Let's not forget our goals in studying calculus. Half of our goal is to maximize / minimize functions. A fancy word to describe this is optimization. Before getting into this, we introduce some tools to study with.
Wolramalpha.com: where we can draw
This book is extremely informal, so pictures are essential. However, it takes too much to draw pictures, so we will address this website to use some drawings. To warm up, go to the website and type "plot y = x^2". Do you see a nice parabola? Good!
Plot of a function
Recall that a function is a machine that if you give me something, it gives you something. For example $x \mapsto x^{2}$ is a function. What does it do? If you give it $1$, it gives you $1$. If you give it $2$ it gives you $4$. What if you give it $10$? It gives you $100$. If you give it $-10$ it gives you $100$ as well. The plot of the function is a painting that captures this. On a paper, you mark the points of the form:
(what you give, what the function give you).
In our case, such points are $(1, 1), (2, 4), (10, 100), (-10, 100)$. Are there more? Yes! For example, you have $(\sqrt{2}, 2), (\pi, \pi^{2}), (1.2345, 1.52399025)$... There are infinitely many points on this plot. Go ahead and go to Wolframalpha and type "plot y = x^2" to see them all.
FooPlot: even more accurate
One setback of Wolframalpha is that you cannot enlarge your plot unless you pay them. For this reason, we shall occasionally address this website to plot functions. Go to the website and under "Function $y(x)$", type "x^2". This has the same effect as typing "plot y = x^2" on Wolframalpha. Notice that you can enlarge / shrink the picture by clicking "+" / "-" button.
What will we study with this plotting skill?
In the next lesson, we will describe a special kind of functions that we can optimize, using the plotting skill we have. They are called continuous functions.
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