ICEs from Math Xb Classes

To realize that given:
- a density function (units of amount of "stuff" per unit volume), and,
- a description of a region or volume,
it is possible to set up an integral that will give the amount of "stuff" contained in that region/volume.
To create a density function that gives concentration of carbon dioxide (in units of metric tons per cubic kilometer) as a function of altitude.
To determine the shape of the Earth's troposphere.
To use the algebraic structure of the density function and the geometrical structure of the Earth's troposphere to divide the troposphere up into "slices" that will have approximately constant density throughout.
To use area and volume formulas from geometry to create an equation for the volume of each tropospheric "slice."
To formulate and evaluate an integral that will give the mass of carbon dioxide gas in the Earth's troposphere.

Wednesday, April 3

Evaluating the safety of the design of an artificial habitat for polar bears

To create a conceptual model for the average value of a function (i.e. the depth of water that is created when a iceberg melts).
To use antiderivatives and definite integrals to calculate the area beneath a curve.
To calculate the average value of a complicated function over an interval.
To use the idea of the average value of a function (together with the interpretation of average value as depth of water) to analyze the safety of a planned polar bear exhibit.

Friday, March 22

How much dough is there in a donut? Mmmm, doo-nuut.

To determine the best way to break up a solid or revolution into convenient shapes.
To use familiar area and volume formulas from geometry to determine a formula for the volume of a "shell" or "thick wall cylinder" shape.
To use the volume formula that you have created to formulate an integral that will give the total volume of a solid shaped like a donut.
To use the integral that you have created, your graphing calculator and measurements from an actual donut to work out the volume of a standard-sized donut.

Wednesday, March 20

Calculating antiderivatives using the technique of u-substitution

To sharpen the instincts that will help you to select a "good" expression for "u" when confronted by an integral.
To practice the various steps involved with calculating a formula for an antiderivative using the method of u-substitution, namely:

Identify an "inside function" and call this u
Calclate the derivative du/dx
Rearrange the derivative to make dx the subject of the equation
Replace as much of the integral as you can using your expressions for u and dx.

Note: If you have made a "good" choice for u then your integral should not have any x's left in it when you are finished this step. If your integral does have some x's left in it then you need to go back to the first step and try another choice for u.
Find a formula for the antiderivative now regarding u as your variable of integration (as opposed to x).

Note: Don't forget the "+C" !
Convert the antiderivative formula back to the x variable by substituting for u in the antiderivative formula.

Monday, March 18

Calculating antiderivatives by reversing the Chain Rule

To recognize that antiderivatives of some quite complicated functions can be easily calculated by reversing the Chain Rule for derivatives.
To realize that the pattern to look for if you would like to try to calculate an antiderivative by reversing the chain rule is:
- The presence of a function, say g(x), located inside another algebraic expression.
- The derivative of this function g(x) should appear as a factor in the symbolic that you are trying to find the antiderivative of.
To practice detecting "inside functions."
To practice calculating formulas for antiderivatives by reversing the Chain Rule for derivatives.

Friday, March 15

Estimating the total concentration of cortisol in a healthy adult in a relatively stress-free environment

To use antiderivative rules to find antiderivatives for:
- Polynomial,
- Exponential, and,
- Rational
functions.
To use antiderivatives to evaluate the area under a graph.
To compare several approximations of the area under a curve and choose the one that will give the estimate that is closest to the actual area under that curve.

Friday, March 15

Five Antidifferentiation Rules

Formulate a rule for creating antiderivatives of constant functions.
Formulate a rule for creating antiderivatives of power functions.
Formulate a rule for creating antiderivatives of exponential functions.
Create an antiderivative for the function f (x) = e^x.
Create an antiderivative for the function f (x) = 1/x.

Wednesday, March 13

Estimating the amount of a greenhouse gas (methane, CH₄) that can be attributed to human activity

To represent the amount of methane (CH₄ released by human activities as an area under a curve.
To use integral notation to represent the amount of methane released using mathematical symbols.
To use data on the rate of methane emissions to create an equation for the rate at which methane is released into the atmosphere as a result of human activity.
To find an formula for the antiderivative of a simple exponential function.
To use antiderivatives to estimate the total amount of methane that has been released (by human activities) into the atmosphere from the beginning of the Industrial Revolution in Europe (around 1860) to the present day.
To use antiderivatives to estimate the total amount of methane (released by human activities) that is currently in the atmosphere.

Monday, March 11

Estimating the year when the world's supplies of petroleum will be exhausted

To use your graphing calculator to create a formula for the derivative of a function.
To work backwards from the formula for the derivative to a formula for the original function (in this case, how much petroleum remains in the world).
To use this original function to determine approximately when the world's supplies of petroleum will be exhausted.

Friday, March 8

Calculating the acoustic power of the American alligator, Alligator mississippiensis.

To interpret the meaning of the area under a curve in a specific setting (in this case, the area under a power density spectrum).
To use integral notation to represent the acoustic power of an American alligator.
To numerically approximate the acoustic power of an American alligator when it makes a high-intensity warning noise.
To determine the acoustic power of alligators in different scenarios.
To use integral notation to represent the different levels of acoustic power produced by american alligators in each of the different scenarios.
To infer (based on the representations of acoustic power by integral notation) some of the formal rules for manipulating mathematical quantities that are expressed using integral notation.

Monday, March 4

Studying the seasonal variation in the number of neurons in the hippocampal formation of a food-storing bird (The Black-Capped Chickadee, Poecile atricapillus).

To practice recognizing patterns in data and to use these patterns to decide what kind of function might do a good job of representing the trends in the data.
To use information about a derivative to reconstruct the graph of the original function.
To recall the relationship between the sign of the derivative and the behavior of the original function:
- When the derivative is positive, the original function is increasing.
- When the derivative is negative, the original function is decreasing.
To use your calculator to approximate the area under a the graph of the derivative, and notice that (at least accoridng to your calculator) it is sometimes possible for area to be negative.
To use the Fundamental Theorem of Calculus to explain why it makes some sense to regard area beneath the x-axis as negative.

Monday, March 4

Handout: How to Use a TI-83 to Approximate Any Area Under a Curve

To visualize the area that is actually being calculated when a TI-83 is used to approximate the area under a curve
To remember the quantities that must be specified in order to calculate the area under a curve (namely the equation for the function defining the curve, the x-coordinates where the area starts and stops, and the number of rectangles
To understand how the area of all of the rectangles can be expressed as a sum using sigma notation
To use the commands that are built into a TI-83 calculator to evaluate the sum that approximates the area under a curve

Answers to Problems on this Handout

Function	Answer you should obtain when using TI-83 to approximate area under curve
y=2x	62.755
y=x²	7.84
y=e^x	4.647459322
y=x( x-1)(x-2)	-0.249375

Friday, March 1

Calculating the growth of natural and genetically modified Atlantic salmon (Salmo salar).

To practice using Euler's method to approximate the value of a function when given one value of the function and information about the rate of change.
To graphically represent the area that a calculator will calculate when attempting to approximate the area under a curve.
To fit a function to given data and in doing so create an equation for a rate of change.
To use your calculator to approximate the area under a graph showing rate of change in salmon weight versus time.
To find a connection between the change in the value of a function (from Euler's Method) and the area beneath a derivative-time graph.

Wednesday, February 27

Calculating the Oral Bioavailability of Oxycodone Delivered by OxyContin^TM Tablets.

To learn how clinical researchers compare the efficiency of different drug delivery mechanisms (such as taking the medication orally versus intravenously)
To use a collection of regular shapes (rectangles) to approximate the area beneath an oxycodone plasma concentration curve
To represent the approximate area under a curve graphically (by drawing the rectangles on the oxycodone plasma concentration curve)
To represent the approximate area under a curve numerically by calculating the ares of the approximating rectangles
To realize that in order to improve the accuracy of the approximate value of the area under the curve, a greater number of "skinnier" rectangles should be used to approximate the area
To learn how the summing capabilities of a graphing calculator can be utilized to approximate the area under a curve (when the curve can be represented by a function that is defined by an equation)
To learn how clinical researchers use areas under curves to calculate the relative efficiency of different methods for drug delivery

Wednesday, February 27

Handout: How to Use a TI-83 to Approximate the Area Under a Curve

To visualize the area that is actually being calculated when a TI-83 is used to approximate the area under a curve
To remember the quantities that must be specified in order to calculate the area under a curve (namely the equation for the function defining the curve, the x-coordinates where the area starts and stops, and the number of rectangles
To understand how the area of all of the rectangles can be expressed as a sum using sigma notation
To use the commands that are built into a TI-83 calculator to evaluate the sum that approximates the area under a curve

Answers to Problems on this Handout

Function	Answer you should obtain when using TI-83 to approximate area under curve
y=2x	99.5
y=x²	2.28
y=e^x	1.7909704738
y=x( x-1)(x-2)	0

Monday, February 25

Drill Problems Using the Ratio Test

To perform the four steps required to use the Ratio Test for convergence/divergence of a series
To practice identifying the general term of an infinite series
To practice creating and simplifying ratios of adjacent or subsequent general terms
To practice taking the limit of the ratio
To practice interpreting the limit of the ratio and using this interpretation to decide whether the infinite series converges or diverges

Wednesday, February 20

Monitoring the Morphine Levels in a Patient's Body

To translate the First Law of Pharmacokinetics into a function that describes how the amount of morphine in a patient's body changes as time goes by
To use this understanding to create a graph showing how the amount of morphine that is in a patient's body changes with time when the patient is receiving the recommended dose of morphine
To use a geometric series to represent the peak amount of morphine that a patient has in their body and the summation formula for a geometric series to create a convenient expression for this peak amount of morphine
To use limits to determine whether the patient is at any risk of an overdose of morphine, even if the treatment is continued indefinitely

Friday, February 15

Calculating the Monthly Mortgage Payment on a Refuge from the Technological Age

To understand how financial institutions calculate the interest and unpaid balance on a loan
To use this understanding to create an algebraic expression for the unpaid balance of a loan as a function of time
To recognize that most of the terms in this algebraic expression can be grouped together to form a geometric series
To use the summation formula for a geometric series to calculate the monthly payment on a mortgage

Wednesday, February 13

The Long Term Storage of Radioactive Medical Waste

To use information about the rate at which radioactive medical waste is collected and the physical properties of the isotope involved (Ir-192) to calculate the mass and volume of Ir-192 that is collected in Denmark every two weeks
To use information about the radioactive decay of Ir-192 to create a geometric series to represent the volume of Ir-192 that accumulates at the Danish facility for the storage of radioactive waste
To use the summation formula for a geometric series and logarithms to calculate the amount of time needed for enough Ir-192 to accumulate to completely fill a storage drum
To use information about the unused storage capacity of the Danish nuclear waste depository to predict how much longer the Danes will be able to use this facility

Friday, February 8

Modeling the physiological response to alcohol consumption

Interpret information presented in verbal form as the values of a function and its derivative
To use information about the rate at which alcohol is entering and leaving the stomach to formulate a differential equation
To use a slope field and some precisely known values of the function to sketch a plausible graph of the function
To use the graph (drawn with the aid of a slope field) to make inferences about the behavior of a function and the behavior of a person who is consuming alcohol as time goes by

Wednesday, February 6

Modeling an outbreak of Herpes gladiatorum at a high school wrestling camp

Interpret information presented in verbal form as the values of a function and its derivative
To use information on where a derivative is equal to zero to select a plausible differential equation to represent the outbreak of a disease
To use the expected slope of the function (at a point where the derivative is not equal to zero) to select a plausible differential equation to represent the outbreak of a disease
To use a slope field and some precisely known values of the function to sketch a plausible graph of the function
To use the graph (drawn with the aid of a slope field) to make inferences about the behavior of the function

Monday, February 4

Trends in Heroin Use for Four European Countries

To remember that it is possible to define a function using the rate of change of the function
To remember how to use an initial value and the rate of change to estimate the values of a function
To determine the effectiveness of different strategies for managing the social problem of heroin addiction