ࡱ> fheVa +bjbjxx =N\N\I#hh8 |oD P ` ` ` y0F23d@oBoBoBoBoBoBo$qytfo3/y033fo` ` a{o@@@3` ` @o@3@o@@fD@ : Key- Math 231 Final Exam Review ______________________________ 1. Find the equation of the line tangent to the curve  EMBED Equation.3  at the point (2, 1). y-1=(-4/5)(x-2) 2. Find the slope of the normal line to  EMBED Equation.3  at the point (0,2). 1 3. Find  EMBED Equation.3  if  EMBED Equation.3 . (ysinxy+8xy)/(-xsinxy-3y^2-4x^2) 4. Find the exact area bounded by  EMBED Equation.3 , the x-axis, x = -2, and x = 2. 8 5. Find the exact area bounded by  EMBED Equation.3  and the x-axis. 5/2 + pi/4 6. Find  EMBED Equation.3 . pi 7. Find  EMBED Equation.3  0 8. Find  EMBED Equation.3  if  EMBED Equation.3 .  EMBED Equation.3  9. Find  EMBED Equation.3  if  EMBED Equation.3 .  EMBED Equation.3  10. Given  EMBED Equation.3 , find  EMBED Equation.3 .  EMBED Equation.3  11. A particle moves along a line with velocity function  EMBED Equation.3 , where v is measured in meters per second. Find (a) the displacement and (b) the distance traveled by the particle during the time interval [0,5]. 175/6 12. Find  EMBED EQUATION  if  EMBED EQUATION . -6^{sqrt{csc x}} ln6 (1/2)(csc x)^{-1/2}cscxcot x 13. For a function g(x), given that  EMBED EQUATION  and  EMBED EQUATION , find all intervals where g(x) is simultaneously decreasing and concave down. (-infty, 0), (3/4,9/8) 14. Consider the function g(x) whose graph is shown below. Find each of the following. If the limit does not exist, describe the outcome as EMBED Equation.3, EMBED Equation.3, or LDNE.  EMBED Word.Picture.6  a)  EMBED EQUATION  ___1__ b)  EMBED EQUATION  __pos_inf__ c)  EMBED EQUATION  ____1_ d)  EMBED EQUATION  ___0__ 15. Compute the EXACT value of each limit, if it exists. If the limit does not exist, describe the outcome as EMBED Equation.3, EMBED Equation.3, or LDNE. a)  EMBED Equation.3  -inf b)  EMBED Equation.3  (-sqrt7)/6 c)  EMBED Equation.3  if  EMBED Equation.3  71 d)  EMBED Equation.3  -11 e)  EMBED Equation.3  1/128 f)  EMBED Equation.3  inf g)  EMBED Equation.3  0 h)  EMBED Equation.3  (sqrt7)/5 16. a) Find the x-coordinate(s) of all point(s), if any, on the graph of the curve  EMBED Equation.3  at which the tangent line is parallel to the line  EMBED Equation.3 . none b) Find the equation of the normal line to the curve  EMBED Equation.3  at x = 1. y-5/3=9(x-1) 17. Use the limit definition of the derivative to find EMBED Equation.3 for the function  EMBED Equation.3 . 18. Use the limit definition of the derivative to find EMBED Equation.3 for the function  EMBED Equation.3 . 19. a) Express the definite integral  EMBED Equation.3  as the limit of a Riemann sum. b) Evaluate the Riemann sum for  EMBED Equation.3  , [1,2] with 4 subintervals taking the sample points to be the left endpoints. a)  EMBED Equation.3  b)  EMBED Equation.3  20. Find the EXACT value of c so that the function f given below is continuous at x = 9.  EMBED Equation.3   EMBED Equation.3  21. State the three points to be satisfied for a function f (x) to be continuous at x = a. 1.__f(a) exists___________________ 2.__limit as x approaches a exists__________________ 3.___above two are equal__________________ Use the above definition to determine whether  EMBED Equation.3  is continuous or discontinuous at a) x = 0 cont. b) x = 1 hole c) x = 2 inf. d) If the function is discontinuous at x = 0,1 and/or 2, describe the type of discontinuity. 22. Sketch the graph of a function  EMBED Equation.3  that satisfies the following conditions.  EMBED Equation.2 f(x) has exactly one point of discontinuity  EMBED Equation.2   EMBED Equation.2  on  EMBED Equation.2 ,  EMBED Equation.2  on  EMBED Equation.2   EMBED Equation.2  on  EMBED Equation.2 ,  EMBED Equation.2  on  EMBED Equation.2   EMBED Equation.2 ,  EMBED Equation.2   EMBED Word.Picture.6  23. Given the function  EMBED Equation.3 , a) Find the linear approximation, L(x), to the function at a = 32. b) Use your result from part (a) to approximate EMBED Equation.2 . Answer to 4 decimal places or exact value. a)  EMBED Equation.3  b)  EMBED Equation.3  24. Given the function  EMBED Equation.3 , a) Find the linear approximation, L(x), to the function at a = 1. b) Use your result from part (a) to approximate EMBED Equation.2 . a)  EMBED Equation.3  b)  EMBED Equation.3  25. A woman standing on a cliff is watching a motorboat through a telescope as the boat approaches the shoreline directly below her. If the telescope is 300 feet above the water level and if the boat is approaching at 5 feet per second, at what rate is the angle, A, of the telescope changing when the boat is 150 feet from the shore? (radian mode)   EMBED Equation.3  26. A long rectangular sheet of metal, 9 inches wide, is to be turned up at both sides to make a horizontal gutter with vertical sides. How many inches should be turned up at each side for maximum carrying capacity?  SHAPE \* MERGEFORMAT  2.25 in 27. Using calculus, determine the absolute minimum and maximum for the function  EMBED Equation.2  on the interval  EMBED Equation.2 . absol min of -5, absol max of -33/8 28. Using calculus, determine all local minimum and maximum for the function  EMBED Equation.2  . (change problem-find x values only) max at -1/2 (sqrt5)/2, min at -1/2 + (sqrt5)/2 29. Suppose a function g(x) has the following derivative:  EMBED EQUATION . On what interval(s) is  EMBED Equation.2  increasing? (-inf,-7/3),(5,inf) 30. Use Newtons method to find all roots of the given equation to 6 decimal places.  EMBED Equation.3  0.641714 31. An object was dropped off a building and hit the ground with a speed of 73 feet per second. What is the height of the building? (use  EMBED Equation.3 ) 5329/64, which is approximately equal to 83.265625 32. Let  EMBED Equation.3  be a position function for an object that moves along a horizontal line. Chart the motion of the object. When is the object speeding up? When is the object moving right? the object is speeding up when t > 2. The object is also moving right when t > 2. 33. Evaluate the Riemann sum for  EMBED Equation.3  , [0,3] with 4 subintervals taking the sample points to be the right endpoints. 6.45375357 a) What integral does the Riemann sum approximate? b) Express the definite integral in part a) as the limit of a Riemann sum. 34. Evaluate each integral. a)  EMBED Equation.3  ln2-11/4 b)  EMBED Equation.3  3pi/2 c)  EMBED Equation.3  -e^cosx+1/9*sin9x+c d)  EMBED Equation.3  -1/4*cot4x + ((arcsinx)^2)/2 +c e)  EMBED Equation.3  ((lnx)^5)/5 +c f)  EMBED Equation.3  (1/6)(6^9-5^9) 35. Find the interval on which  EMBED Equation.3  is concave up. The function is concave up when x < -1/2. 36. Let  EMBED Equation.3  where f is the function whose graph is shown. At what values of x do the local maximum and minimum values of g occur?  SHAPE \* MERGEFORMAT  Local max at x=2,6 Local min at x=4 37. Find all values of c in the given interval that satisfy the Mean Value Theorem for :  EMBED Equation.3  , [0,2] +- 2/sqrt3 38. Let  EMBED Equation.3 . How large can  EMBED Equation.3  possibly be? 26 39. Graph  EMBED Equation.3 . Label all asymptotes, local max/mins, inflection points, if they exist. 40. The diameter of a sphere is measured to be 6 inches with a possible error of 0.05 inches. Use differentials to estimate the maximum error in the calculated volume. 36pi(0.025) This review may not cover every topic on the final exam. It is essential to reread the text, go over your instructors class notes, review all assigned HW then find similar problems in the chapter reviews for more drill. Begin your final exam review at least 2 weeks prior to the final.     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