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Due date: 16 March 2021 before 11.00 PM 1. Find the inverse function (if it is defined) of the following functions. (4 marks) a. f(x) = (2x-3)/(x+5) (x in R, x ≠ -5) b. f(x) = x" + 2. (x in R) 2. Three different functions, f(x), g(x) and h(x), have the same graph on [0, 2] as shown in Figure. On separate diagrams, sketch their graphs for [-4, 4] given that a. f(x) is periodic with period 2; b. g(x) is periodic with period 4 and is even; c. h(x) is periodic with period 4 and is odd. (6 marks) 3. Calculate the rate of change of the linear functions given by (2 marks) a. f(x) = 1/2 x - 4 b. f(2) = 5 and f(-3) = 10 4. Determine which of the following quadratic functions are irreducible (show your working) (3 marks) a. f(x) = x" + 3x + 4 b. f(x) = 4x" - 12x + 9 c. f(x) = 9 - 5x - 2x" 5. For what values of x are the values of the quadratic function below greater than zero? (2 marks) a. f(x) = x" - 3x - 6 6. Show that the roots α, β of the equation x" + 4x + 1 = 0 satisfy the equations. (8 marks) α" + β" = 14 α³ + β³ = -52 Hence find the quadratic equations whose roots are a. α" and β" b. α³ and β³ 7. Express as partial fractions. (6 marks) (a) (x + 2) / ((x - 1)(x - 4)) (b) (x" + 4) / ((x + 1)(x - 3)) (c) (x" - 2x + 3) / ((x + 2)"(x - 1)) 8. Sketch the graph of the function given below locating asymptotes (4 marks) y = (x" - 8x + 15) / x 9. Plot the curve whose parametric equations are x = t(t + 4), y = t + 1. Show that it is a parabola. (5 marks) 10. Sketch for -3π ≤ x ≤ 3π. (4 marks) a. y = sin 2x b. y = cos 1/2 x 11. Solve the following equations for 0 ≤ x ≤ 2π. (6 marks) a. 3sin"x + 2sinx - 1 = 0 (b) 4cos"x + 5cosx + 1 = 0 (c) 2tan"x - tanx - 1 = 0 12. Express as a single logarithm: (1 mark) 4 ln 3 - 1/2 ln 36 13. Sketch carefully the graphs of the functions (4 marks) (a) y = 2^x, y = log₂x (on the same axes) (b) y = e^x, y = ln x (on the same axes)
Sri K.
Use Cramer's rule to compute the solution of the system. 1x1 + 5x2 = 13 4x1 + 7x2 = 18 7x1 + 4x2 = -4 2x1 + 5x2 = 11 What is the solution of the system? x1 = x2 = (Type integers or simplified fractions.) 3. Use Cramer's rule to compute the solution of the system. 10x1 + 3x2 + x3 = 4 4x1 + 4x3 = 4 10x1 + x2 = 2 x1 = x2 = x3 = (Type integers or simplified fractions.) 4. Compute the adjugate of the given matrix, and then use the Inverse Formula to give the inverse of the matrix. A = [ 0 -5 -1 2 0 0 -1 1 1 ] The adjugate of the given matrix is adj A = 5. Compute the adjugate of the given matrix, and then use the Inverse Formula to give the inverse of the matrix. A = 1 3 4 3 1 -1 0 1 2 The adjugate of the given matrix is adj A = 6. Find the area of the parallelogram whose vertices are listed. a. (0,0),(5,9),(6,4),(11,13) b. (-1,0),(0,7),(8,-4),(9,3) The area of the parallelogram is square units. 7. Let S be the parallelogram determined by the vectors b1 = -3 7 and b2 = -3 10 , and let A = 7 -7 -2 7 . Compute the area of the image of S under the mapping x -> Ax. The area of the image of S under the mapping x -> Ax is 8. Let S be the tetrahedron in ℝ3 with vertices at the vectors 0, e1, e2, and e3, and let S' be the tetrahedron with vertices at vectors 0, v1, v2, and v3. Complete parts (a) and (b) below. a. Describe a linear transformation that maps S onto S'. If T is a linear transformation that maps S onto S', then the standard matrix for T, written in terms of v1, v2, and v3, is A =
Adi S.
Sketch the graph of the following function. List the coordinates of where extrema or points of inflection occur. State where the function is increasing or decreasing, as well as where it is concave up or concave down: f(x) = 3x^2 + 36x What are the coordinates of the relative extrema? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The coordinates of the relative extrema are (Simplify your answer. Type an ordered pair. Type an exact answer using radicals as needed. Use integers or fractions for any numbers in the expression. Use comma to separate answers as needed.) B. There are no relative extrema. What are the coordinates of the inflection points? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The coordinates of the inflection points are (Simplify your answer. Type an ordered pair. Type an exact answer using radicals as needed. Use integers or fractions for any numbers in the expression. Use comma to separate answers as needed.) B. There are no inflection points. On what interval(s) is the function increasing or decreasing? A. The function is increasing on (-∞, 0) and (2, ∞). The function is decreasing on (0, 2). B. The function is never increasing. The function is decreasing on (-∞, 0). C. The function is increasing on (-∞, ∞). The function is never decreasing. D. The function is increasing on (0, ∞). The function is decreasing on (-6, 0).
Carson M.
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