Standards‎ > ‎

Accumulating Change (AC)

AC.1

RIEMANN SUMS
I can approximate the area underneath a graph using a Riemann Sum or a trapezoidal approximation, and can determine the accuracy of my approximation compared to other approximations. 
Video about Riemann sums. Also available here and on Al-Khazneh

AC.2

INDEFINITE INTEGRAL
I can calculate basic indefinite integrals (antiderivatives) for power and trigonometric functions. 

AC.3

 
 INITIAL CONDITIONS
I can determine a function given its derivative and an initial condition (finding a specific antiderivative) and can do so in applied conditions. 
 Practice from the "flip up for the answer" activity we did in class:
(also available in the attachments)

AC.4

 
 DEFINITE INTEGRAL
I can evaluate a basic definite integral using the First Fundamental Theorem of Calculus (both by hand and with a graphing calculator).
 

AC.5

 
 AREA vs. INTEGRAL
I can distinguish between "an integral" and "the area" and can apply the same concept to an applied situation with displacement and distance.
 

AC.6

 
 NET CHANGE THEOREM
I can interpret the integral of a derivative as the net change of that function and can use this idea to solve problems, especially in applied situations including units.
See AC.8 below for that packet we worked on in class that started with the Net Change Theorem.
 

Video about the Net Change Theorem. Also available here and on Al-Khazneh.
 


Problem from class:

A factory is polluting a lake in such a way that the rate of pollutants entering the lake at any time t, in months, is given by R(t)=280t^(3/2). measured in kilograms per month. Define N(t) to be the total number of kilograms at any time t.

a. How many kilograms of pollutants enter the lake in the first 16 months?
b. How many kilograms of pollutants enter the lake in the next  16 months after that?
c. The lake must be cleaned when 50,000 kg of pollutants have entered the lake from now. How long until this occurs?
d. If the lake originally had 1,000 kg of pollutants, determine an equation for N(t). How many kilograms of pollutants are in the lake after 16 months and after 32 months?


AC.7

 
 AVERAGE VALUE
I can use an integral to determine the average value of a function over an interrval, and can appply this idea to applied situations with units.
 

AC.8

 FUNCTIONS DEFINED BY INTEGRALS
I can interpret the notation of a function defined by an intergral both algebraically and graphically, and can use the Second Fudamental Theorem of Calculus to solve problems involving these functions.
 Video of me solving an AP problem about this topic. Also available here and on Al-Khazneh.



Answers to the packet we worked on in class about this topic (below are the answers to the AP Problems from the back:
(also available in the attachments)

Answers to the AP problems from the back of the packet we worked on in class:
(also available in the attachments)


AC.9

 
 INTEGRATION BY SUBSTITUTION
I can use U-substitution to integrate a certain type of function, and can recognize when the U-substitution is applicable.
Definite Integral U-sub Practice:

AC.10

 
AREA BETWEEN CURVES
I can determine the area bounded between two curves. 
 Class notes on area between curves:

 Answers to the area between curves puzzles that we did:

AC.11

 
VOLUME: CROSS SECTIONS
I can calculate the volume of a solid generated with its base a region in the xy plane with cross sections of a known shape.
 Class notes on solids of known cross section:

 Answers to the solids of known cross section puzzles that we did:

AC.12

 
VOLUME: REVOLUTION
I can calculate the volume of a solid generated by revolving a region around a line, both in the x and the y direction and both around the coordinate axes and other lines.
 Video of me solving an AP problem:

Answers to all of the volume "strips" we did in class (including one at the end for practice that we did NOT do together:

AP Problems and Answers for AC.10-AC.12 (also available in the attachments):


AC.13

 
INTEGRALS
GRAB BAG
I can comfortably integrate a wide range of functions and can figure out which of my many differentiation tools to use where.
  Rules and techniques to know and to be able to distinguish from each other:
-power rule
-1/x (logarithmic integrals)
-exponential functions (of all bases)
-trig functions including basic identities
     -basic trig relationships
     -double angle for sin and cos
     -pythagorean for sin/cos, sec/tan and csc/cot
-substitution
-change of variables
-basic algebraic manipulation to clear a quotient or a product


Č
Ċ
Bowman Dickson,
Feb 2, 2013, 7:22 AM
Ċ
Bowman Dickson,
Feb 20, 2013, 9:35 AM
Ċ
Bowman Dickson,
Apr 21, 2013, 8:45 AM
Ċ
Bowman Dickson,
Apr 21, 2013, 8:45 AM
Ċ
Bowman Dickson,
Apr 21, 2013, 8:45 AM
Ċ
Bowman Dickson,
Apr 21, 2013, 8:45 AM
Comments