PR.1
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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)
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PR.2
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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)
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PR.3
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LOGARITHMIC DERIVATIVES
I can differentiate functions involving the natural logarithmic function
and can apply the technique of logarithmic differentiation to
differentiate algebraically complicated expressions.
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Video about logarithmic differentiation. Also available here and on Al-Khazneh.
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PR.4
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LOGARITHMIC INTEGRALS
I can use the Log Rule of Integration to integrate a rational function and can use this to integrate trigonometric functions.
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PR.5
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EXPONENTIAL CALCULUS
I can differentiate and integrate natural exponential functions. |
Deriving the fact that d/dx[e^x] = e^x
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PR.6
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"UNNATURAL" BASES
I can differentiate and integrate exponential functions of a base other than e. |
Class Notes for "Unnatural Bases":
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PR.7
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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.
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PR.8
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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)
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PR.9
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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)
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Video about an important exponential models. Also available here and on Al-Khazneh.
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PR.10
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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:
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PR.11
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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:
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