Polar coordinates and parametric equations pdf

Here is a set of practice problems to accompany the polar coordinates section of the parametric equations and polar coordinates chapter of the notes for paul dawkins calculus ii course at lamar university. Parametric equations and polar coordinates problem set 11. Projectile motion sketch and axes, cannon at origin, trajectory mechanics gives and. Indicate with arrows the direction in which the curve is traced as t increases.

Find materials for this course in the pages linked along the left. Sketching graphs of parametric equations converting between polar coordinates and cartesian coordinates sketching graphs of polar equations sections 10. Aguilar 1 parametricequations consider the following curve c in the plane. Suppose that and are both given as functions of a third variable called a parameter by the equations called parametric equations. Until now we have been representing a graph by a single equation involving two variables. For instance, example 2 finding slope and concavity for the curve given by and. Using polar coordinates and polar equations duration. The first two of these equations uniquely determine the cartesian coordinates and given the polar coordinates and. Jim and bob are racing from the origin to the point 5, 10. Prelude to parametric equations and polar coordinates in this chapter we also study parametric equations, which give us a convenient way to describe curves, or to study the position of a particle or object in two dimensions as a function of time. Let t be the number of seconds after the start of the race.

Like cartesian coordinates, polar coordinates are used to identify the locations of points in the plane. The x and y components of the motion and derive one equation for yx. Parametric equations and polar coordinates here are a set of practice problems for the parametric equations and polar coordinates chapter of the calculus ii notes. Then write a second set of parametric equations that represent the same function, but with a faster speed and an opposite orientation. Here we begin to study situations in which three variables are used to represent a curve in the rectangular coordinate plane. We will look at polar coordinates for points in the xyplane, using the origin 0. However, as we noticed in the previous section not every solution to a di erential equation is a function meaning. Parametric equations and polar coordinates so far in this course we have been describing curves as functions yof xi. Usually, we use cartesian coordinates, which are directed distances from two perpendicular axes. Jims position at any time t is given by the parametric equations x t, y 2t. May 24, 2017 this precalculus video provides a basic introduction into parametric equations. If youd like a pdf document containing the solutions the download tab above contains links to pdf s containing the solutions for the full book, chapter and section. You may use your calculator for all sections of this problem.

Parametric equations and polar coordinates coordinate. Parametric equations, polar coordinates, and vectorvalued. Polar functions are graphed using polar coordinates, i. For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. Just as we describe curves in the plane using equations involving x and y, so can we describe curves using equations involving r and most common are equations of the form r f.

Surface area with polar coordinates in this section we will discuss how to find the surface area of a solid obtained by rotating a polar curve about the x or y axis using only polar coordinates rather than converting to cartesian coordinates and using standard calculus techniques. Chapter 9 polar coordinates and plane curves this chapter presents further applications of the derivative and integral. Here, we describe a coordinate system introduced by newton, called the polar coordinate system. Parametric equations and polar coordinates mathematics. A coordinate system represents a point in the plane by an ordered pair of numbers called coordinates.

Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with stepbystep explanations, just like a math tutor. You may assume that the culve traces out exactly once for the given range of for problems 2 and 3 set up, but do not evaluate, an integral that gives the length of the given polar cuve. It explains the process of eliminating the parameter t to get a rectangular equation of y in terms of an x variable. Calculus with algebra and trigonometry ii lecture 5 more. A projectile subject only to the force of gravity probability chung pdf is launched at an angle. The plane, being two dimensional, requires two numbers to uniquely identify a given point. Tues nov 12 today we worked on converting between polar and rectangular forms of equations. This section contains problem set questions and solutions on parametric curves, polar coordinates, and graphing. Find parametric equations for the line segment joining the points p3,1q and p5,4q. Plane curves, parametric equations, and polar coordinates. Find parametric equations for this curve, using a circle of radius 1, and assuming that the string unwinds counterclockwise and the end of the string is initially at\1,0\. Here are a set of practice problems for the parametric equations and polar coordinates chapter of the calculus ii notes. We then introduce a new coordinate system called polar coordinates which often shows up in physical applications and analyze polar graphing.

Then, are parametric equations for a curve in the plane. Suppose an object is propelled into the air at an angle of 45. If and are given, the third equation gives two possible choices f xy r xy. We then discuss calculus in polar coordinates, and solve the tangent line, arclength, and area problems for polar curves. Calculus with parametric equations let cbe a parametric curve described by the parametric equations x ft. Although the point is unique, these numbers, or coordinates, are far from unique and depend entirely on the coordinate system in use.

If the function f and g are di erentiable and y is also a di erentiable function of x, the three derivatives dy dx, dy dt and dx dt are related by the chain rule. Polar equations of conics in this chapter you have seen that the rectangular equations of ellipses and hyperbo. The polar coordinate system will be useful for many problems you encounter at mit, such as those involving circular motion or radial forces. And polar coordinates, it can be specified as r is equal to 5, and theta is 53. We are used to working with functions whose output is a single variable, and whose graph is defined with cartesian, i. Calculus with algebra and trigonometry ii lecture 5 more parametric equations, polar coordinates and lhopitals rule feb 3, 2015 calculus with algebra and trigonometry ii lecture 5more parametric equations, polar coordinates and lhopitals rulefeb 3, 2015 1 18. Understand and use keplers laws of planetary motion. You will also have an opportunity to demonstrate your understanding of parametric equations, vectors, and complex numbers. Thecurve cannot be expressedas the graph of a func. Use the graphs of the parametric equations x ft and y gt below to sketch the parametric curve in terms of x and y. Thecurve cannot be expressedas the graph of a function y f x because there are points x associated to. Calculus examples parametric equations and polar coordinates. Understand and use vectorvalued functions, as well as velocity. For example, vectorvalued functions can have two variables or more as outputs.

Write the equations of the parametric equations and graph then in your calculator. Students will sketch curves, eliminate the parameter, and write parametric equations. Parametric equations, vectors, and polar coordinates the material in this chapter is tested only on the bc exam. Parametric equations introduction, eliminating the paremeter. This precalculus video provides a basic introduction into parametric equations. For each, 0,0, there is a unique 0,2 satisfying the first two equations, each then giving a polar coordinate. Polar coordinates, parametric equations whitman college.