Download 27 Questions on Calculus and Analytic Geometry with Answer key - Exam | MATH 221 and more Exams Analytical Geometry and Calculus in PDF only on Docsity! Exam 1 A. Miller Fall 91 Math 221 1. (2 %) Define what it means to say that limx→a f(x) = L. 2. (2 %) Explain what is means to say that a limit has the form 00 . Give an example of a limit of this form which turns out to be 2. 3. (2 %) Give an example of a limit which does not exist. 4. (2 %) Give an example where limx→a+ f(x) = L1, limx→a− f(x) = L2, and L1 6= L2. 5. (2 %) Define what it means to say that the function is continuous. 6. (2 %) Give an example of a function which is not continuous at the point 0. 7. (2 %) Define what it means to say that the function is differentiable. 8. (2 %) Give an example of a function which is continuous at every point but not differentiable at 0. 9. (2 %) Is it always the case that a differentiable function is continuous? Why or why not? 10. (2 %) Is it possible to have a function which is defined for every real number but is not continuous at any point? Why or why not? 11. (2 %) Suppose that a differentiable function has both negative values and positive values. Must it have zeros? Why or why not? 12. (2 %) Is it true that every function defined on an interval takes on a minimum value and a maximum value in that interval? If not, what must be true of that function and that interval? 13. (8 %) Give a direct proof without using any rules of differentiation that d dxx 2 = 2x 14. (8 %) Show how to derive the quotient rule for differentiation: d dx ( u v ) = du dxv − u dv dx v2 15. (8 %) Explain how to derive the formula xn+1 = xn − f(xn)f ′(xn) which is used in Newton’s method to solve the equation f(x) = 0. 16. (5 %) limx→∞ 1−x 2 1+4x+3x2 = 17. (4 %) limx→∞ 12x2 = 18. (4 %) limx→0+ √ 3x x = 1 19. (4 %) ddxπ 3 = 20. (6 %) d 2 dx2 (2x 4 − x2 + x + 3) = 21. (4 %) ddx ( 3 √ x) = 22. (4 %) ddx (sin 8(x)) = 23. (4 %) ddx ( tan(x) x3+1 ) = 24. (4 %) ddx (sec(3x) cos(x)) = 25. (4 %) Suppose x = t2 and y = sin(t). dydx = 26. (5 %) Suppose y + x2y3 = 10. Find dydx when x = 1 and y = 2. 27. (4 %) If Newton’s method is used to solve the equation x3 − 1 = x2 with starting value x1 = 1, what will be x3? 2
