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Exercises

Functions

A function is much like a computer program. The function is a rule (e.g., an equation) that takes some inputs (numbers) and produces some output, another number. One kind of function is the type...

Magnitude

With an understanding that vectors are simply mathematical equivalents of mythological hybrid beasts, we will investigate their parts to see if we can understand them better. Surely, with the help...

Direction

Since we know all about the bodies, err, magnitudes of vectors, we need to look at the directions of vectors. We can equate the direction of the vector to the head of the griffin or the minotaur. T...

Scaling Vectors

Quiz time. Already? Yep.What's one apple plus another apple? Two apples. What's two oranges less one orange? One orange. What's one griffin plus one minotaur? Any mythology buff could tell you it's...

Unit Vectors

We've been through apples, oranges and mythological creatures unlike anyone has ever seen. But we still don't have a quick way to compare the magnitude of a vector before and after we scale it.To d...

Vector Functions

Recall that functions are much like computers or machines that take in one or several input numbers and put out a single number. And recall that vectors are mathematical entities composed of two pi...

Graphing Parametric Equations

Graphing parametric equations is similar to graphing vector functions.One way to graph parametric equations is to find points for certain values of t, graph those points, then play connect-the-dots...

Points on Graphs of Parametric Equations

We can describe trajectories of things with parametric equations.Before we go forward, we should review parabolas via Super Mario Brothers. When Mario jumps, we can parameterize his trajectory d(x,...

Parametrizations of the Unit Circle

One of the first shapes we learn as a toddlers is the circle. It probably didn't take long to realize that pizza is shaped this way. Now we will discuss the unit circle, the circle with radius equa...

Parameterization of Lines

If circles equal pizzas, then lines equal hot dogs. Welcome to Calculus Analogies 101.What kind of toppings we can put on hot dogs to suit our tastes?Sample ProblemBegin with mustard and ketchup, t...

Polar Coordinates

It's a hot summer evening. Bob has been training for an Iron Man and needs major calorie intake. He finished his dinner of pizza and hot dogs, but he still has room for dessert. He eyeballs that ke...

Graphing Polar Functions

It's time to get friendly with our graphing calculators. First, we need to understand how to graph polar functions by hand. This is one of the many instances in calculus where it's helpful to...

Rules of Graphing We Do (or Don't) Have

The best way to graph polar functions is by using a graphing calculator or a computer program. We can wave our hands and pull a rabbit out of a hat. That's because there aren't as many rules about...

Bounds on Theta

Polar function plots may leave your head spinning like you got off the Tilt-a-Whirl ride at the amusement park. But using them, it's possible to model that Tilt-a-Whirl ride, and they make some of...

Intersections of Polar Functions

We need one more tool to solve for intersections of polar functions. We need to know how to solve a system of two equations.We can find where two rectangular functions y = f (x) and y = g(x) i...
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