University:
Rice University
Course:
MATH 102 | Single Variable Calculus II
Academic year:

2023
Views:

18

Pages:

7
Author:

MathMonk

Questions and Answers Sheet 5 Parametric Equations Question #1 Find the parametrization of the function 𝑓(𝑥) = 2𝑥 3 − 7 Answer: Question #2 Find y'(x) and y''(x) of the function y = f(x) defined parametrically as 𝑥 = 𝑡 2 − 3, 𝑦 = 𝑡 3 − 2𝑡, 𝑡 = 1 Answer: Question #3 The Cartesian coordinates of a point are given. (-9, 9) (i) Find polar coordinates (𝑟, 𝜃) of the point, where r > 0 and 0 ≤ 𝜃 < 2? Answer: Question #4 Find all points of interception of the pair of polar equations in polar form r=3 and 𝑟 = −4 cos 𝑥 Answer: Question #5 Replace the polar equations with equivalent Cartesian equations. Then describe or identify the graph. 𝑟 2 = −6𝑟 sin 𝜃 Answer: 𝑟 2 = −6𝑟 sin 𝜃. The Cartesian equation is 𝑥 2 + 𝑦 2 = −6𝑦[∵ 𝑥 = 𝑟 cos 𝜃 , 𝑦 = 𝑟 sin 𝜃 & 𝑟 = √𝑥 2 + 𝑦 2 ] ⇒ 𝑥 2 + 𝑦 2 + 6𝑦 = 0 ⇒ 𝑥 2 + 𝑦 2 + 6𝑦 + 9 = 9 ⇒ 𝑥 2 + (𝑦 + 3)2 = 9 2 ⇒ (𝑥 − 0)2 + (𝑦 − (−3)) = 32 Now its the form of of ⇒ (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2 Hence it represents a circle with centre at (h,k)(0,-3) & radius 3 Question #6 Replace the polar equations with equivalent Cartesian equations. Then describe or identify the graph. 𝑟 2 = −4𝑟 cos 𝜃. Answer: 𝑟 2 = −4𝑟 cos 𝜃. Hence, 𝑥 2 + 𝑦 2 = −4𝑥 Hence, 𝑥 2 + 4𝑥 + 𝑦 2 = 0 Or, 𝑥 2 + 4𝑥 + 𝑦 2 + 4 = 0 + 4 = 4 Or, (𝑥 + 2)2 + 𝑦2 = 22 Hence, the equivalent cartesian equation: (𝑥 + 2)2 + 𝑦 2 = 22 And it represents the circle of radius 2 centred at (-2, 0) Question #7 Find polar equation for the circle. Sketch circle in the coordinate plane and label it with both its Cartesian and polar equations. 𝑥 2 + 2𝑥 + 𝑦 2 = 0 Answer: Given that: 𝑥 2 + 𝑦 2 + 2𝑥 = 0 To covert Cartesian coordinates to polar coordinates, 𝑥 = 𝑟 cos(𝜃) , 𝑦 = 𝑟 sin(𝜃) 𝑥 2 + 𝑦 2 = 𝑟 {2] 𝑥 2 + 𝑦 2 + 2𝑥 = 0 𝑟 2 + 2𝑟 cos(𝜃) = 0 𝑟(𝑟 + 2 cos(𝜃)) = 0 r=0 𝑟 + 2 cos(𝜃) = 0 𝑟 = −2 cos(𝜃) Question #8 Find the point(s) of intersection of the following two parametric curves, by first eliminating the parameter, then solving the system of equations. 1 𝑥 = 𝑡 + 5, 𝑦 = 𝑡 2 and 𝑥 = 𝑡, 𝑦 = 𝑡 16 a. (25, 400) b. (1, 16) c. (400, 16) d. A and B e. A and C Answer: Question #9 Polar coordinates of point P are given. Find all of its polar coordinates. 𝑃 = (1, −𝜋/4) Answer: A polar coordinate (r, θ) can be written in the form: (𝑟, 𝜃 + 2𝑛𝜋) or (−𝑟, 𝜃 + (2𝑛 + 1)𝜋) If θ is in radians or: (𝑟, 𝜃 + 360°𝑛) or (−𝑟, 𝜃 + 180° + 360°𝑛) If θ is in degrees. So we write P as: (1, −𝜋/4 + 2𝑛𝜋) and (−1, −𝜋/4 + (2𝑛 + 1)𝜋) Where n is an integer. Question #10 The letters r and θ represent polar coordinates. Write each equation using rectangular coordinates (x, y). r = sinθ - cosθr = sinθ - cosθ. Answer: We know that r cos θ = x and that r sin θ = y, so x/r = cos θ and y/r. Thus, r = y/r – x/r. Multiplying both sides by r gives: 𝑅2 = 𝑦– 𝑥. We know that 𝑟 = √𝑥 2 + 𝑦 2 , then 𝑥 2 + 𝑦2 = 𝑦 − 𝑥