# MATH1013 Calculus I, 2013-14 Fall Week 12 Eleventh ... machiang/1013/2013-14_Fall/Worksheets/...MATH1013 Calculus I, 2013-14 Fall Week 12 Eleventh tutorial Worksheet: Mean Value theorem, L’H^opital’s rule, Newton’s method, Anti-derivatives (L10, T11A) Name: ID No.: Tutorial Section: Complete at least TWO questions from ...

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<ul><li><p>MATH1013 Calculus I, 2013-14 Fall</p><p>Week 12 Eleventh tutorial Worksheet: Mean Value theorem, LHopitals rule,Newtons method, Anti-derivatives (L10, T11A)</p><p>Name: ID No.: Tutorial Section:</p><p>Complete at least TWO questions from the following questions!</p><p>(Solution of this worksheet will be available at the course website the week after.)</p><p>1. (Demonstration) (p.288, Q. 27) Without evaluating derivatives, which of the functions g(x) = 2x10,h(x) = x10 + 2, and p(x) = x10 ln 2 have the same derivative as f(x) = x10?</p><p>2. (Demonstration) (page 289, Q. 31) A state patrol officer saw a car start from rest at a highwayon-ramp. She radioed ahead to a patrol officer 30 miles along the highway. When the car reached thelocation of the second officer 28 mins later, it was clocked going 60 mile/hour. The car driver was given aticket for exceeding the 60 mile/hour speed limit. Why can the officer conclude that the driver exceededthe speed limit?</p><p>3. (Demonstration) (p.300, Q. 18) Using LHopitals rule to evaluate limx14 arctanx </p><p>x 1.</p><p>4. (Demonstration) (p.309, Q. 11) Use Newtons method to approximate the root of f(x) = sinx+x 1with the initial approximation x0 = 1.5. Calculate up to seven iterations.</p><p>5. (Demonstration) (page 321, Q.92) Given the acceleration a(t) = 2 cos t of an object moving along aline, find the position function with the initial velocity and position: v(0) = 1, and s(0) = 0.</p><p>6. (Demonstration) (page 321, Q. 97) A softball is popped up vertically from the ground level with avelocity of 30m/s. We assume that the acceleration due to gravitation is 9.8m/s2.</p><p> Find the velocity of the object as a function of time t; Find the position of the object as a function of time t; Find when the object reaches the maximum height and at find this height; Find the time when the object strikes the ground.</p><p>7. (Class work) (page 300, Q. 46, 52) Evaluate limx1(1 x) tan(x</p><p>2</p><p>)and limx(x</p><p>x2 + 1)</p><p>Answer</p><p>8. (Class work) (p.288, Q. 27) Without evaluating derivatives, which of the functions f(x) = lnx, g(x) =ln 2x, and h(x) = lnx2 and p(x) = ln(10x2) have the same derivative as f(x) = x10?</p><p>Answer</p><p>9. (Class work) (page 289, Q. 32) A state patrol officer saw a car start from rest at a highway on-ramp.She radioed ahead to a patrol officer 30 miles along the highway. When the car reached the location ofthe second officer 30 mins later, it was clocked going 60 mile/hour. The car driver was given a ticket</p><p>1</p></li><li><p>for exceeding the 60 mile/hour speed limit. Can the officer conclude that the driver exceeded the speedlimit?</p><p>Answer</p><p>10. (Class work) (p.300, Q. 17, 20) Using LHopitals rule to evaluate limx1lnx 1x e</p><p>and limx1x sinx+ x2 42</p><p>x 2.</p><p>Answer</p><p>11. (Class work)(page 300, Q. 47, 54) Evaluate limx0(csc 6x sin 7x) and limx(xx2 + 4x)</p><p>Answer</p><p>12. (Class work) (p.309, Q. 13) Use Newtons method to approximate the root of f(x) = tanx 2x withthe initial approximation x0 = 1.5. Calculate up to seven iterations.</p><p>Answer</p><p>13. (Class work) (page 320, Q.49, 51) Find primitives of (i)</p><p>6</p><p>25 x2dx, (ii)</p><p>dx</p><p>xx2 100</p><p>.</p><p>Answer</p><p>14. (Class work) (page 320, Q.49, 51) Find primitives of (i)</p><p>6</p><p>25 x2dx, (ii)</p><p>dx</p><p>xx2 100</p><p>.</p><p>2</p></li><li><p>Answer</p><p>15. (Class work) (page 321, Q.93) Suppose a(t) = 3 sin 2t is the acceleration of an object moving along aline, find the position function with the given initial velocity and position: v(0) = 1, and s(0) = 10.</p><p>Answer</p><p>16. (Class work) (page 321, Q. 100) A payload is released at an elevation of 400m from a hot-air balloonthat is rising at a rate of 10 m/s. We assume that the acceleration due to gravitation is 9.8m/s2.</p><p> Find the velocity of the object as a function of time t; Find the position of the object as a function of time t; Find when the object reaches the maximum height and at find this height; Find the time when the object strikes the ground.</p><p>Answer</p><p>3</p></li></ul>