(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
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 91E
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Question
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
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab0e1c6e-2cd5-492d-90bb-1a9c33e07d97%2F31988877-7237-46a9-b124-20e4ca1965e4%2F3pg70le_processed.jpeg&w=3840&q=75)
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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