AP Calculus Topics Real World Applications

Calculus Topics In The Real World

1. Optimization & Particle Motion Nuha Elhassani AP Calculus AB - Final Project

2. Optimization

3. What is Optimization & Its Real World Applications Optimization in calculus can be defined as the process of finding an absolute maximum or minimum value of a function given a set of limitations. Optimization is useful when trying to maximize volume, area , or profit with a given restriction. Specific real world applications include finding the minimum amount of material to make a soda can holding a certain amount of liquid or trying to maximize the area of a corral with a set amount of fencing.

4. Optimization Example Explanation Find the dimensions of a can that holds 128 cubic centimeters of soda and that minimizes the amount of material used, assuming a cylindrical shape. First Step: state the quantity to be optimized. Here, we are minimizing the surface area of the can. Second Step: make a sketch Third Step: write the formula for the surface area: A = 2πr 2 + 2πrh. Fourth Step: write the formula in terms of one variable through substitution of a secondary equation: the formula for the volume - 128=πr 2 h. We can solve for h and write the formula in terms of r. h = 128/πr 2 → sub into A = 2πr 2 + 2πr(128/πr 2 ) = A = 2πr 2 + 256/r h r

5. Optimization Example Explanation Find the dimensions of a can that holds 128 cubic centimeters of soda and that minimizes the amount of material used, assuming a cylindrical shape. Fifth Step: differentiate - A’ = 4πr - 256/r 2 Sixth Step: solve for the critical points of A - 0 = 4πr - 256/r 2 , r≈2.731 Seventh Step: confirm that r≈2.731 is a relative minimum of A. Because A’ changes from negative to positive at this point, it is a relative minimum of A and also the absolute minimum. (There are no endpoints to compare to) Eighth Step: find h when r≈2.731. h = 128/π(2.731) 2 → 5.463 Final Step: state dimensions with units - r≈2.731 centimeters, h≈5.463 centimeters. h r

6. Material Engineer Material engineering is a field that heavily relies on optimization . Material engineers may focus on minimizing material costs or maximizing manufacturing profits . As demonstrated in the previous example of a soda can, professionals in this field use optimization techniques to reduce expenses associated with materials such as aluminum, by determining the minimum amount required to produce a can capable of holding 128 cubic centimeters of soda.

7. Particle Motion

8. What is Particle Motion & Real World Applications Particle motion refers to the analysis of individual particle movement within a system, considering factors like trajectory , velocity , and acceleration . This concept finds practical application in various fields like engineering and sports . Knowledge of particle motion is essential for designing missiles or analyzing the trajectory of thrown objects. It’s also useful when finding total distance traveled, as well as particle displacement which can be analyzed through the use of particle motion formulas in Calculus.

9. Particle Motion Demonstration A mechanical system has a piston that moves up and down within a cylinder. The position of the piston at any time t seconds can be described by the function s(t) = 3 t 2 - 2 t + 5 , where s is measured in meters and t in seconds. At t=0, is the piston moving up or down? Find s’(t) = v(t) = 6t - 2, 6(0)-2=-2 → v(0) = -2 . At t=0, the piston is moving down. At t=1.5, is the velocity of the piston increasing or decreasing? s’’(t) = a(t) = 6, a(1.5) = 6 . At t=1.5 , the velocity of the piston is increasing. What time does the piston change direction? 6t-2=0 → t= ⅓. At time t=⅓ seconds, the piston changes direction. (sign change) What is the total distance traveled by the piston from t=0 to t=2 using v(t) ? ∫|v(t)|dt = 8.667 meters 2

10. Mechanical Engineer One career that utilizes particle motion is mechanical engineering . It aids in analyzing and optimizing systems like engines and turbines. By examining trajectories, velocities, and accelerations , engineers enhance system efficiency and reliability . Additionally, engineers use particle motion principles to solve problems like determining the motion direction at given time points, as shown in the previous piston problem. Particle motion guides engineers in creating innovative solutions across industries, ensuring mechanical system efficiency and safety.