ABC 6. (a) Find constants A, B and C such that (3 marks) (b) Hence determine...
(a) (6 marks) Determine the values of A, B and C such that 1 x(x
2 + 1) = A x + Bx + C x 2 + 1 . Hence evaluate Z 1 x(x 2 + 1)dx.
(b) (6 marks) Find the area under the curve y = xe−x for x ≥ 0.
3. (a) (6 marks) Determine the values of A, B and C such that 1 А + Bir + 12+1 (22+1) 1 Hence evaluate dr. (2+1)...
b) Using implicit function theorem, find the slope of the following utility functions. 6 marks) e parameters 0, μ, π, c, b and q are positive constants. What is the name for all of these slopes in economics? Th (1 marks) c) Determine the level of homogeneity (and hence, return to scale) for the following production function:
b) Using implicit function theorem, find the slope of the following utility functions. 6 marks) e parameters 0, μ, π, c, b and...
9. A vector field is defined as: 2 marks (a) Sketch this field on the below axis. 0.5 0.5 0.5 2 marks (b) Evaluate ▽ . F. (c) Evaluate ▽ × F and hence conclude whether F is a conservative vector field 2 marks (d) Demonstrate whether there is a scalar function φ such that F and 3 marks] hence conclude whether Fis a conservative vector fiel Recall dx and that this can be used to determine tan dx
9....
6.a) Let -2 5 -6 10 ), 2) an Evaluate the matrix 8 marks) b) Find the area enclosed between the following curves and sketch the region. 5x2-2. f(x)--3x2 + 30 and g(x) marks) c) Evaluate the following integrals. Give your answers to 2 decimal places. dac i) 2 (3+2 dx. (8 marks) [25 marks]
6.a) Let -2 5 -6 10 ), 2) an Evaluate the matrix 8 marks) b) Find the area enclosed between the following curves and sketch...
2. Let 6 marks (a) Find f(x),f"(x), and f"(x). (b) Find the second order Taylor expansion of f at 1, namely f(r) = ao + ala-1 ) + a2(z-1)2 + R2(x), where Ra is the remainder. You should find ao, a, a2, and R(p). 8 marks that the error in this estimation (i.e., R2(0.9)1) is at most 10-3. 6 marks (c) Use the Taylor expansion found above to estimate the value of f(0.9). Show Find f(x), f"(), and f" (b)...
1(a) Find the square roots of the complex number z -3 + j4, expressing your answer in the form a + jb. Hence find the roots for the quadratic equation: x2-x(1- 0 giving your answer in the form p+ q where p is a real number and q is a complex number. I7 marks] (b) Express: 3 + in the form ω-reje (r> 0, 0 which o is real and positive. θ < 2π). Hence find the smallest value of...
for the following function (5 Marks) dy 5. (b) Use implicit differentiation to find dx 3x* - 6 ln(2xy) = 7-5e24-sy dy 1. (b) Evaluate: lim r? + 4x -12 (3-12x +1 (5 Marks)
Question 4 6-69) Consider the following three points: a-0 b-3and c-1 a) Find the parametric vector equation of the plane which passes through these three [3 marks] b) Find the vector cartesian equation of the plane passing through the three points listed [2 marks] c) Hence, or otherwise, find the non-vector cartesian equation of the plane passing through 3 marks] points above. the points above.
Question 4 6-69) Consider the following three points: a-0 b-3and c-1 a) Find the parametric...
Consider the 3 x 3 matrix A-1-ovvT where a R, 1 is the identity matrix and v the vector (a) Determine the eigenvalues and eigenvectors of A (b) Hence find a matrix which diagonalises A. (c) For which a is the matrix A singular? (d) For which α is the matrix A orthogonal ?
Consider the 3 x 3 matrix A-1-ovvT where a R, 1 is the identity matrix and v the vector (a) Determine the eigenvalues and eigenvectors of...
($ ?) 4 2. (a) Find the eigenvalues and eigenvectors of the matrix 3 Hence or otherwise find the general solution of the system = 4x + 2y = 3x - y 195 marks 5. (a) Give a precise definition of Laplace transform of a function f(t). Use your definition to determine the Laplace transform of 3. Osts 2 6-t, 2 <t f(t) = [20 marks] (b) A logistic initial value problem is given by dP dt kP(M-P), P(0) -...