Math 0220 Sample Final 1(10 pts.)1a. A particle moves with speed 2 around a circle of radius 4 centered at (x, y) =(1, 0). Assume that the particle is at (x, y) = (5, 0) at time t = 0. Find the vectorequation describing the motion of the particle if it moves clockwise around thecircle as t increases.(15 pts.)1b. The trajectory of an object is described by the vector function¯r = (4 + 7t3)¯i + (1 − 2t)¯j, −∞ < t < ∞Eliminate t and find an equation in x and y that describes the curve on whichthe object moves.(15 pts.)2. Use a tangent line to the function f(x) = (8x)1/3to find an approximate valuefor (8.08)1/3.(10 pts.)3. Using Newton’s method, find x2, the second iterate, to approximate the solutionof x5+ x3= 1. Assume that x1= 1.4. Given the function:f(x) =1x2, −∞ < x ≤ −1−x2, −1 < x ≤ 01 + x2, x > 0Determine:(5 pts. each)4a. limx→−1+f(x)4b. limx→0−f0(x)4c. Sketch the graph of the function.(6 pts. each)5. Find the first derivative of the following functions:5a. f(x) = tan−1(x3+ 2x)5b. s(x) = sin2(x) −3x1/3, x 6= 0.5c. y =xln(x).5d. h(x) = 3tan(x).5e. y = x3ln(x2)(6 pts. each)6. Determine the following limits:6a. limh→0+|− 2 + h| − |− 2|h6b. limx→0tan−1(2 + x) − tan−1(2)x.6c. limx→0+x2ln(x3)6d. limt→0(1 + 3t)1t6e. limx→0x2(1 − cos(2x))−1(15 pts.)7. Find the equation for the line tangent to the graph of the equation√y + x −√y − x = 2 at Q = (10, 26).(15 pts.)8. A spherically shaped balloon is being inflated by pumped air. The area of itssurface is S = 4πr2square inches, and its volume is V =43πr3cubic inches,where r is the radial distance from the center of the balloon to its surface. Asair is pumped into the balloon, assume that the area of the surface is increasingat a rate of 8 square inches per second. How fast is its radius increasing whenthe volume reaches32π3cubic inches.(15 pts.)9. A wire 16 feet long has to be formed into a rectangle. What dimensions shouldthe rectangle have to maximize its area?(10 pts. each)10a. Find the area under the curve: y = 2xbetween x = 0 and x = 5.10b. EvaluateZ(x + 1)1 + 2x2dx11. Consider the function f = x2e−xwhere −∞ < x < ∞.(7 pts.)11a. Find all values of x where f attains a relative maximum or a relative minimum.Justify your answer.(3 pts.)11b. Sketch the graph of the