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Optimization
Calculus and derivatives may be used to solve problems, sometimescalled word problems or verbal problems or applications, whicheffectively call for finding a maximum or minimum of a function.
These problems should be approached in the same way that wordproblems that come up in algebra, geometry or trigonometry areapproached. The only real difference is that whereas the solutionsto problems in those areas are usually ultimately found by solvingequations that arise when analyzing the information given, theseproblems will involve optimization, that is, finding a minimum ormaximum, so besides solving equations one will generally have tocalculate derivatives and usually solve an equation or equationsobtained when one sets a derivative to 0.
Alan H. SteinUniversity of Connecticut
Optimization
Calculus and derivatives may be used to solve problems, sometimescalled word problems or verbal problems or applications, whicheffectively call for finding a maximum or minimum of a function.
These problems should be approached in the same way that wordproblems that come up in algebra, geometry or trigonometry areapproached. The only real difference is that whereas the solutionsto problems in those areas are usually ultimately found by solvingequations that arise when analyzing the information given, theseproblems will involve optimization, that is, finding a minimum ormaximum, so besides solving equations one will generally have tocalculate derivatives and usually solve an equation or equationsobtained when one sets a derivative to 0.
Alan H. SteinUniversity of Connecticut
Guidelines
I Read the question!
I Read the question!
I Read the question!
Has the point been made? Everything that must be done generallybecomes apparent if one reads the question!
Alan H. SteinUniversity of Connecticut
Guidelines
I Read the question!
I Read the question!
I Read the question!
Has the point been made? Everything that must be done generallybecomes apparent if one reads the question!
Alan H. SteinUniversity of Connecticut
Guidelines
I Read the question!
I Read the question!
I Read the question!
Has the point been made? Everything that must be done generallybecomes apparent if one reads the question!
Alan H. SteinUniversity of Connecticut
Guidelines
I Read the question!
I Read the question!
I Read the question!
Has the point been made? Everything that must be done generallybecomes apparent if one reads the question!
Alan H. SteinUniversity of Connecticut
Ideas to Keep in Mind
I What do you know?
I What don’t you know?
I What can you figure out or infer?
I What do you want?This is actually the least important!
Alan H. SteinUniversity of Connecticut
Ideas to Keep in Mind
I What do you know?
I What don’t you know?
I What can you figure out or infer?
I What do you want?This is actually the least important!
Alan H. SteinUniversity of Connecticut
Ideas to Keep in Mind
I What do you know?
I What don’t you know?
I What can you figure out or infer?
I What do you want?This is actually the least important!
Alan H. SteinUniversity of Connecticut
Ideas to Keep in Mind
I What do you know?
I What don’t you know?
I What can you figure out or infer?
I What do you want?This is actually the least important!
Alan H. SteinUniversity of Connecticut
What You Know
Every fact translates into a mathematical statement, generally aformula, equation or statement that a particular variable takes on acertain value when another variable takes on a certain value.
The key to writing down an appropriate formula corresponding to agiven fact is to write down that fact in plain language and thenrewrite the fact using the descriptions of variables already definedand a verb such as is or equals which indicates that two quantitiesare equal.
Alan H. SteinUniversity of Connecticut
What You Don’t Know
This is often the key. Any unknown quantity can often beprofitably represented by a variable.
Related to unknown quantities are variable quantities. Thesealmost always need to be represented by variables.
Alan H. SteinUniversity of Connecticut
What You Don’t Know
This is often the key. Any unknown quantity can often beprofitably represented by a variable.
Related to unknown quantities are variable quantities. Thesealmost always need to be represented by variables.
Alan H. SteinUniversity of Connecticut
What You Want
This needs to be kept in the back of your mind. One commonmistake is to concentrate too hard on what you want; it’s generallymore advantageous to concentrate on what you know, what youdon’t know and what you can figure out from what you know.
Just remember to notice when you’ve actually figured out whatyou ultimately want.
Alan H. SteinUniversity of Connecticut
What You Want
This needs to be kept in the back of your mind. One commonmistake is to concentrate too hard on what you want; it’s generallymore advantageous to concentrate on what you know, what youdon’t know and what you can figure out from what you know.
Just remember to notice when you’ve actually figured out whatyou ultimately want.
Alan H. SteinUniversity of Connecticut
Suggestions
I Draw pictures, charts, graphs or anything visual that may helpyou understand the problem. The key is understanding andtranslating facts to mathematics.
I Look for variables and unknowns. Represent them by symbols.Write down what each stands for and make sure you don’t usethe same symbol to represent two different quantities.
I Write down known facts in terms of the variables andunknowns you’ve defined. These will generally be in the formof equations and formulas.
I Solve equations where possible.
I If there’s a variable that needs to be optimized, solve for it interms of other variables, find its derivative and see when itsderivative is equal to 0 or undefined. Remember: A derivativeis always undefined at the endpoints!
Alan H. SteinUniversity of Connecticut
Suggestions
I Draw pictures, charts, graphs or anything visual that may helpyou understand the problem. The key is understanding andtranslating facts to mathematics.
I Look for variables and unknowns. Represent them by symbols.Write down what each stands for and make sure you don’t usethe same symbol to represent two different quantities.
I Write down known facts in terms of the variables andunknowns you’ve defined. These will generally be in the formof equations and formulas.
I Solve equations where possible.
I If there’s a variable that needs to be optimized, solve for it interms of other variables, find its derivative and see when itsderivative is equal to 0 or undefined. Remember: A derivativeis always undefined at the endpoints!
Alan H. SteinUniversity of Connecticut
Suggestions
I Draw pictures, charts, graphs or anything visual that may helpyou understand the problem. The key is understanding andtranslating facts to mathematics.
I Look for variables and unknowns. Represent them by symbols.Write down what each stands for and make sure you don’t usethe same symbol to represent two different quantities.
I Write down known facts in terms of the variables andunknowns you’ve defined. These will generally be in the formof equations and formulas.
I Solve equations where possible.
I If there’s a variable that needs to be optimized, solve for it interms of other variables, find its derivative and see when itsderivative is equal to 0 or undefined. Remember: A derivativeis always undefined at the endpoints!
Alan H. SteinUniversity of Connecticut
Suggestions
I Draw pictures, charts, graphs or anything visual that may helpyou understand the problem. The key is understanding andtranslating facts to mathematics.
I Look for variables and unknowns. Represent them by symbols.Write down what each stands for and make sure you don’t usethe same symbol to represent two different quantities.
I Write down known facts in terms of the variables andunknowns you’ve defined. These will generally be in the formof equations and formulas.
I Solve equations where possible.
I If there’s a variable that needs to be optimized, solve for it interms of other variables, find its derivative and see when itsderivative is equal to 0 or undefined. Remember: A derivativeis always undefined at the endpoints!
Alan H. SteinUniversity of Connecticut
Suggestions
I Draw pictures, charts, graphs or anything visual that may helpyou understand the problem. The key is understanding andtranslating facts to mathematics.
I Look for variables and unknowns. Represent them by symbols.Write down what each stands for and make sure you don’t usethe same symbol to represent two different quantities.
I Write down known facts in terms of the variables andunknowns you’ve defined. These will generally be in the formof equations and formulas.
I Solve equations where possible.
I If there’s a variable that needs to be optimized, solve for it interms of other variables, find its derivative and see when itsderivative is equal to 0 or undefined. Remember: A derivativeis always undefined at the endpoints!
Alan H. SteinUniversity of Connecticut
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