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.
