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 )
Tuesday, May 30, 2017
Wednesday, May 24, 2017
Why Calculus ?
So you may have heard that I took the AP Calculus BC exam a little while back. Your reaction could have been ugggh ! that stuff is so boring and complicated , why would you ever do that?
( or it could have been Awesome that stuff is so cool! If it was something like that then you do not have to read this but you still can ) I mean when would you EVER have to use that in regular life ?
My article today strives to answer at least part of this question " why calculus " and how it is awesome!
Calculus , like any topic, can be made boring and dissatisfying through mental prejudice and unenthusiasm. When you put it into your brain ( quite unfairly ) that this is not going to be fun and it will never be fun , in a nutshell that you dislike it. This approach must not be the way you look at calculus if you ever want to come to love it and have fun doing it, but rather you at least have to have a open approach to the subject. Preferably you should have a positive, optimistic approach to the subject having a hope that it will be fun and exciting.
What makes Calculus fun is that with knowledge you can solve problems, thereby thereby using your brainpower to get answers. Calculus is really not fun when you do not know how to do it, so learning it is an essential part of the process. Learning can be hard at times and so you need to press forward continue trying and rely on the infinite grace and mercy of our Heavenly Father to learn and understand calculus. Once you learn the material , answering problems becomes easy ( or at least doable) and the excitement of being able to find the answer emerges.
Learning Calculus can be fun to though , once you finally understand a new definition , or theorem it enables you to do something new in math that you did not know how to do before. It opens new doors in your ability to solve problems , and things that seemed impossible now become second nature.
For example , say you knew basic algebra and how to find the slope of a graph, but then you think of a curved graph ( say ln(x) ) and wonder how to find out its slope at a defined point. You can increasingly approximate the slope at the point but not quite reach it using algebra, but with differential calculus you can literally find the slop of a point of a curved graph. For example at x=2 dy/dx of y=ln(x) or the infinemestial slope at the point x=2 of the natural log of x is equal to the 1/2.
While I was learning how to take the volume of irregular solids, I wondered if you could somehow use related rates to find the rate of change of the volume of an irregular solid where the upper integrand is changing. So what this means is that say you have a graph of a function ( for example y=x2) and then you rotate the graph around an axis ( say the x ) so what you get is a solid with a volume that can be determined using integration and the property that each slice of the solid is a circle. The area of each of the circles is 𝛑r² and the one thing that changes between the each of the circles is the radius. The radius at each point or for each circle is equal to the value of the function at that point ( since the diameter is twice the function value ) or x2
If you sum the areas of every circle in the range that you want ( say from 0 to 2 , which numbers are called the lower and upper bounds of integration, or the integrands ) then you get the volume. To sum the areas of every circle you have to use an integral which is and the limit as the number of circles approaches infinity or the limit as the width of the cylinders approaches zero of the sum of each of the cylinders. It would be 𝛑∫(x²)²dx since 𝛑(x²)² is the area of one circle and the sum of all of them is the integral. Using integration rules ( ∫xn=
xn+1
from 0 to 2. The rate of change of this volume was a bit hard for me to determine how to find when given that the upper integrand was changing at some rate, but I eventually got it , with Heavenly Father's help. So the fundamental theorem of Calculus state that the derivative of an integral of a function with respect to the changing upper integrand is equal to the function of the integrand.
By the chain rule the derivative of a composite function f(g(x)) with respect to x is equal to the derivative of f(g(x)) with respect to g(x) multiplied by the derivative of g(x) with respect to x. So say that you were given the rate of change of the upper integrand and asked for the rate of change of an irregular volume produced by rotating f(x) around the x axis. The way to do this is multiply f(x) by dx/dt or the function of x multiplied by the rate of change in x. This is a simpler way than what I did. I used a differential equation to solve for the upper integrand in terms of t and then plugged that into the integral as the integrand and than applied the fundamental theorem and the chain rule.
Once you have mastered Calculus answering problems gives you a "zing" to the brain and it can get mildly addictive, so do not do it to much.
( or it could have been Awesome that stuff is so cool! If it was something like that then you do not have to read this but you still can ) I mean when would you EVER have to use that in regular life ?
My article today strives to answer at least part of this question " why calculus " and how it is awesome!
Calculus , like any topic, can be made boring and dissatisfying through mental prejudice and unenthusiasm. When you put it into your brain ( quite unfairly ) that this is not going to be fun and it will never be fun , in a nutshell that you dislike it. This approach must not be the way you look at calculus if you ever want to come to love it and have fun doing it, but rather you at least have to have a open approach to the subject. Preferably you should have a positive, optimistic approach to the subject having a hope that it will be fun and exciting.
What makes Calculus fun is that with knowledge you can solve problems, thereby thereby using your brainpower to get answers. Calculus is really not fun when you do not know how to do it, so learning it is an essential part of the process. Learning can be hard at times and so you need to press forward continue trying and rely on the infinite grace and mercy of our Heavenly Father to learn and understand calculus. Once you learn the material , answering problems becomes easy ( or at least doable) and the excitement of being able to find the answer emerges.
Learning Calculus can be fun to though , once you finally understand a new definition , or theorem it enables you to do something new in math that you did not know how to do before. It opens new doors in your ability to solve problems , and things that seemed impossible now become second nature.
For example , say you knew basic algebra and how to find the slope of a graph, but then you think of a curved graph ( say ln(x) ) and wonder how to find out its slope at a defined point. You can increasingly approximate the slope at the point but not quite reach it using algebra, but with differential calculus you can literally find the slop of a point of a curved graph. For example at x=2 dy/dx of y=ln(x) or the infinemestial slope at the point x=2 of the natural log of x is equal to the 1/2.
While I was learning how to take the volume of irregular solids, I wondered if you could somehow use related rates to find the rate of change of the volume of an irregular solid where the upper integrand is changing. So what this means is that say you have a graph of a function ( for example y=x2) and then you rotate the graph around an axis ( say the x ) so what you get is a solid with a volume that can be determined using integration and the property that each slice of the solid is a circle. The area of each of the circles is 𝛑r² and the one thing that changes between the each of the circles is the radius. The radius at each point or for each circle is equal to the value of the function at that point ( since the diameter is twice the function value ) or x2
If you sum the areas of every circle in the range that you want ( say from 0 to 2 , which numbers are called the lower and upper bounds of integration, or the integrands ) then you get the volume. To sum the areas of every circle you have to use an integral which is and the limit as the number of circles approaches infinity or the limit as the width of the cylinders approaches zero of the sum of each of the cylinders. It would be 𝛑∫(x²)²dx since 𝛑(x²)² is the area of one circle and the sum of all of them is the integral. Using integration rules ( ∫xn=
xn+1
from 0 to 2. The rate of change of this volume was a bit hard for me to determine how to find when given that the upper integrand was changing at some rate, but I eventually got it , with Heavenly Father's help. So the fundamental theorem of Calculus state that the derivative of an integral of a function with respect to the changing upper integrand is equal to the function of the integrand.
By the chain rule the derivative of a composite function f(g(x)) with respect to x is equal to the derivative of f(g(x)) with respect to g(x) multiplied by the derivative of g(x) with respect to x. So say that you were given the rate of change of the upper integrand and asked for the rate of change of an irregular volume produced by rotating f(x) around the x axis. The way to do this is multiply f(x) by dx/dt or the function of x multiplied by the rate of change in x. This is a simpler way than what I did. I used a differential equation to solve for the upper integrand in terms of t and then plugged that into the integral as the integrand and than applied the fundamental theorem and the chain rule.
Once you have mastered Calculus answering problems gives you a "zing" to the brain and it can get mildly addictive, so do not do it to much.
Monday, May 15, 2017
AP Physics C:Mechanics
I took the AP Physics C:Mechanics exam last week, and it was a bit challenging, but I am pretty sure I passed ( for BYU that means get at least a 4 ). You see I had not studied it nearly as weel or as much or as enthusiastically as Calculus and I did not understand it as well and Calculus BC. The Physics exam had more than just math ( unlike the AP Calculus BC exam did ) it had theory and laws and you had to know how to solve the problems. Which luckily I did ( most of them ) but there were still some that had me a bit confused. The Free response especially , which had a whole question on the rolling motion of a ball , was confusing. But I believe that I wrote enough and got enough of the Multiple choice correct to get a 4.
The Exam is separated into two sections: multiple choice and free response , where the MC part has ~35 questions which you have to fill in a letter on a answer sheet, and the FR part has 3 questions where you have to write. You are given 45 minutes for both parts, and so an hour and a half for the whole exam.
The Exam tests kinematics, forces and newtons laws, work and energy, linear momentum , impulse and center of mass, rotational motion, gravitation, and oscillations.
Kinematics deals mainly with the study of motion, how things move ( velocity, displacement, acceleration and the equations that relate these quantities and time ) and is only up to two-dimensional in AP Physics c mechanics with projectile motion. In this course both kinematics under constant acceleration and kinematics under varying acceleration are covered ,where in the first case the kinematic equations are used and in the second case calculus is used.
Forces and newtons laws deal with why objects move and the forces that make them move. The method of drawing free body diagrams and solving newtons second law equations (F=ma) are covered as are the 3 laws ( inertia , F net = ma, action reaction pairs ). Inclined planes and force trigonometry is used to find components of forces in different directions. Different types of force, such as the normal force, the gravitational force, the frictional force and centripetal force are covered.
Linear Momentum is the mass of an object times its velocity or the integral of force with respect to time for varying forces. Impulse if equal to momentum as stated in the impulse-momentum theorem, and calculations are done on these equations. Conservation of linear momentum is covered and so are the different types of collisions ( elastic , inelastic and completely inelastic ). The method for calculating the center of mass of both a system of particles and a continuous body are also covered.
Work and energy deal with the integral of force with respect to distance ( for varying force), Mechanical , Potential , and Kinetic energy, potential energy graphs, conservation of mechanical energy, non conservative energy, basic gravitational and spring potential energy, work energy theorem , and power.
Rotational motion deals with the rotational analogous of the fore covered topics of kinematics and dynamics when rotating around a fixed axis ( so it can only rotate in two directions ), and how to solve problems with this. These include but are not limited to angular displacement , velocity, and acceleration, arc length , torque , rotational inertia, and rotational kinetic energy.
Gravitation deals with Kepler's 3 laws,Newtons law of gravity, actual gravitational acceleration, the equating of centripetal force and gravitational force for circular orbits, the finding of the gravitational force inside a sphere, the actual gravitational potential energy and the discovery of escape velocity, and the like
Oscillation deals with springs, the restoring force for simple harmonic motion, potential energy for springs, period and frequency, the model for sinusoidal simple harmonic motion of position, velocity and acceleration, pendulums and the simple harmonic motion equations for ones that are displaced a small angle of theta.
It is so awesome how newton discovered these laws that rule the motion of objects ! Our Heavenly Father loves it when we strive to find the truth and with his help we do. He helped me learn Physics and I am so thankful for this.
Kinematics deals mainly with the study of motion, how things move ( velocity, displacement, acceleration and the equations that relate these quantities and time ) and is only up to two-dimensional in AP Physics c mechanics with projectile motion. In this course both kinematics under constant acceleration and kinematics under varying acceleration are covered ,where in the first case the kinematic equations are used and in the second case calculus is used.
Forces and newtons laws deal with why objects move and the forces that make them move. The method of drawing free body diagrams and solving newtons second law equations (F=ma) are covered as are the 3 laws ( inertia , F net = ma, action reaction pairs ). Inclined planes and force trigonometry is used to find components of forces in different directions. Different types of force, such as the normal force, the gravitational force, the frictional force and centripetal force are covered.
Linear Momentum is the mass of an object times its velocity or the integral of force with respect to time for varying forces. Impulse if equal to momentum as stated in the impulse-momentum theorem, and calculations are done on these equations. Conservation of linear momentum is covered and so are the different types of collisions ( elastic , inelastic and completely inelastic ). The method for calculating the center of mass of both a system of particles and a continuous body are also covered.
Work and energy deal with the integral of force with respect to distance ( for varying force), Mechanical , Potential , and Kinetic energy, potential energy graphs, conservation of mechanical energy, non conservative energy, basic gravitational and spring potential energy, work energy theorem , and power.
Rotational motion deals with the rotational analogous of the fore covered topics of kinematics and dynamics when rotating around a fixed axis ( so it can only rotate in two directions ), and how to solve problems with this. These include but are not limited to angular displacement , velocity, and acceleration, arc length , torque , rotational inertia, and rotational kinetic energy.
Gravitation deals with Kepler's 3 laws,Newtons law of gravity, actual gravitational acceleration, the equating of centripetal force and gravitational force for circular orbits, the finding of the gravitational force inside a sphere, the actual gravitational potential energy and the discovery of escape velocity, and the like
Oscillation deals with springs, the restoring force for simple harmonic motion, potential energy for springs, period and frequency, the model for sinusoidal simple harmonic motion of position, velocity and acceleration, pendulums and the simple harmonic motion equations for ones that are displaced a small angle of theta.
It is so awesome how newton discovered these laws that rule the motion of objects ! Our Heavenly Father loves it when we strive to find the truth and with his help we do. He helped me learn Physics and I am so thankful for this.
Saturday, May 13, 2017
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:
2. Derivatives
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.
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:
- Limits
- definition,
- evalutaion,
- L'Hoplitals rule,
- asymptoes ,
- continutity,
- types of discontinuty
- theorems
- 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)
- 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
- 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.
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