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Pitt MATH 0220 - Math 0220 Final

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Math 0220 Sample Final 21. Let ~a = h2, 1i and~b = h1, 3i.(3 pts.)1a. Evaluate |~a +~b|(3 pts.)1b. Find the unit vector in the direction of~b.(4 pts.)1c. Find all values of t such that ~a is perpendicular to ~c = h−4, 8ti.1(5 pts.)2a. Give a parametric vector equation for a circle of radius 9 with the center at thepoint (1, −2).(5 pts.)2b. The trajectory of an object is determined by~r(t) = h2t, −2t2+ 16ti where − ∞ < t < ∞.Eliminate the parameter t and find an equation in x and y that describes thecurve on which the object moves.23. Let f(x) = x(x − 1)2, −∞ < x < ∞.(10 pts.)3a. Find all points where f has a local maximum or local minimum. Justify youranswers.(10 pts.)3b. Find all inflection points. Justify your answer.(10 pts.)3c. Graph the function.3(10 pts.)4. Find x2, the second iterate in Newton’s method, to find an approximate valuefor the negative solution of x4= 10100. Assume that x1= −10. Show all details.4(10 pts.)5a. Find the equation for the line tangent to y = x1/4at x = 10000. Hint:(10000)1/4= 10.(5 pts.)5b. Use the tangent line found in part (a) to obtain an approximate value for(10100)1/4.5(10 pts.)6a. Evaluate: limx→0x√x + 4 − 2(10 pts.)6b. Evaluate: limx→3−|x −3|x − 3(10 pts.)6c. Evaluate: limx→1arctan(tan(2x − 3))2x −5(10 pts.)6d. Evaluate: limx→−∞ln1 +3x2sin4x2(10 pts.)6e. Evaluate: limx→0x2ln(x2)6(10 pts.)7a. Evaluate:Zdx4 + 25x2(10 pts.)7b. Evaluate:Z(12x+ x1/2)dx(10 pts.)7c. Let f(x) =Z2x0dt√1 + t2. Finddfdx.7(10 pts.)8a. Finddydxat the point (x, y) = (0, 1) on the curve defined by the equation y2+xey2= 1.(10 pts.)8b. Let y = arctan(3 sin2(x)). Find y0π4.(10 pts.)8c. Let y = x(2x). Finddydx.8(15 pts.)9. A particle moves along the curve 2x2− xy + 3y2= 24. If at a given time, theparticle is at position (−3, 1) and the x coordinate of its velocity at this point is5 then find the y coordinate of the

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