MATH 141 Study Guide - Quiz Guide: Ope, Product Rule, Royal Netherlands Academy Of Arts And Sciences

We know that if r1 and r2 are the roots of ar2 + br + c = 0 and with r1 r2, then er1t and er2t are solutions of the differential equation ay" + by' + cy = 0. Show that W = er2t - er1t / r2 - r1 is also a solution of the equation when r1 r2. Think of r1 as a fixed constant and use L'Hospital rule to evaluate the limit of W as r2 approaches r1, thereby obtaining the second solution in the case of equal characteristic roots.

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.

4. Recall that when we write an indefinite integral, we always attach a "+C" at the end. The point of this exercise is to show us that we have to be careful doing that sometimes. We will calculate the following integral in two different ways: 2x +1 2 dx 1 2 dx 1 Notice that C, and C2 are arbitrary constants. If we were to set C1 C2-0, then However, substitution of x =-1/2 into the original integrand causes division by 01 Make an argument that C, and G2 cannot simultaneously equal 0, avoiding this paradox. Hint: Set国and ® equal to each other, then find a relationship between C1 and C2