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Optimization real life examples calculus9/13/2023 Step 1: Translate the problem using assign symbols, variables, and sketches, when applicable, by finding two equations: one is the primary equation that contains the variable we wish to optimize, and the other is called the secondary equation, which holds the constraints. Solving Optimization Problems (Step-by-Step) It is our job to translate the problem or picture into usable functions to find the extreme values. Optimization is the process of finding maximum and minimum values given constraints using calculus.įor example, you’ll be given a situation where you’re asked to find: Or, on the flip side, have you ever felt like the day couldn’t end fast enough?īoth are trying to optimize the situation! Since we know the Acceleration function a(t) = -16 (we’re decelerating so the acceleration is negative), we can take the antiderivative to find the Velocity function v(t).Jenn, Founder Calcworkshop ®, 15+ Years Experience (Licensed & Certified Teacher) Let s(t) = Position Let v(t) = Velocity Let a(t) = Acceleration Braking Begins Now let’s define these functions and set up our equation. Position (is the antiderivative of) Velocity (is the antiderivative of) Acceleration These three functions are related, as they are the derivatives and antiderivatives of each other.Īcceleration (is the derivative of) Velocity (is the derivative of) Position How fast were you going when you first braked?įirst, it’s important to note that there are three functions all dependent on time ‘t’ in this equation: position, velocity, and acceleration. Your go-cart travels 200 feet after you hit the brakes before it finally stops. When you cross the finish line (in 1st place), you hit the brakes and decelerate constantly at 16 feet per second squared. You’re driving a high-speed go-cart, racing your friend. Let’s look at a basic example of Calculus in kinematics using go-carts. The applications include all matter of kinematics, electrodynamics, fluid dynamics, and more. This is because time is virtually always a factor in physics and because modeling change is the primary function of Calculus. The applications of Calculus in Physics are abundant. NASA: Navier-Stokes Equations How Is Calculus Used In Physics Read more about the Navier-Stokes equations and the Millenial prize problems here: The Navier-Stokes existence of smoothness equation is part of the Millenium Prize Problems: 7 unsolved problems that will earn any person $1,000,000 if they solve just one of them. However, the Navier-Stokes equations aren’t completely solved. These equations are typically taught in Calculus 3 (Vector / Multivariable Calculus) and beyond. Each character represents the following:ĭ v / Dt = The derivative of ‘ v‘ with respect to time, ‘ t‘ Few people can look at the above and understand the implications, let alone what each of the characters represents and how they relate to each other. If the equation above looks like an alien language, then you’re not alone. Also, Calculus can be used to calculate the rate of change in cost or the marginal revenue for an interest-bearing account. This includes maximizing profits, minimizing cost, and maximizing or minimizing production. In Business, Calculus is mainly used for optimization. Another example is meteorologists using Calculus to predict the weather patterns. For example, a manufacturer could use Calculus to optimize production costs. What Is Calculus Used For?Ĭalculus is used for optimization, summation, and predicting trends through modeling change over time. Today, you can find Calculus within all manner of sciences such as chemistry, physics, computer science, and engineering, and even business and economics. I did some research and found out many different functional uses that Calculus has had since its discovery. Now that I’m at University taking Calculus, I really want to know how it’s useful in real life. How many times have you found yourself sitting in a math class asking: “When is calculus actually used for in real life?” That’s exactly what I thought when I was in high school.
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