MAT 21A Quiz: MAT21A Quiz 4a 2014 Winter

Create an equation with x on both sides of the equation for each of the following. a. Every real number is a solution. b. No real number is a solution. c. 0 is a solution. a. Choose the correct answer below. A. x = 2x - 2 B. 5x = x + 5 C. 2(x + 5) = 2x + 10 D. 3(x + 3) = 3x + 3

Consider the differential equation dy/dx = e^2x + (1 + 2e^x) y + y^2(*) Show that y_1(X) = -e^x is a solution of equation(*)Solve (*) Solve the differential equation 3(1 + x^2) dy/dx = 2xy(y^3 - 1) Solve the differential equation y - (x + 4) y' = (y')^2 Solve the initial value problem -xy' + y = (y' + 1)^2, y(0) = 0 Determine whether the following set of functions are linearly independent for all x where the function is defined {x, e^2x + 1, e^3x - 4} Prove that f_1(x) = x^2 and f_2(x) = |x| are linearly independent on (-infinity, +infinity) Show that W(f_1(x), f_2(x)) = 0 for every real number. Find a general solution of the differential equation 2y" + 5y' + 2y = 0, y(0) = pi, y'(0) = 1

Find all solutions of the equation 3x - 7 = 4 = 0. Explain why you have found all the roots. Discuss this problem in class. Find all solutions of the quadratic equations x^2 + x --4 = 0. Explain why you have found all the roots. Discuss this problem in class. Apply the quadratic formula to the equation x^2 + x + 4. What difficulty do you encounter? Does this equation have any real roots? Draw the graph of p(x) = x^2 + x + 4 and discuss in class why there are no real roots. If a polynomial is to have real roots then its graph must cross the x-axis. Discuss in class why this is true. Then discuss why a cubic equation will always have at least one real root while a quadratic equation may not. Discuss in class whether a quartic (i.e., a fourth-degree) equation will always have a real root. Look at some examples. What about p(x) = x^4 + 1? What about x^4 - 2x^2 + 1? Use Cardano's method to find a root of the polynomial x^3 - x - 6. Use Cardano's method to find a root of the polynomial 3x^3 - 10x^2 + 9. Can you write down a polynomial whose roots are -1, 3, 5? Can you give an example of a polynomial of degree 2 that has no real roots? How about degree 4? Explain why a polynomial of odd degree at least 1 will always have at least one real root.