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This video shows the cycle of mathematical modeling using the Nuna solar car as an example: A real-world problem is described mathematically using functions. Equations help to solve these problems in order to make predictions in the real-world. This video is part of the edX course: pre-university Calculus (Calc001x)

Hello! Welcome to the first week of the pre-university

Calculus course! Do you remember this from our course promotion

video? The World Solar Challenge. A race for cars using only solar power to

cross Australia. Racing three thousand kilometers from Darwin

to Adelaide. Students of the Delft University of Technology built the Nuna7 and won this

world championship of Solar cars. How are mathematics like Calculus, important to building a winning car? The students of the Nuon Solar team have a

clear challenge: Design a solar car that can drive the long

distance as fast as possible, using only the sun as a power source. This is what we would call a real-world problem. In terms of energy:

you want to build a car that maximizes power input and minimizes power loss. The first step with such a problem or challenge

is analysis: You try to break down the question in smaller

and more specific questions, like:

how much power can you harvest from the sun? What is the influence of power losses like

air friction, roll friction and battery loss? Here starts the modeling cycle that is used

in science, design and engineering. You have a real-world problem and you try

to formulate it in terms of a mathematical model. A mathematical model describes the relation

between the different parameters that are of influence in this real-world problem. Such relations are described by mathematical

functions. Just a small example of such relations.

The power input of the Nuna car is described by a function relating the efficiency of the

solar cells, the area of solar cells,

the height of the sun and other quantities. Power loss depends on roll and air friction,

which both depend on speed and other parameters, and loss of electrical power. Once you modeled your problem in terms of

functions, you can start making calculations. This often involves solving equations. In particular if you try to find an optimal

situation. For example:

what is the highest speed the Nuna car can attain given certain weather conditions? After solving the mathematical questions,

you have to interpret the results to predict how a design or phenomenon will work in real-life. And,

ultimately, you will have to test your predictions and

see whether they are correct, or whether you need another or more refined

model. This cycle of mathematical modeling is used

in medicine, economics,

science and engineering to tackle real world problems. Think of questions like:

What is the optimum dose for this medicine? How can you maximize the profit of your company? How can you design a thrilling roller coaster

ride? How can you reduce energy consumption? In particular,

it shows how important it is to have a thorough understanding of mathematics if you want to

be a scientist or engineer. This starts with understanding the building

blocks of any model: mathematical functions. In first two weeks of this course we will

review the standard functions like polynomials, exponential functions and trigonometric functions. You will have lots of opportunity to practice

with these functions, their properties and their applications. Have fun!