794 words - 4 pages

Calculus 2 (BT)

Homework Assignment #1

Spring 2015

Deadline: 8:00 AM on Thursday, March 26, 2015 Remarks: -Students hand in the solutions to the ﬁrst 25 problems listed below: 20 problems of Chapter 1 and problems 1-5 of Chapter 2. -Students should show the solutions in A4 paper size. The solutions must be in the order as listed. Chapter 1: (20 problems) Section 9.1: 45. Section 9.2: 18, 27, 33, 53, 55. Section 9.3: 4, 5, 39, 42. Section 9.5: 4, 11, 21, 22. Section 9.6: 11, 22, 30, 37, 40, 53. Chapter 2: 1. Write down the augmented matrix, and then solve the following systems of linear equations. Comment on each case (e.g. the number of equations and unknowns, whether there is inconsistency, whether the solutions are unique, many, trivial, etc.). (a) x − 2y − z − w = −4 3x + y + z − 2w = 11 x + 12y + 7z + w = 31. (b) 2x + y + 5z = 1 x − 3y + 6z = 2 3x + 5y + 4z = 0 (c) 2x + 5y − z = 15 x − y + 3z = 4 3x + 3y − 5z = 2. (d) x1 + 2x2 + ...view middle of the document...

The corresponding units for An Giang are 31, 24, and 11. How many bags of each brand of fertilizer should be used per acre for each farm? Hint: a. Dong Nai: (x, y, z) = (5, 2, 3). b. An Giang: (x, y, z) = (2, 4, 7). 3. Find values of a, b, and c such that the graph of the polynomial p(x) = ax2 + bx + c passes through the points (1, 2), (−1, 6), and (2, 3). Hint: a = 1, b = 2, and c = 3. 4. A dietician is preparing a meal consisting of foods A, B, and C. Each ounce of food A contains 2 units of protein, 3 units of fat, and 4 units of carbohydrate. Each ounce of food B contains 3 units of protein, 2 units of fat, and 1 unit of carbohydrate. Each ounce of food C contains 3 units of protein, 3 units of fat, and 2 units of carbohydrate. If the meal must provide exactly 25 units of protein, 24 units of fat, and 21 units of carbohydrate, how many ounces of each type of food should be used? 5. A= 1 0 −2 4 , 2 1 B= 3 5 −1 , 0 −2 1 C = 5 3 , 4 0

D=

4 −2 . −2 1

State whether the followings exist. Find those which do exist. (a) A + B (b) B + D (c) 3AT C (d) AB 2 (e) BA (f) A = AA (g) AAT (h) BACD. 6. Evaluate the following determinants (a) a 0 0 1 b 0 3 d c (b) 5 7 −1 1 2 0 3 1 3 7. Let A= 2 −1 4 2 ,B = −3 4 1 −1 2

Calculus 2 (BT)

Homework Assignment #1

Spring 2015

(a) Compute |A|, |B|, A−1 , and B −1 . (b) Verify that (AB)−1 = B −1 A−1 and |AB| = |A||B|. 8. (a) Find the inverse of matrix A, and (b) use it to solve the given system of equations. 1 0 −2 4 A = −3 1 2 −3 4 x − 2z = 1 −3x + y + 4z = 2 2x − 3y + 4z = −1 Hint: A−1 8 3 1 = 10 4 1

7 2 3 2 1 2

9. Find the eigenvalues and eigenvectors of the following matrices: (a) 1 5 A= 3 −1 (b) 1 0 0 B= 0 1 2 0 0 2 10. Balance the following chemical equations using the vector equation approach discussed in the lecture: (a) Limestone, CaCO3 , neutralizes the acid, H3 O, in acid rain by the following unbalanced equation: H3 O + CaCO3 → H2 O + Ca + CO2 (b) Alka-Seltzer contains sodium bicarbonate (N aHCO3 ) and citric acid (H3 C6 H5 O7 ). When a tablet is dissolved in water, the following reaction produces sodium citrate, water, and carbon dioxide (gas): N aHCO3 + H3 C6 H5 O7 → N a3 C6 H5 O7 + H2 O + CO2 -END

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