This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Welcome back kiran — been a while! Let's pick up where you left off. To find the level curves f(x, y) = c, we set the given function equal to each value of c and simplify the resulting equation. Since I am a text-based AI, I will provide the equations for the level curves and describe their shapes, but I cannot sketch them. 13. f(x, y) = x + y - 1, c = -3, -2, -1, 0, 1, 2, 3 Step 1: Set f(x, y) = c. x + y - 1 = c Step 2: Rearrange the equation to solve for y. y = -x + 1 + c These are equations of parallel lines with a slope of -1. For c = -3: y = -x + 1 - 3 y = -x - 2 For c = -2: y = -x + 1 - 2 y = -x - 1 For c = -1: y = -x + 1 - 1 y = -x For c = 0: y = -x + 1 + 0 y = -x + 1 For c = 1: y = -x + 1 + 1 y = -x + 2 For c = 2: y = -x + 1 + 2 y = -x + 3 For c = 3: y = -x + 1 + 3 y = -x + 4 14. f(x, y) = x^2 + 4y^2, c = 0, 1, 4, 9, 16, 25 Step 1: Set f(x, y) = c. x^2 + 4y^2 = c These are equations of ellipses centered at the origin, or a point if c=0. For c = 0: x^2 + 4y^2 = 0 x=0, y=0 (a single point at the origin) For c = 1: x^2 + 4y^2 = 1 (x^2)/(1) + (y^2)/(1/4) = 1 (an ellipse with semi-axes a=1, b=1/2) For c = 4: x^2 + 4y^2 = 4 (x^2)/(4) + (y^2)/(1) = 1 (an ellipse with semi-axes a=2, b=1) For c = 9: x^2 + 4y^2 = 9 (x^2)/(9) + (y^2)/(9/4) = 1 (an ellipse with semi-axes a=3, b=3/2) For c = 16: x^2 + 4y^2 = 16 (x^2)/(16) + (y^2)/(4) = 1 (an ellipse with semi-axes a=4, b=2) For c = 25: x^2 + 4y^2 = 25 (x^2)/(25) + (y^2)/(25/4) = 1 (an ellipse with semi-axes a=5, b=5/2) 15. f(x, y) = xy, c = -3, -2, -1, 0, 1, 2, 3 Step 1: Set f(x, y) = c. xy = c These are equations of hyperbolas, except for c=0. For c = -3: xy = -3 (a hyperbola in the second and fourth quadrants) For c = -2: xy = -2 (a hyperbola in the second and fourth quadrants) For c = -1: xy = -1 (a hyperbola in the second and fourth quadrants) For c = 0: xy = 0 (the x-axis and y-axis) For c = 1: xy = 1 (a hyperbola in the first and third quadrants) For c = 2: xy = 2 (a hyperbola in the first and third quadrants) For c = 3: xy = 3 (a hyperbola in the first and third quadrants) 16. f(x, y) = sqrt(25 - x^2 - y^2), c = 0, 1, 2, 3, 4 Step 1: Set f(x, y) = c. sqrt(25 - x^2 - y^2) = c Step 2: Square both sides of the equation. Note that c must be non-negative. 25 - x^2 - y^2 = c^2 Step 3: Rearrange the equation to the standard form of a circle. x^2 + y^2 = 25 - c^2 These are equations of concentric circles centered at the origin. For c = 0: x^2 + y^2 = 25 - 0^2 x^2 + y^2 = 25 (a circle with radius 5) For c = 1: x^2 + y^2 = 25 - 1^2 x^2 + y^2 = 24 (a circle with radius sqrt(24) = 2sqrt(6)) For c = 2: x^2 + y^2 = 25 - 2^2 x^2 + y^2 = 21 (a circle with radius sqrt(21)) For c = 3: x^2 + y^2 = 25 - 3^2 x^2 + y^2 = 16 (a circle with radius 4) For c = 4: x^2 + y^2 = 25 - 4^2 x^2 + y^2 = 9 (a circle with radius 3) Drop the next question!