MATH135 Midterm: 2011 Midterm Study Guide with every proposition and proofs for them

Exercises and Problems for Section 17.1 18. A ie segment hetween (2, 1,3) and (4.3,2 he plane 5 20. A line perpendicular to the plane --y+7 and through the point (1,1,6) 21. The circle of fabus 2 inthe 'y-plane, centered the on- gin, clockwise. 22. The ciecle of radis 2 parallel to the ry-plane, centered the point (0,0, 1), and traversed countenclockwise when viewed from below 23. The circle of radius 2 in the za-plane, cenered at the ori- gin. 24. The circle of radias 3 parallel to the ry-plane, centered at the point (0,0,2) 25. The circle of redius 3 in the ya-plane, centered a the point (0,0,2) 26. The circle of radius 5 parallel to the ya-plane, centered at the poinn (-1,0,-2) 28. The curve y - 27. The curve zy in the zy-plane zy-plane. in the 29, The curveエーー3? in the za-plane. 30.The curve in which the plane 1, 2 cuts the surface -2 z 31. The curve y-4-5' through the point (0,4,4), par. allel to the ry-plane. the line. 32. The elipse of major diameter 5 parallel to the y-axis la Exercises 7-17, find parametric equations for 7. The line in the direction of the vector - and through and minor diameter 2 parallel to the z-axis, centered at the point (0, 1,0). &. The line in the direction of the vector ? +2 33. The ellipse of major diameter 6 along the z-axis and mi- and or diameter 4 along the y-axis, centered at the origin. through the point (3, 0,-4). 9. The line 34·The ellipse of major diameter 3 parallel to the z-axis The line in the direction of the vector 53 + 2k and In Exercises 35-42, find a parametric equation for the curve dhrough the point (5,-1,1) segment. (There are many possible answers.) 36. Line from凡= (-1,-3) to h-(5,2). 38. Semicircle from (0,0,5) to (0,0,-5) in the yz-plane The line in the direction of the vector 37 -3 +E and 35. Line from (-1,2,-3) to (2,2,2). through the point (1,2,3). 12. The line in the direction of the vector ai +2j-3k and 37. Line from P(1,-3,2 to P (4.1.-3). 13. The line through (-3,-2,1) and (-1,-3, -1) through the point (-3,4, -2) with y 2 0. The line through the points (1,5,2) and (5,0,-1).39. Semicircle from (1,0,0) to (-1,0,0) in the zy-plane 5. The line through the points (2,3, -1) and (5,2,0) with y 2 0. 40. Graph of y VE from (1,1) to (16,4). 41. Are of a circle of radius 5 from P=(0,0) to Q IS The line through (3-2,2)and intersecting the y axis at v 2 (10,0). 庂The line intersecting the z-axis at z = 3 and the z-axis 42. Quarter-ellipse from (4,0 3) to (0 -3,3) in the plane

At what value of z will the curve be at its highest point? (a) (b) Graph this function with a graphing utlityto (a) Find the derivatives of the functions in Problems 1-34. 2, y=x2-3ex 4, f(x) = 4ex-In x 6. h(x) 750e004 . fx) -e 5·g(x) = 500(1-e-0.1x) 7, y=e 9, y=6e verity your answer Find the mode of the normal distribution. given by 38, 10. y = 1-2c" 2π (b) What is the mean of this normal distribution? (c) Use a graphing utility to verify your answer In x 15. y e 16, y = 2ev's /x+e In Problems 39-42, find any relative maxima and 19. ste 21, y=er_ (ex)' 23,y=ln (e4x + 2) 25, y=e-川n (2x) 27, y= 20. p4e 22, y = 4(ex)"-4er' 24. y In (e 1) 26. yeIn (4x) 28, y = minima. Use a graphing utility to check your results. 40,y= 41, y=x-e APPLICATIONS 43. Future value If $P is invested for n years at 10% 42, y=ex 29, y = (ex + 4)io 31 33, y=4" 35. (a) What is the slope of the line tangent to yxe "at compounded continuously, the future value that resul after n years is given by the function 32. y 3 (a) At what rate is the future value growing at any ih Write the equation of the line tangent to the graph ofy = xe-x at x = What is the slope of the line tangent to y=c"/(1 + e-x) atx = 0? (b) (for any nonnegative n)? (b) At what rate is the future value growing 1. after I yell 36. (a) (b) Write the equation of the line tangent to the graph (c) Is the rate of growth of the future value after 1 ofy = e-x/(1 + e-x) at x-0 greater than 10%? why? 37. The equation for the standard normal probability dis- 44. Future value The future value that accrues when is invested at 9%, compounded continuously. S(t) 700e tribution is where t is the number of years. The mode occurs at the highest point on normal curves thl and