MATH 140A MIDTERM EXAMINATION II October 27, 2003Name ID # Section #There are 8 multiple choice questions, 10 True/False questions, and 4 free response questions.To receive full credit for free response questions (problems 10, 11, 12 and 13) allwork must be shown.THE USE OF CALCULATORS IS NOT PERMITTED IN THISEXAMINATION.There are 13 problems on 10 pages, including this one.Check your booklet now.The space below is for the instructor’s use.MCT/F10.11.12.13.TotalMATH 140A MIDTERM EXAMINATION II PAGE 21. (5 pts.) tan(−π/3) =a) −1b) −1/2c) −1/√2d) −√3/2e) −√32. (5 pts.) If a ball is thrown vertically upward with a velocity of 96 ft/s, then its height aftert seconds is s = 96t − 16t2in feet. What is the maximum height reached by the ball?a) 56 ftb) 64 ftc) 72 ftd) 144 fte) 288 ftMATH 140A MIDTERM EXAMINATION II PAGE 33. (5 pts.) If θ is an obtuse angle with sin θ = 3/5, thena) tan θ = 3/4b) tan θ = −4/5c) cos θ = −3/5d) cos θ = −4/5e) cot θ = 3/44. (5 pts.) limx→0x2− sin2(3x)2x2=a) −1b) −2c) −3d) −4e) Does not exist.MATH 140A MIDTERM EXAMINATION II PAGE 45. (5 pts.) Let x be such that sin x =2√67andπ2< x < π. Then sin 2x =a) 20√6/49b) −5/49c) 1/49d) −10√6/49e) −20√6/496. (5 pts.)ddx(sec2x − tan2x) =a) 2 sec2x tan xb) 2 sec x − 2 tan xc) (sec3x − tan3x)/3d) 1e) 0MATH 140A MIDTERM EXAMINATION II PAGE 57. (5 pts.) limh→0cos(π3+ h) −12h=a) −1/2b) 1/2c)√3/2d) −√3/2e) Does not exist.8. (5 pts.) In a circle of radius 2, the area of the sector with central angle 450isa) π/2b) πc) π/4d) π/3e) 2π/3MATH 140A MIDTERM EXAMINATION II PAGE 69. (10 pts. 2 pts. each) True or False: (Circle the appropriate letter.)a) T Fddx(3π2) = 6πb) T Fsin xx= 1 for any real number x.c) T F limx→0cos xx= 1d) T F 300= π/3e) T F cos π = −1f) T F If f0(a) exists, then limx→af(x) = f(a).g) T F If the graph of f has a vertical tangent at a then f is differentiable at a.h) T F If g(x) = x5, then limx→2g(x) − g(2)x − 2= 80.i) T F If f and g are differentiable, thenddx[f(x) + g(x)] = f0(x) + g0(x).j) T F If f and g are differentiable, thenddx[f(x)g(x)] = f0(x)g0(x).MATH 140A MIDTERM EXAMINATION II PAGE 710. (10 pts.)(a)(4 pts.)State the limit definition of the derivative f0(x) of a function f(x). (Your answermust be exact. Partial credit will not be given for this part.)(b)(6 pts.) Use the definition of derivative to compute f0(x) if f(x) = 1 − x2. (Warning: Nocredit will be awarded for just giving the answer. You must use the definition from part(a)).MATH 140A MIDTERM EXAMINATION II PAGE 811. (10 pts.) Let f (x) =2x2x + 1.(a) Compute f0(x).(b) Find an equation of the tangent line to the graph of f at the point (1, 1).MATH 140A MIDTERM EXAMINATION II PAGE 912. (10 pts.) Find all the solutions of2 cos2x + 7 cos x − 4 = 0on the interval [0, 2π).MATH 140A MIDTERM EXAMINATION II PAGE 1013. (10 pts.) The position x of a bee flying along a straight line is given by x = 2t3− 6t2+ 3,where t represents time.(a) Derive the expression for the velocity v of the bee as a function of t.(b) When does the bee change the direction of motion, if ever? Give the x−coordinate atthe place where it changes direction.(c) Find the distance travelled by the bee between t = 0 and t =