Application of Derivatives
Now that we have a solid understanding on how to find derivatives both numerically and by using formulas, we can now apply that knowledge to real world situations. Optimization and related rates are both use applications of derivatives.
Optimization
The idea behind optimization is to maximize or minimize a certain value. This can come in really useful in real life. For example we can use optimization to find the maximum area possible if given limited fencing materials to make a garden. Or figure out how to maximize the volume of a box. After expressing the quantity we want to maximize or minimize as a function, we can use calculus to solve the rest. It is important to under the steps to solving an optimization problem. The following steps are useful when solving optimization word problems. (Steps taken from Mr. Latimer's "Optimization (Word) Problems" handout.
1) Identify unknown's
2) Write down the quantities that need to be maximized or minimized
3) Draw diagrams to help you visualize the problem.
4)Explicitly define each unknown using variables. Write down what the variable represents.
5) Write an algebraic equation or equations involving your defined variables. Use additional information from scenarios to clarify or simplify relationships among your variables.
- Your goal is to isolate the variable you are trying to solve for, this is the variable that you are trying to maximize or minimize, defined in terms of only one other variable. In other words we are looking for an equation of the form Q=f(x), where Q is the variable being optimized and x, is a related variable.
6) Identify the domain of function f(x).
7) Under consideration of the domain identified, determine the absolute/global maximum for f(x), which should solve the optimization problem.
Now that we know what optimization problems are and their steps, lets solve an example problem.
1) Identify unknown's
2) Write down the quantities that need to be maximized or minimized
3) Draw diagrams to help you visualize the problem.
4)Explicitly define each unknown using variables. Write down what the variable represents.
5) Write an algebraic equation or equations involving your defined variables. Use additional information from scenarios to clarify or simplify relationships among your variables.
- Your goal is to isolate the variable you are trying to solve for, this is the variable that you are trying to maximize or minimize, defined in terms of only one other variable. In other words we are looking for an equation of the form Q=f(x), where Q is the variable being optimized and x, is a related variable.
6) Identify the domain of function f(x).
7) Under consideration of the domain identified, determine the absolute/global maximum for f(x), which should solve the optimization problem.
Now that we know what optimization problems are and their steps, lets solve an example problem.
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Example Problem:
1) An open rectangular box with a square base has a surface area of 36ft squared. What dimensions can give us the largest possible volume?
1) An open rectangular box with a square base has a surface area of 36ft squared. What dimensions can give us the largest possible volume?
We know that 36 is the surface area so therefore the following is true. Our unknown values are x,y and V(volume).
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Related Rates
In addition to using optimization, we can also use related rates to solve real world application problems. Related rates problems involve at least two quantities that change with respect to time(t), and asks you to find out the rate at which one is changing by giving you enough information about the other quantities. Related rates are solved using implicit differentiation with respect to time(t). Like in optimization, there are several steps to solving a related rates problem. Steps are from the University of California math Davis.
1)Draw a diagram and label all numbers with variables. All rates correspond to a derivative.
2)Clearly state the goal of the problem.
3)Make your related rates equation.
4)Isolate your goal. Make sure to solve for all other variables before this step.
5)Substitute numbers and solve. DON't FORGET UNITS!
Now let's do an example problem.
1)Draw a diagram and label all numbers with variables. All rates correspond to a derivative.
2)Clearly state the goal of the problem.
3)Make your related rates equation.
4)Isolate your goal. Make sure to solve for all other variables before this step.
5)Substitute numbers and solve. DON't FORGET UNITS!
Now let's do an example problem.
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Example Problem:
1) A 20 ft ladder is leaning against a building. The building is perpendicular to the ground. The bottom of the ladder is being pulled out by a kid 5-inches per second. How fast is the top of the ladder falling when the ladder is 10-feet high?
First thing you have to do is draw a picture.
1) A 20 ft ladder is leaning against a building. The building is perpendicular to the ground. The bottom of the ladder is being pulled out by a kid 5-inches per second. How fast is the top of the ladder falling when the ladder is 10-feet high?
First thing you have to do is draw a picture.
Our goal is to find out the rate at which y is changing.