Applications+of+Derivatives

__**SUMMARY**__ Summary of the Entire Unit and More~TPayne In Unit 3: Application of Derivatives, you will use what you learned in unit 2 (how to take the first and second derivative of a function) to interpret graphs and find their maximums and minimums. Also, you will need you know the slope formulas: the slope of a normal line, and also the slope of a parallel line, and the equation of a line. It will be helpful to recall trig functions, and also the definition of derivative. For Related Rates, you will need to recall past formulas for area of a cylinder, sphere, and rectangular prism. NOTES -find critical points by setting the derivative of F’ equal to zero. -Evaluate F at each critical number -Evaluate F at the endpoints -The smaller of the numbers is the minimum; largest is the maximum -Where the max/min is=x-value -What the max/min is=y-value -take the derivative of F -find critical points -set them up in a sign chart and find the signs of F' to the left and right of the critical numbers -F is increasing where F' is positive -F is decreasing where F' is negative
 * Finding Absolute Extrema on the Closed Interval:**
 * To find intervals of increase and decrease:**

Rolle's Theorem Guarantees both min and max in interior of interval which is open (a,b) __IF__ f(a)=f(b) and differentiable on (a,b) then some C -> a<c<b ;so therefore f"(c)=0 Mean Value Theorem Formula Particle Motion Formulas --** MARISSA POWELL **
 * v(t) = s'(t)
 * a(t) = v'(t)

__**VIDEOS**__ [|Motion Particle Helpful Video -Troy Askew] Unit 3: Applications of Derivatives[|Motion Problems 2 Helpful Video With Examples -Troy Askew]

Overview of The Mean Value Theorem video - Mernuelita Florissant

Helpful video about Concavity, Point of Inflections, and Second Derivatives- Mernuelita Florissant [|Optimization] - Mernuelita Florissant

[|Great video explaining related rates] - Josh Stowe

__**SITES**__ The Mean Value Theorem -Mernuelita Florissant

Helpful Rolle's Theorem and Mean Value Theorem Examples- Mernuelita Florissant

Extremely helpful for difficulty with Graphing using 1st & 2nd Derivative! [|Marissa Powell] This site gives you tips/facts to remember when graphing the first and second derivative, such as knowing when the original graph is increasing or decreasing or concave up or down. Also, helpful information about relative and absolute extrema are presented. Following the directions and rules to graphing a function and its first and second derivative, there are some sample problems that allow you to try to graph the functions yourself and then provide a detailed solution.

[|Great information on finding max and min values]- Josh Stowe [|Related Rates]help on Khan Academy- Josh Stowe

__**EXAMPLES AND PRACTICE**__ Marissa Powell & Katie M.

Graphing a Function Help~Triston Payne

Derivatives as a Rate of Change~Triston Payne

Max/Min Help~Triston Payne

Practice Test~Triston Payne

[|Related Rates Practice]-Marissa Powell This site offers information on how to calculate related rates problems. It is helpful in breaking down the use of prerequisite skills and explaining how to use them in derivatives. It is also full of examples and practice problems along with the solutions. [|Particle Motion Help]-Marissa Powell- This example helped me a lot! This site gives a particle motion problem and walks through each step on how to find the different parts. It helped me a lot in understanding the relationships between velocity and acceleration. It is helpful to see the problem broken down and what each piece means.

[|Various Max/Min practice (includes optimization)]-Marissa Powell-Awesome Site!

[|MVT Notes & Practice- Marissa Powell] This further explains the Mean Value Theorem and breaks it down into parts. It also provides an example and walks you through the steps to make it easier to understand. It also provides problems you can work out yourself and a detailed, explained solution to each problem. [|Using graphs of Derivative-Marissa Powell] This is a website solely based on graphing with derivatives. It explains intervals of increase and decrease. It also explains how to read a first derivative graph and be able to tell what its original and second derivative looks like. There are many practice problems and solutions to each. [|First & Second Derivative Test Notes & Practice]- Marissa Powell (great explanation for graphing!!)

__**ANNOTATED BIBLIOGRAPHY**__- Josh Stowe Chapter 5 of the 2002 5 Steps to a 5 book uses examples and problems to help students use derivatives to solve problems involving optimization, related rates, relative and absolute extrema, and particle motion.
 * // 5 Steps to a 5: AP Calculus // . Chapter 5. Mcgraw-Hill, 2002. Print.
 * __CAREERS/USES__**


 * 1) Application of Derivatives is widely used in finance to calculate the inflation of money at the present time and also what the inflation will be in (x) years from now. Inflation is defined as the rate of change of prices over a period of time. When calculating the inflation derivative, the following formula is used where P=price, Pi=some number, and t=time, and delta=change in time.[[file:math.docx]](Marissa)
 * 2) Apartment complexes use optimization to calculate how many apartments they should rent out to maximize their profits. They take into account their maintenance costs, create a constraint, and then find the number of apartments they can rent out to reach a maximum profit. However, their maximum will fall short of full capacity because of renters moving out, etc. (Marissa)
 * 3) Chemical engineering is a career that uses the application of derivatives to measure how a system changes with time.
 * 4) Graphing the earnings of a business and analyzing the maximum and minimum values can help a business see where they need to improve- Josh Stowe