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Proportional Rates (PR)


PR.1

RELATED RATES
I can solve applied problems involving related rates, including setting up a problem based on a written description, differentiating implicitly and solving for a given quantity or rate.
 Answers to Related Rates Packet 
(available in the attachments at the bottom too)



PR.2

 
OPTIMIZATION
I can solve basic optimization problems including setting up equations for the volume and area of simple shapes and solids.
 Answers to Related Rates Packet 
(available in the attachments at the bottom too)

PR.3

 
 LOGARITHMIC DERIVATIVES
I can differentiate functions involving the natural logarithmic function and can apply the technique of logarithmic differentiation to differentiate algebraically complicated expressions.
 Video about logarithmic differentiation. Also available here and on Al-Khazneh.


PR.4

 
 LOGARITHMIC INTEGRALS
I can use the Log Rule of Integration to integrate a rational function and can use this to integrate trigonometric functions.
 

PR.5

 
 EXPONENTIAL CALCULUS
I can differentiate and integrate natural exponential functions.
 Deriving the fact that d/dx[e^x] = e^x


PR.6

 
 "UNNATURAL" BASES
I can differentiate and integrate exponential functions of a base other than e.
 Class Notes for "Unnatural Bases":


 

PR.7

 SLOPE FIELDS
Given a differential equation, I can draw a slope field and sketch a particular solution through a given point. I can also match a slope field with a differential equation and vice-versa.
 Video about slope fields. Also available here and on Al-Khazneh.

PR.8

DIFFERENTIAL EQUATIONS
Given a separable differential equation, I can solve for the general solution and plug in initial conditions to find a particular solution.
 The three most important tricky situations you might encounter in solving diffEQs:

1. Flipping a whole expression
   ex: dy/dx = x^3y^2   through (0,2)

Answer: y = 4/(2-x^4)

2. Choosing the correct branch of a solution when you take a square root
   ex: dy/dx = x^3/y   through (1,-1)

Answer: y = -sqrt(x^4/2+1/2)

3. Dealing with constants and exponential functions
   ex: dy/dx = y/x^2   through (-1,4e)

Answer: y = 4e^(-1/x)

PR.9

 MODELS
I can use specific differential equations to model exponential decay and growth, and can perform a range of tasks with the equation that I create (including solving for variables)
 Video about an important exponential models. Also available here and on Al-Khazneh.

PR.10

 
 INVERSE FUNCTIONS
I can prove that a function has an inverse by showing that it's monotonic and can find the inverse in many cases. Also, given information about the slope of a function, I can calculate the slope of its inverse.
 Drill on finding the tangent line to a given point for an inverse function:

http://archives.math.utk.edu/visual.calculus/3/inverse.1/index.html

Notes from class on review/pre-cal topics:

PR.11

 
 INVERSE TRIG FUNCTIONS
I can evaluate values of arcsin, arccos and arctan by hand, noting their domain and range, and can differentiate functions involving these three functions
Arctrig evaluation practice (not derivatives!)




 Multiple Choice Review/Practice for PR.10 and PR.11:


Ċ
Bowman Dickson,
Nov 16, 2012, 7:23 AM
Ċ
Bowman Dickson,
Mar 13, 2013, 4:30 AM
Ċ
Bowman Dickson,
Mar 4, 2013, 3:34 AM
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