Math 0220 Sample Final 31a. (10 pts.) Given f (x) = x2+ 1, use the definition of the derivative to show thatf0(x) = 2x.1b. (5 pts.) Find the equation of the line tangent to the curve of y = x1/3atx = 1000.(Note: (1000)1/3= 10.)1c. (5 pts.) Use the tangent line in part (b) to obtain an approximation for (1005)1/3.1d. (5 pts.) Approximate√2 by applying Newton’s Method to approximate thepositive zero of the function f (x) = x2− 2 with x1= 1.5. Find x2. (You dohave to show your work.)12a. (9 pts.) Let y =ln xx3+ 1. Finddydx.2b. (9 pts.) Let y = (1 + 2x − x3)100. Finddydx.2c. (9 pts.) Let y = arctan(x2+ 3). Finddydx.2d. (9 pts.) Let y = xsin x. Finddydx.2e. (9 pts.) Given y3+ xy + e2x= 2, finddydxat (0, 1).23. Let f (x) = 2x3− 3x2− 12x + 5, −∞ < x ≤ 4. Then f0(x) = 6x2− 6x − 12 =6(x − 2)(x + 1).3a. (5 pts.) Find the intervals on which f(x) is increasing or decreasing.3b. (5 pts.) Find the local maxima and local minima of f(x).3c. (5 pts.) Find the intervals on which the graph of f (x) is concave upward orconcave downward.3d. (5 pts.) Find the points of inflection.3e. (5 pts.) Sketch the graph of f(x) on (−∞, 4].3f. (5 pts.) Find the global (absolute) maximum and the global (absolute) minimum.34. (10 pts.) The owner of a nursery center wants to fence in 1600 square feet of landin a rectangular plot to be used for different shrubs. The plot is to be dividedas follows:xWhat is the least number of feet of fence needed?5a. (5 pts.) Find limx→∞3x5+ x − 15x5+ 3x2+ 2.5b. (5 pts.) limx→0sin(8x)56x.5c. (5 pts.) limt→0(1 + 3t)1t.46. Let ~v = h−1, 4i and ~u = h2, −3i be two vectors in a plane.6a. (5 pts.) Find ||3~v − 2~u||.6b. (5 pts.) Find the unit vector in the direction of ~u.6c. (5 pts.) Find the vector of ~v projected on the vector ~u.6d. (5 pts.) Let w = ht2, 9i. Find t such that ~w is perpendicular to ~v.57. The position vector of a particle traveling on the x − y plane at time t is~r(t) = ht, 4t − t2i, 0 ≤ t ≤ 4where t is measured in seconds and the distance is measured in meters.7a. (5 pts.) Find the average velocity of the particle during the time from t = 0 tot = 2.7b. (5 pts.) Find the velocity and speed of the particle at t = 1.7c. (5 pts.) Find the acceleration of the particle at t = 1.7d. (5 pts.) Find the equation (by eliminating t) in x and y which describes the pathcurve of the particle and sketch the path curve from t = 0 to t = 4.68a. (5 pts.) Find the Riemann sum R4=4Pi=1f(ci)(xi− xi−1) forZ80x2dx withregular partition points. xi= 2i for i = 0, 1, 2, 3, 4 and the midpoint rule ci=12(xi−1+ xi) for i = 0, 1, 2, 3, 4.8b. (6 pts.) State the definition ofZbaf(x)dx. (You do have to explain yournotations.)78c. (6 pts.)Z20(3x2+√x) dx.8d. (6 pts.)Zsin2x cos x dx8e. (6 pts.)Zx4 + x2dx8f. (6 pts.)Z14 +