Learn parametric calculus with free interactive flashcards. Determine derivatives and equations of tangents for parametric curves. Find the area of a surface of revolution parametric form. Calculus and parametric equations math 211, calculus ii j. As you work through the problems listed below, you should reference chapter 10. If the function f and g are di erentiable and y is also a di erentiable. Introduction to parametric equations calculus socratic. A curve c is defined by the parametric equations x t2, y t3. The collection of points that we get by letting \t\ be all possible values is the graph of the parametric equations and is called the parametric curve. Solving for ydoes not give you one but two functions y p a2 x2 and the implicit equation. Then write a second set of parametric equations that. Calculus with parametric curves mathematics libretexts.
Bc calculus manual revised 52016 this page provides the bc calculus manual for the classroom all chapters of this manual are provided as free downloads. Calculus 2 lia vas parametric curves in the past, we mostly worked with curves in the form y fx. Parametric curves general parametric equations we have seen parametric equations for lines. Calculus with parametric equationsexample 2area under a curvearc length. In order to transform a parametric equation into a normal one, you need to do a process called eliminating the parameter.
If a curve cis described by the parametric equation x ft, y gt for t. Calculus parametric functions introduction to parametric equations. I described a surface as a 2dimensional object in space. A curve c is defined by the parametric equations x t 2 t 1, y t3 t2. Calculus bc parametric equations, polar coordinates, and vectorvalued functions defining and differentiating parametric equations parametric equations differentiation ap.
Parametric equations read calculus ck12 foundation. Second derivatives of parametric equations with concavity duration. Calculus bc lower moreland township school district. Math 232 calculus iii brian veitch fall 2015 northern illinois university 10. Choose from 85 different sets of parametric calculus flashcards on quizlet. This is called a parametrization of the surface, or you might describe s as a parametric.
Find the slope of a tangent line to a curve given by a set of parametric equations. We have already seen how to compute slopes of curves given by parametric equationsit is how we computed slopes in polar coordinates. Instead of one equation relating say, x and y, we have two equations, one relating x with the parameter. 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. Parametricequationsofcurves the parametric equationsfor a curve in the plane consists of a pair of equations x ft, y gt, a. As you probably realize, that this is a video on parametric equations, not physics. Apply the formula for surface area to a volume generated by a parametric curve. Parametric form refers to a relationship that includes and. Calculus is designed for the typical two or threesemester general calculus. In what direction is the graph traced out as the value of t increases. The following links are pdf files of notes we took inclass for each section. Use point plotting to graph plane curves described by parametric equations. Themes for advanced placement calculus 57 theme 15 parametric equations and vector functions summary parametric equations and vectorvalued functions are very similar. In mathematics this third quantity is called a parameter.
We need to be able to do the same when functions are defined parametrically. Solve one of the parametric equations for t and substitute that into the other equation. In this section we will introduce parametric equations and parametric curves i. This section is a complete high school course for preparing students to take the bc calculus. It depends on the curve youre analyzing, in general, finding the parametric equations. To help visualize just what a parametric curve is pretend that we have a big tank of water that is in constant motion and we drop a ping pong ball into the tank. And time tends to be the parameter when people talk about parametric equations. A curve c is defined by the parametric equations x 2cost, y 3sint. Fifty famous curves, lots of calculus questions, and a few. Use the equation for arc length of a parametric curve. Find the parametric equation for the unit circle in the plane. Volume 3 covers parametric equations and polar coordinates, vectors, functions of several variables, multiple integration, and secondorder differential equations. Here are a set of practice problems for the parametric equations and polar coordinates chapter of the calculus ii notes. Eliminating the parameter is a phrase that means to turn a parametric equation.
Calculus with parametric equations let cbe a parametric curve described by the parametric equations x ft. Length of a curve calculus with parametric equations let cbe a parametric curve described by the parametric equations x ft. Robert buchanan department of mathematics fall 2019. Length of a curve calculus with parametric equations let cbe a parametric curve described by the parametric equations. Sketch the graph determined by the parametric equations.
Parametric equations differentiation practice khan academy. We then discuss calculus in polar coordinates, and solve the tangent line, arclength, and area problems for polar curves. Chapter 1 functions and equations lecture 1 explicit, implicit and parametric equations 14 a popular explicit function the explicit function. If u and v are the input variables often called parameters and x, y, and z are the output variables, then s can be written in component form as. Analyze and write equations of parabolas using properties of parabolas. Notice in this definition that x and y are used in two ways. Once you understand the nature of parametric equations, their calculus applications are similar to what you already know. Find an equation of the tangent line to c at the point where t 2. Then write a second set of parametric equations that represent the same function, but with a faster speed and an opposite orientation.
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. If a curve c is described by the parametric equation x ft, y gt for. Find the area of a surface of revolution parametric. To this point in both calculus i and calculus ii weve looked almost exclusively at functions in the form \y f\left x \right\. Parametric surfaces millersville university of pennsylvania. However, this format does not encompass all the curves one encounters in applications. Find the arc length of a curve given by a set of parametric equations. First make a table using various values of t, including negative numbers, positive numbers and zero, and determine the x and y values that correspond to. Calculus ii parametric equations and polar coordinates. Let c be a parametric curve described by the parametric equations x ft,y gt. Parametric equations and calculus if a smooth curve c is given by the equations.
Calculus and parametric equations classwork when we studied functions, we were able to determine the slope of the tangent to a curve at a point by taking the derivative. Calculus and parametric equations mathematics libretexts. Now we will look at parametric equations of more general trajectories. Polar coordinates, parametric equations whitman college. A 1 only b 0 only c 2 only d 1 and 2 only e 1, 0, and 2. Find materials for this course in the pages linked along the left. Each value of the parametert gives values for x and y. How do you find the parametric equations of a curve.
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