10.1 Curves Defined by Parametric Equation

Transcription

1 10.1 Curves Defined by Parametric Equation 1. Imagine that a particle moves along the curve C shown below. It is impossible to describe C by an equation of the form y = f (x) because C fails the Vertical Line Test. But the x- and y- coordinates of the particle are functions of time and so we can write x = f (t), y = g (t). Such a pair of equations is often a convenient way. In conclusion, suppose that x and y are both given as functions of a third variable t (called a parameter) by equations x = f (t) y = g (t) called parametric equations. Each value of t determines a point (x,y), where we can plot in a coordinate plane. As t varies, the point (x, y) = (f (t), g (t)) varies and traces out a curve C which is a parametric curve. In general, the curve with parametric equations x = f (t) y = g (t) a t b has initial point (f (a), g (a)) and the terminal point (f (b), g (b)). 2. (Cycloid). The curve traced out by a point P on the circumference of a circle as the circle rolls along a straight line is called a cycloid. The parametric equations of the cycloid are x = r (θ sinθ) y = r (1 cosθ) θ R 1

2 Some Homework Questions: 1. Select the curve generated by x = e t + t, y = e t t, 2 t 2 2. Consider x = 1 t 2, y = t 5, 2 t 2. Sketch the curve. Find a Cartesian equation of the curve. 2

3 3. Consider x = t, y = 2 t. Sketch the curve. Find a Cartesian equation of the curve. 4. Match the graphs of parametric equations x = f (t) and y = g (t) in (a) (d) with the parametric curve labeled I - IV. 3

4 5. Use the graphs of x = f (t) and y = g (t) to sketch the parametric curve. 4

5 6. Match the graphs with their parametric equations. 5

6 1. (Tangents) 10.2 Calculus with Parametric Curves Suppose f and g are both differentiable. If we want to find the tangent line at a point on the parametric curve x = f (t) and y = g (t), then the chain rule gives Also 2. (Areas). The area under the curve x = f (t) and y = g (t), a t b is 3. (Arc Length). A = a d y dx = d 2 y dx 2 = d y dt dx dt d dt ( d y dx ) dx dt g (t)f (t)dt or A = The arc length of the curve x = f (t) and y = g (t), a t b is a b g (t)f (t)dt 4. (Surface Area). L = a ( dx dt )2 + ( d y dt )2 dt If you rotate x = f (t) and y = g (t), a t b, about the x-axis, then the surface area is S = a 2πy ( dx dt )2 + ( d y dt )2 dt 10.3 Polar Coordinates 1. (Polar coordinate system). We choose a point in the plane that is called the pole (origin) and is labeled O. Then we draw a ray (half-line) starting at O called the polar axis. If P is a point in the plane, let r be the distance from O to P and let θ be the angle between the polar axis and the line OP. Then the point P is represented by the ordered pair (r,θ) and it is called polar coordinates of P. The angle is positive if measured in counterclockwise direction from the polar axis and negative in the clockwise direction. If P=O, then r=0, the (0,θ) represents the pole for any θ. 2. ( r,θ) and (r,θ) lie on the same line though O and at the same distance r from O, but on the opposite side of O. ( r,θ) = (r,θ + π) 6

7 3. (Connection between Cartesian Coordinate and Polar Coordinate). x = r cosθ, y = r sinθ,r 2 = x 2 + y 2,tanθ = y x cosθ = x r,sinθ = y r 4. (Polar Curves). The graph of a polar equation r = f (θ), or more generally, F (r,θ) = 0, consists of all points P that have at least one polar representation (r,θ) whose coordinates satisfy the equation. 5. (Cardioid) r = 1 + sinθ 6. (Four-leaved rose) r = 1 + sinθ 7. (Symmetry). Four rules (a) If a polar equation is unchanged when θ is replaced by θ, the curve is symmetric about the polar axis. (b) If a polar equation is unchanged when r is replaced by r, the curve is symmetric about the pole. (c) If a polar equation is unchanged when θ is replaced by θ + π, the curve is symmetric about the pole. (b) and (c) means that the curve remains unchanged if we rotate it though π about the origin. (d) If a polar equation is unchanged when θ is replaced by π θ, the curve is symmetric about the vertical line θ = π/2. 7

8 8. (Tangents to Polar Curves). TO find a tangent line to a polar curve r = f (θ), we have to be the slope. d y dx = dr dθ sinθ + r cosθ dr dθ cosθ r sinθ We locate horizontal tangents by finding the points where d y/dθ = 0. We locate vertical tangents at the points where dx/dθ = Areas and Lengths in Polar Coordinates 1. Area of a sector of a circle is A = 1 2 r 2 θ, where r is the radius and θ is the radian measure of central angle. 2. Let R be the region bounded by the polar curve r = f (θ) and by the rays θ = a and θ = b, where f is a positive continuous function and 0 < b a 2π. Then the area is 1 b A = a 2 r 2 1 dθ == a 2 f (θ)2 dθ 3. The area of the region bounde by two polar curves, r = f (θ), r = g (θ), θ = a and θ = b, where f (θ) g (θ) 0 and 0 < b a 2π. 8

9 The area is 1 A === a 2 [f (θ)2 g (θ) 2 ]dθ 4. (Arc Length). The length of the curve with polar equation r = f (θ), a θ b is L = a r 2 + ( dr dθ )2 dθ 9