ArringtonSampleMath 0220Examination #1SampleName (Print)____________________________ PeopleSoft# _____________. Signature____________________________________________Score .TA (Circle one) Instructions:1. Clearly print your name and Peoplesoft number and sign your name in the space above.2. There are 8 problems, each worth the specified number of points, for a total of 100 points. There is also an extra credit problem worth 7 points.3. Please work each problem in the space provided. Extra space is available on the back of each exam sheet. Clearly identify the problem for which the space is required when using the backs of sheets.4. Show all calculations and display answers clearly. Unjustified answerswill receive no credit.5. Write neatly and legibly. Cross out any work that you do not wish to be considered for grading. 6. No calculators, headphones, tables, books, notes, or computers may be used. All derivatives are to be found by learned methods of calculus.ArringtonSample 2 .1. (a) (5 pts.) Write the definition of ),x(fthe derivative of the function )x(f . (b) (7 pts.) Use this definition to find )x('f forx21)x(f 2. (21 pts.) Find the indicated derivatives of the following functions. You must use the correct notation, but you need not simplify: (a)y2tan)2cos(xxex. Find 2dxy2d. (b)9x2)x2sin()x(f. Find )x(fdxd. (c) )3csc()2(cot)(3xxxg . Find )x(g.ArringtonSample 33. (24 pts.) Do any 4 of the following 5: (a) Find a function f and a number a such that ).a(f5x322limx5x(b) Find 64x8x2x32 4x lim (c) The graph of y = f(x) is given. Sketch the graph of y = -2f(x-2). (d) x351x4lim2x (e) Find xxxsin2tan 0x lim You may earn 7 extra points by stating the Intermediate Value Theorem.ArringtonSample 44. (7 pts.) The displacement (in meters) of a body moving in a straight line is given by s(t) =t where t is measured in seconds. (a) Find the average acceleration over the time interval [1, 4]. (b) Find the instantaneous acceleration when t = 9.5. (12 pts.) .)sin( yxy Use implicit differentiation to find y and equations of the tangent and normal lines at2x , y = 1.6. (8 pts.) Use a linear approximation (or differentials) to approximate .145ArringtonSample 57. (8 pts.) The beacon on a lighthouse 50 m from a straight shoreline rotates twice per minute, or 4 radians per minute. How fast is the beam moving along the shoreline at the moment when light beam and the shoreline are at right angles ( = 0)? 8. (8 pts) A rock is thrown into a still pond. The circular ripples move from the point of impact of the rock so that the radius of the circle formed by a ripple increases at a rate of 2 ft. per minute. Find the rate at which the area is changing at the instant the radius is 4 ft. The area and the radius of a circle are related by A =
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