Math 11A FINAL Version 112/11/2008 , Dr. Frank B¨auerle, UCSCNote: Show your work. In other words, just writing the answer,even if correct, may not be sufficient for full credit. Scientific calcula-tors are allowed, but no programmable and/or graphing calculators.Your Name:Your TA’s Name:Problem 1: out of 30Problem 2: out of 10Problem 3: out of 10Problem 4: out of 10Problem 5: out of 10Problem 6: out of 10Problem 7: out of 10Problem 8: out of 30Problem 9: out of 15Problem 10: out of 15Problem 11: out of 10Total: out of 160Good luck and have a good winter break!11. (30 points) Compute the derivativesdydxof the following functions (noneed to simplify):(a) y = e−2x2+3x−1(b) y = (3x2− 1)(e1−x)(c) y =3!1 + 2 tan2xThe problem continues on the next page !2(d) y =1 + sec x1 − cos3x(e) y =21x(f) y =(x +(x + sin2x)3)432. (10 points)(a) Give a precise definition for ”f (x) is differentiable at x = a”(b) Find the value for m such that g(x) is differentiable at x = 1.g(x)=x2if x ≥ 1m(x − 1) + 1 if x<13. (10 points) Assume that a particle travels on the curve given byx2y + xy4=2where x = x(t) and y = y(t) are functions of time. If at a certain time t1the particle is at the point P (1, 1) and the x-coordinate of the particlechanges at a rate of 2 ft/sec, at what rate does the y-coordinate of theparticle change at that instant?44. (10 points) Find the equation of the tangent line to the curve given byy =41+x2at the point (−1, 2).5. (10 points) Find the equation of the tangent line to the curve given byx ln y = y ln xat the point (1, 1).56. (10 points) Show that tan 2x ≈ 2x for x VERY close to zero.7. (10 points) Use a linear approximation to estimate10√1.0568. (20 points) For this problem let y = f(x) be the function given byf(x) = 20x3− 3x5. Use calculus to do the following.(a) (6 points) Find f%(x) and f%%(x).(b) (6 points) Find and classify the critical points of f(x).(c) (3 points) Find the intervals where y = f (x) is increasing ANDwhere y = f(x) is decreasing.The problem continues on the next page !7Problem continued from the previous page !(d) (6 points) Find all inflection points of f(x).(e) (3 points) Find the intervals where y = f(x) is concave up ANDwhere y = f(x) is concave down.(f) (6 points) Sketch the graph of y = f(x)89. (15 points) A rectangular flat field is bounded on one side by a river andthe other three sides by a fence. Find the dimensions of the field thatwill maximize the enclosed area if the fence has total length of 320 ft.Justify your answer.910. (15 points) Compute the following limits or explain why they don’t exist.Justify your steps. You can, but you do not need to use L’Hospital’s rule.a) limx−→03x5− x4+12x5+7x2− 1b) limx−→∞3x5− x4+12x5+7x2− 1c) limx−→0sin2xx(d) (Extra Credit, 5 points) Use L’Hospital’s Rule to find limx−→0ex− 1 − xx21011. (10 points) Suppose that you are studying reproduction in moss. A rea-sonably accurate scaling relation has been found (Niklas, 1994) betweenthe number of moss spores N and the capsule length L, namelyN = kL2.11where k>0 is a constant of proportionality depending on the particularmoss. Assume that you want to estimate the number of moss spores bymeasuring the capsule length. If you wish to estimate the the numberof moss spores within an error of 5%, how accurately must you measurecapsule length? (Justify your