Consider the following function 6 f(x, y,z)=z - x? cos(my) + xy? (i) Find the gradient of the function f(x, y, z) at the point P,(2,-1,-7). (ii) Find the directional derivative of f(x, y, z) at P,(2,-1,-7) along the direction of the vector ū = 2î+j+2k. (iii) Find the equation of the tangent plane to the surface given below at the point P,(2,-1, -7). 6 :- xcos(ty) + = 0 xy
1. Let {ü, 7,w, i}, where u = (3,-2), v = (0,4), ū = (-1,5) and i = (-6,4). Find the components of the resultants obtained by doing the following linear combinations. a. r = 2ū - 40 b. š= 3ū – +20 +
+ y +z- 0 y-1 5. Solve: ANS: -(2) -(0) 6. Suppose: W Calculate and H -5W+4H ANS: 7. Write the system in #5 as a MATRIX EQUATION and a VECTOR EQUATION Matrix equation: Vector Equation: Find all h so that the system with the augmented matrix is INCONSISTENT 8. 1-1 7 a. h=7 b. h-7 c. h0 d. h 0 is: 9. The SPAN of the vector d. none of these c. the point: (0, 0) b. the line...
It is given that the vectors ū = [11,0, 6] and ✓ = (-2,0,–27) lie in a linear subspace W of R'. It follows that, also ū = 29, 0, -36) lies in W. This can be seen by writing was a linear combination of ū and V. Determine the numbers x and y so û = x ·ū+y. V. Give your answer in the form x = a 1y=b for two numbers a and b.
Question 10 (1 point) If ū = (-4,-2) and W= (-6,-12), find 2ū + aw 0 (-10,-16) O(-10,-14) (-14,-6) O(-9,-14) O 1-9,-6) (-14,-16)
Consider the following. z = x2 + y2, z = 36 − y, (6, -1, 37) (a) Find symmetric equations of the tangent line to the curve of intersection of the surfaces at the given point. x − 6 12 = y + 1 −2 = z − 37 −1 x − 6 1 = y + 1 12 = z − 37 −12 x − 6 = y + 1 = z − 37 x − 6 12 =...
1 1 0 -1 Exercise 2. Let A = 0 1 0 in M3,R, and ✓ = 0 in R3. -1 0 For every vector W E R”, set g(ū) = WT AV E R. (i) Show that g: R3 → R defines a linear transformation. What is the matrix [g]c,b in the bases C = {1} and B = { 9 8 B |}? (ii) Let f: R3 + R be the function defined by f(w) = vſ Aw...
8. 4 marks] (a) Suppose that a is the reflection across the line z y +2-0 and β is the reflection ac oss the line y = 2. i. Show that the compositions a(β(z)) and β(a(z)) have a common fixed point in the intersection of two lines. Find this point. ii. Find complex numbers a, b, c, and d for which a(z) = a5+b and β(z) =cz+d. iii. Classify the compositions a(β(z)) and β(a(z)). (b) Let z = 21 and...
there were no solutions to this past paper question. Question 1 (24 marks]. (a) Rewrite the following linear program in standard inequality form: minimise 78 - 12 +503 - 1034 subject to -11x1 - 12.12 - 13 + 14 > 1, 11-472-873 = 12 21 +672 +31, 57. 11,1220, 13 50 I, unrestricted (b) Consider the following linear program in standard equation form: maximise + 212-383 + 70s subject to I +212 +2.13 + 14 = 3 I + 2x2...
and z2 = 1 1 + 3i 3-i a) Given that zı = find z such that z = 2 + i 4- ¿ 22 Give your answer in the form of a + bi. Hence, find the modulus and argument of z, such that -- < arg(2) < 7. (6 marks) b) Given w = = -32, i. express w in polar form. (1 marks) ii. find all the roots of 2b = -32 in the form of a...