10.2 plane curves and parametric equations PDF

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plane curves and parametric equations...

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MATH 252: 10.2, Plane Curves and Parametric Equations 2

(1) Motivation. Consider a ball that is thrown that traces a path given by y = − x72 + x. This certainly tells us how high the ball is when it is a certain distance across the ground given along the x-axis. However, for this type of problem, it does not indicate at a specific point (x, y). √ at what time we are√ For this reason, we consider the following: x = 24 2t and y = −16t2 + 24 2t, for a time t. In this case, t is called a parameter, and together, these form a set of parametric equations that when we plot the graph we get a plane curve. (2) Definition of a Plane Curve If f and g are continuous functions of t on an interval I, then the equation x = f (t) and y = g(t) are called parametric equations and t is called the parameter. The set of points (x, y) obtained as t varies over the interval I is called the graph of the parametric equation. Taken together, the parametric equations and the graph are called a plane curve, denoted by C . (3) When we plot these curves, we don’t just plot the points. We also have to show the trajectory, or the direction the curve is heading. That is what makes parametric equations so interesting. It doesn’t just give you the point, it also gives you the direction of the path so you know how to connect the points. (4) Example A: Sketch the curve described by x = t2 − 4, and y = t/2 for −2 ≤ t ≤ 3. Solution: We find a few points. for t = −2, x = 0, and y = −1. For t = −1, x = −3 and y = −1/2. We continue in this pattern.

(5) It often happens that two different parametric equations have the same graph. For example x = 4t2 − 4, and y = t for −1 ≤ t ≤ 3/2 looks like The difference is the rate at which the graph is traced

out (be thinking what this means for the derivative). (6) Eliminating the Paramater. We solve for t in one equation. Substitute into the other. Then simplify. 1

(7) Example B: Sketch the graph of by eliminating the parameter and adjusting the domain. 1 t fort > −1. x= √ and y = t+1 t+1 Solution: Using the x equation, we solve for t, and we get 1 − x2 1 t = 2 −1= x x2 Substitute this into the parametric equation for y which gives us y = 1 − x2 .

However, this is not just the entire parabola. So restrict x > 0.

(8) Example C: Sketch the graph of x = 3 cos θ and y = 4 sin θ for 0 ≤ θ ≤ 2π . Solution: We can find the cartesian form of this equation. (x/3) = cos θ and (y/4) = sin θ. Since cos2 θ + sin2 θ = 1 which yields x2 /9 + y2 /16 = 1, which is the graph of an ellipse.

(9) Definition of Smooth Curve. A curve C represented by x = f (t) and y = g (t) on an interval I is called smooth if f ′ and g ′ are continuous on I and not simultaneously 0 except possibly at the endpoints of I. The curve C is called piecewise smooth if it is smooth on each subinterval of some partition of I . Homework: See handout, it is not on WebAssign. This is due next class. Remember, homework is not an option, it is part of your grade.

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