Homework Answers
1. Using F(r) = -y/r2, show that the equation of motion in a central force, namely...
1. Using F(r) = -y/r2, show that the equation of motion in a central force, namely a na = -u(0) - EF (), produces the solution u(b) = (-1(1 + e coso) What are the expressions for C and e in terms of the parameters of the problem? 2. Evaluate the entries for the following table for the given values of e (assume l,u,y are known) E (T + U) Pmax/rmin Shape of orbit 0 .5
Consider the minimisation and maximisation of the objective function f : R2 + R given by...
Consider the minimisation and maximisation of the objective function f : R2 + R given by f(x,y) = (1 - 1)2 + y2 + 3 on the feasible region D C R2 consisting of the boundary and interior of the right-angled trian- gle whose vertices are the points (0,0), (3,0) and (0,4). (a) Write down a Lagrangian function L(x, y, 1) whose only stationary point (x*, y*, \*) corre- sponds to the point of tangency (r*, y*) between the line...
4 please In each of Problems 3 to 7 (a) sketch the graph of the given...
4 please
In each of Problems 3 to 7 (a) sketch the graph of the given function for three periods, (b) find the Fourier series, and (c) sketch the graph of the function to which the series converges for three periods. 3. f(x) = -1, -L <r<L; f(x + 2L) = f(x) 4. f(x) = { }; -L<=<0, 0, 052<L: f(x+2L) = f(x) 5. Ls - 1 +1, -13&<0,1 <: (x + 2) = f(x) 0 < 1; ) +...
Consider the minimisation and maximisation of the objective function f : R2 + R given by...
Consider the minimisation and maximisation of the objective function f : R2 + R given by f(x, y) = (x - 1)2 + y2 + 3 on the feasible region DC R2 consisting of the boundary and interior of the right-angled trian- gle whose vertices are the points (0,0), (3,0) and (0,4). (iv) Find the coordinates (x*, y*, 1*) of the stationary point of your function L(x, y, 1). (v) State if the point (x*,y*) is a constrained minimum of...
QUESTION 4 Given the equation of a point, r(t) ( I)i ( -I)j Sketch the graph of r(r) = (1 + l)i + (r2-Dj fr-2 2. Dr...
QUESTION 4 Given the equation of a point, r(t) ( I)i ( -I)j Sketch the graph of r(r) = (1 + l)i + (r2-Dj fr-2 2. Draw the (a) t 4 marks) position vector r(0) on the same diagram. b) Find the unit tangent vector of the point at 0 and show it on the same diagram in (a). Explain what you understand about the direction of the tangent (5 marks)
Question 86 - 100: Look at the graph of function f(r) . A -5 -1 3...
Question 86 - 100: Look at the graph of function f(r) . A -5 -1 3 5 7 9 11 1--1 86. lim f(x) = 87. lim f() = 88. Is f(x) continuous at I = 1? 89. Is f(x) continuous at x = 0? 90. Is f(x) continuous at x = - 3 from the left hand side ? 91. limf() = 92. Does limf (x) exist ? 93. Is f(x) continuous at I = 8? 94. Is f()...
Part E Let 1-1.50 A, R.-1.00 cm, R-260 an, ard R3-2.50 crn Graph B from R-010 R = 3.00 cm No elements selected 3....
Part E Let 1-1.50 A, R.-1.00 cm, R-260 an, ard R3-2.50 crn Graph B from R-010 R = 3.00 cm No elements selected 3.0 2.5 2.0 1.0 0.5 1.0 25 R (em) Select the elements from the list and add them to the canvas setting the appropriate attributes Press TAD to get to the main menu Provide Feedback Part B Determine the magnetic field at a distance R trom the axs tor R<R<R Express your answer in terms of some...
(a) Consider the minimisation and maximisation of the objective function f : R2 + R given...
(a) Consider the minimisation and maximisation of the objective function f : R2 + R given by f(x,y) = (x - 1)2 + y2 +3 on the feasible region D C R2 consisting of the boundary and interior of the right-angled trian- gle whose vertices are the points (0,0), (3,0) and (0,4). (0) Make a sketch on the coordinate plane Rd of the region D and add to your sketch a few contours of the objective function f. (ii) Obtain...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...
1. (2 pts each) The graph of some unknown function f is given below. 10 6/ 8-64-2 624 10 12 Use the graph to estimate the following quantities: (0 f (9) (g) f(4) b) lim (a) lim (e) (d) lim ( 6)...
1. (2 pts each) The graph of some unknown function f is given below. 10 6/ 8-64-2 624 10 12 Use the graph to estimate the following quantities: (0 f (9) (g) f(4) b) lim (a) lim (e) (d) lim ( 6) (e) lim f(x) (c) lim f(x) if g(x)f(x) 6) a value of r where f is continuous but not differentiable (k) a value of r where f"(x) 0 and f"(x)>0 (1) the location of a relative maximum value...
1. Using F(r) = -y/r2, show that the equation of motion in a central force, namely a na = -u(0) - EF (), produces the solution u(b) = (-1(1 + e coso) What are the expressions for C and e in terms of the parameters of the problem? 2. Evaluate the entries for the following table for the given values of e (assume l,u,y are known) E (T + U) Pmax/rmin Shape of orbit 0 .5
Consider the minimisation and maximisation of the objective function f : R2 + R given by f(x,y) = (1 - 1)2 + y2 + 3 on the feasible region D C R2 consisting of the boundary and interior of the right-angled trian- gle whose vertices are the points (0,0), (3,0) and (0,4). (a) Write down a Lagrangian function L(x, y, 1) whose only stationary point (x*, y*, \*) corre- sponds to the point of tangency (r*, y*) between the line...
4 please
In each of Problems 3 to 7 (a) sketch the graph of the given function for three periods, (b) find the Fourier series, and (c) sketch the graph of the function to which the series converges for three periods. 3. f(x) = -1, -L <r<L; f(x + 2L) = f(x) 4. f(x) = { }; -L<=<0, 0, 052<L: f(x+2L) = f(x) 5. Ls - 1 +1, -13&<0,1 <: (x + 2) = f(x) 0 < 1; ) +...
Consider the minimisation and maximisation of the objective function f : R2 + R given by f(x, y) = (x - 1)2 + y2 + 3 on the feasible region DC R2 consisting of the boundary and interior of the right-angled trian- gle whose vertices are the points (0,0), (3,0) and (0,4). (iv) Find the coordinates (x*, y*, 1*) of the stationary point of your function L(x, y, 1). (v) State if the point (x*,y*) is a constrained minimum of...
QUESTION 4 Given the equation of a point, r(t) ( I)i ( -I)j Sketch the graph of r(r) = (1 + l)i + (r2-Dj fr-2 2. Draw the (a) t 4 marks) position vector r(0) on the same diagram. b) Find the unit tangent vector of the point at 0 and show it on the same diagram in (a). Explain what you understand about the direction of the tangent (5 marks)
Question 86 - 100: Look at the graph of function f(r) . A -5 -1 3 5 7 9 11 1--1 86. lim f(x) = 87. lim f() = 88. Is f(x) continuous at I = 1? 89. Is f(x) continuous at x = 0? 90. Is f(x) continuous at x = - 3 from the left hand side ? 91. limf() = 92. Does limf (x) exist ? 93. Is f(x) continuous at I = 8? 94. Is f()...
Part E Let 1-1.50 A, R.-1.00 cm, R-260 an, ard R3-2.50 crn Graph B from R-010 R = 3.00 cm No elements selected 3.0 2.5 2.0 1.0 0.5 1.0 25 R (em) Select the elements from the list and add them to the canvas setting the appropriate attributes Press TAD to get to the main menu Provide Feedback Part B Determine the magnetic field at a distance R trom the axs tor R<R<R Express your answer in terms of some...
(a) Consider the minimisation and maximisation of the objective function f : R2 + R given by f(x,y) = (x - 1)2 + y2 +3 on the feasible region D C R2 consisting of the boundary and interior of the right-angled trian- gle whose vertices are the points (0,0), (3,0) and (0,4). (0) Make a sketch on the coordinate plane Rd of the region D and add to your sketch a few contours of the objective function f. (ii) Obtain...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...
1. (2 pts each) The graph of some unknown function f is given below. 10 6/ 8-64-2 624 10 12 Use the graph to estimate the following quantities: (0 f (9) (g) f(4) b) lim (a) lim (e) (d) lim ( 6) (e) lim f(x) (c) lim f(x) if g(x)f(x) 6) a value of r where f is continuous but not differentiable (k) a value of r where f"(x) 0 and f"(x)>0 (1) the location of a relative maximum value...
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