceland. The name of the island came from a fire giant from Norse mythology, named Surtr ( Iceland was populated by Norsemen before 1000 AD , and has Norse culture ). It started erupting above surface in November 1964, and has been dormant since about 1966. The first sighting of it erupting was made by a cook on a nearby ship. Iceland claimed it and it is part of Iceland. It has been used as a scientific researching area, by volcanologists ( volcano geologists ) and biologists. The volcanologists studied the eruption and how the island fought the erosion caused by the sea ( among other things ). The biologists studied the founder populations of living organisms that populated the island coming from the outside. The sea continually eroded the Island, but the Eruptions added more and more rock to the island so it grew . After 3 or so years though , the eruptions abated ( and then stopped )  and the erosion damage started making an impact. The Island is getting smaller every year, but once the sea gets to the hard strong ( almost ) non-erodible rock at the core of the Island, the erosion rate will decrease. The living organisms the biologists studied include insects, plants ( including non vascular and vascular ) and birds. The insects and plants were first, and then the birds came, and the way they are connected in a food web is amazing. The Insects eat the plants, then the birds come and the insects provide food for them. There were at first few plant species, but when the birds came and fertilized the soil more , more species were able to survive. Also dead birds provided food for carnivorous insects and decomposers. The Island became protected and only scientists were allowed to go on shore. Everyone who goes on shore now is checked to make sure they are not bringing any new seeds to the island.