Name: _______________ LoE: ______ hours ECE 3054 Homework 01 Fall 2017 This particular homework assignment is not typical of the assignments in this course, because it is intended to test your background from physics and math rather than to practice new skills. Due to this difference, some special rules apply: All of your work should appear on this assignment. You may not discuss the problems with anyone else. This assignment will be weighted the same as any other. 1. Write a mathematical equation and verbal description for Gauss's Law. 2. Write a mathematical equation and verbal description for Faraday's Law of Induction. 3. Consider the capacitor shown to the right. a. How much charge is stored in the capacitor? 20 V 5 F b. How much energy is stored in the capacitor? 1 For Problems 4 through 6, consider two lines x 2 y 3 and y 3x 1 . y 4 4. Sketch both lines on the set of axes to the right. 5. Determine the coordinates of the point of intersection for the two lines. 3 2 1 0 x -1 -2 6. Express the first line in slope-intercept form y mx b . -3 -4 -4 For Problems 7 and 8, consider the graph shown to the right. 7. Write an expression for Line Segment 1. -3 -2 -1 0 1 y 4 2 3 4 Line Segment 1 3 2 Line Segment 2 1 0 8. 0 Write an expression for Line Segment 2. 2 2 4 6 8 x 2 0.5 For Problems 9 through 12 consider the matrix A 1 1 9. What is the value of the product Ax when xT = [1 2]? 10. What is the value of the inverse of A? 11. What is the value of the solution x to the equation Ax = b when bT = [ 10]? 12. What are the eigenvalues of A? 3 13. Simplify the following expression by performing the indicated derivative. x(t ) 14. Simplify the following expression by performing the indicated derivative. Assume that , Fa, Fb, and are parameters (constants). x(t ) 15. d 2t 2 3t 1 dt d exp(t )( Fa cos(t ) Fb sin(t )) dt Simplify the following expression by performing the indicated integration. Assume that t is the independent variable and that t0, , and F are parameters (constants). Your answer should be written such that t and t0 appear only together as the difference t t0. t x (t ) e ( t ) F d t0 4 For Problems 16 and 17, consider the function f (x ) shown on the graph to the right. y 16. Sketch the derivative 4 d f (x ) . 2 dx 0 x -2 f(x) 2 1 0 -1 -2 0 1 2 -4 0 17. Sketch the integral x F ( x ) f ( ) d . 0 1 2 3 4 y 3 2 1 0 x -1 0 18. 1 2 Find the maximum value of the function f ( x ) x [0, ) . 5 3 4 x over the semi-infinite interval ( x 1) 2 3 4 x 19. Consider the following differential equation: d x(t ) ax(t ) F dt where a and F are constants. Later in the course you will learn how to find the solution x (t ) to such a problem; for now, we will give you the solution: x (t ) F Ae at a where A is an arbitrary constant. Show that the given solution satisfies the differential equation by differentiating it and then substituting dx / dt and x into the original differential equation. 6 20. What are the roots of the polynomial p( s) s 2 2s 8 ? 21. What are the roots of the polynomial p( s) s 2 2s 5 ? 22. The expression f (t ) K cos(t ) can also be written f (t ) A cos(t ) B sin( t ) . Determine the expressions for A and B in terms of K and by applying an appropriate trigonometric identity. 7 For problems 23 through 27 consider the two complex numbers c1 = 3 j4 and c2 = 2 + j. 23. What is the complex conjugate of c1? 24. Rewrite c1 in polar form. 25. What is the value of the sum c1 + c2? 26. What is the value of the product c1c2? 27. What is the value of the ratio c1/c2? 8 For Problems 28 through 30, construct the desired vector using only a straight edge and compass. 28. Construct the sum v1 v2 . v2 v1 29. Construct the difference v1 v 2 . v2 v1 30. Construct the projection of v 2 onto v1 . v2 v1 9