Answer these in detail with work shown

Instructions: Show all your work for full credit. Indicate your answers clearly. 2-3 Problem 1. (5 pts) Discuss the curve y = 1 2 with respect to intervals of increasing and decreasing, local maximal and minimal points, intervals of concavity, and points of inflection. Use this information to sketch the curve. Problem 2. (7 pts) Use mathematical optimization to prove that the shortest distance from a fixed point A to any point on a given line is the length of the line segment joining the point A to the line and is perpendicular to the line. Problem 3. (5 pts each) Evaluate the following two integrals. () -V36- 22 da (ii) cost do sin2 0 "sin a Problem 4. (6 pts) If f(x) =/ Vi + 12 dt and g(y) = / f(x) dx, find g"(7/ 6). Problem 5. (5 pts each) Find the indefinite integrals. (ii) / sin x sec (cos x) da Problem 6. (5 pts each) Calculate each of the following three definite integrals. 1 (1 + 7 cost x) 1/3 COS" x sin x da (in ) (25 - 620 + sind 1 + 1 2 dx 3n Problem 7. (5 pts) Evaluate the expression lim n- too Problem 8. (5 pts) If f(x) = batx7 +er, find (f-1)' (1). Problem 9. (5 pts) Use logarithmic differentiation to find the derivative of the function y = (5x + 1)20 (24 - 7)49 Problem 10. (5 pts each) Differentiate each of the following functions. (i) f(x) = sin (5x + 1) (ii) g(x) = xsing, for each r E (0, too) Problem 11. (5 pts each) Compute the following three integrals. (i) 22523 dx (iii) 2 In x d.x Problem 12. (7 pts) Evaluate the following limit. lim 1 - 7 / 4 . cos Jw/4 1+ o de Note that you are not allowed to use a technique that was not introduced or discussed in this course