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(a) Given is the function h(x, y) = (y- 1)(x + 1). (1) Draw the level curves for h(x, y) = c for c= -1,0, 1 in the zy-plane. Label curves clearly with the appropriate value of c. Show all your working. (ii) In the drawing of the previous part, clearly mark points P₁ = (-1,3). P₂ = (-3,1) and P3 =(-2,-1). Draw a direction in which h neither increases nor decreases at P₁. Draw the gradient vector of h at P2. Draw the direction of the steepest decrease at P3. Show your working and justify your choices.

(a) Given is the function h(x, y) = (y- 1)(x + 1). (1) Draw the level curves for h(x, y) = c for c= -1,0, 1 in the zy-plane. Label curves clearly with the appropriate value of c. Show all your working. (ii) In the drawing of the previous part, clearly mark points P₁ = (-1,3). P₂ = (-3,1) and P3 =(-2,-1). Draw a direction in which h neither increases nor decreases at P₁. Draw the gradient vector of h at P2. Draw the direction of the steepest decrease at P3. Show your working and justify your choices.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 91E
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Part A needed in 10 minutes please
(a) Given is the function h(x, y) = (y − 1)(x + 1). (1) Draw the level curves for h(x, y) = c for c= -1,0, 1 in the zy-plane. Label curves clearly with the appropriate value of c. Show all your working. (ii) In the drawing of the previous part, clearly mark points P₁ = (-1,3), = (-3,-1). Draw a direction in which h neither increases nor ( P₁ = (-2, 1) and P₁ = decreases at P₁. Draw the gradient vector of h at P2. Draw the direction of the steepest decrease at P3. Show your working and justify your choices. (b) Consider the following second-order differential equation d'y dy + + y = 0. dx² dx (c) (i) Find the general solution of the equation. (ii) Use the general solution from part (i) to evaluate lim y(x). 84+∞ (iii) Find the particular solution satisfying y(0) = 0 and y' (0) = √3. Calculate the particular solution of d'y dr² - (0) = 0. d³ y dx³ = sin(x) with y(0) = 0, d (0) = 0 and =
Transcribed Image Text:(a) Given is the function h(x, y) = (y − 1)(x + 1). (1) Draw the level curves for h(x, y) = c for c= -1,0, 1 in the zy-plane. Label curves clearly with the appropriate value of c. Show all your working. (ii) In the drawing of the previous part, clearly mark points P₁ = (-1,3), = (-3,-1). Draw a direction in which h neither increases nor ( P₁ = (-2, 1) and P₁ = decreases at P₁. Draw the gradient vector of h at P2. Draw the direction of the steepest decrease at P3. Show your working and justify your choices. (b) Consider the following second-order differential equation d'y dy + + y = 0. dx² dx (c) (i) Find the general solution of the equation. (ii) Use the general solution from part (i) to evaluate lim y(x). 84+∞ (iii) Find the particular solution satisfying y(0) = 0 and y' (0) = √3. Calculate the particular solution of d'y dr² - (0) = 0. d³ y dx³ = sin(x) with y(0) = 0, d (0) = 0 and =
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