MATH 222 Midterm: Midterm Solutions.PDF

College Algebra 32203 > Assignments> Midterm Review Ch 1-3 Midterm Review Ch 1-3 MATH 1828 College Algebra Summer 2018 Session 2 Jessie Andrasek Homework: Midterm Review Ch 1 3 Score: 0 of 1 pt HY Score: 37.68%, 15.07 of 40 pts 2.2.47-GO EQuestion Help Total proft i defined as total revenue Rr. mnus total cost. ?(x), and s gnen by the furet on P(-R(x)-C(x) G en RC-58-04x2 and C(x-5x+13, fr d each ofthe a) P(x) b) R(90), C(90), and P(90) r box and then All remaining

(a) At what value of z will the curve be at its highes (b) Graph this function with a graphing utility to Find the derivatives of the functions in Problems 1-34. 3、 f(x) = ex-x" 5. gx) 500(1-01x) point verify your answer. given by 6. h(x) - 750e04 38. (a) Find the mode of the normal distribution 10, y=1-2e 12, y=e 14, y=e, + elnx 16, y = 2 18, y= 2x + 2 20, p=4qe 22, y = 4(ex)3-4 24, y = ln (c" + 1) 26, y-er' In (4 ) 28. y =- 2x 9, y= 6e 11, y=2e(x1+1), (b) What is the mean of this normal distribution? (c) Use a graphing utility to verify your answer. 15, y=e-1/x 17, y = e-1/x' + e-r 19, s = 12e, 21, y=er_ (ex)' 23. y In (e + 2) 25, y= e-kin (2x) In Problems 39-42, find any relative maxima and minima. Use a graphing utility to check your results. 40, y =- 41 y = x-ex APPLICATIONS 43. Future value If SP is invested fornyearsat 10% 42, y=ex 1 + e ix 29, y = (r" + 4)10 30, y = 32.y-3 34, y = 52-1 compounded continuously, the future value that after n years is given by the function 33, y=4" 35. (a) What is the slope of the line tangent to y xe "at (b) Write the equation of the line tangent to the graph 0.1n S -PeIn (a) At what rate is the future value growing at any ofy = xe-x at x = 1. What is the slope of the line tangent to y = e (1 + e") at x = o? (for any nonnegative n)? (b) At what rate is the future value growing after 36. (a) (e) Is the rate of growth of the future value after (b) Write the equation of the line tangent to the graph greater than 10%? why? is invested at 9%, compounded continuously, is where t is the number of years. of y-r-x/(1 x) at x = 0. 44. Future value The future value that accrues when + e 37. The equation for the standard normal probability dis- tribution is S(t) 700e 09 The mode occurs at the highest point on norm curves and equals thr

Justify your work, communicate clearly and mathematically. calculation not allowed. Unless decimal approximations are requested, numerical answers must Label your problems clearly and mark the work you want graded/not graded; also, box in your answers. Using techniques from this class, approximate the value of exy - cosx at (.1,2.3) Consider the vector field F(x, y) = (cos(sin x + y) cosx ex, cos(sin x + y) + y) the work done as you traverse the Archimedes spiral (r = Theta) from (x,y) = (0,0) to (x,y)= (2 pi. 0). (Hint: check to see if the vector field is conservative.) Find the absolute maximum and minimum of y ex-z on the ellipsoid9x2+4y2+36z2=36 Compute Consider the region bounded by 2 = 4 - x2 - y2 and 2 = 0. Let S be the boundary of this region. Compute the flux of F = (xy2z - xz, yz + ev, - zev) across S. If F represents a fluid flow, explain what the above computation means. First, sketch and describe the surface 5: r(u. v) - (ucosv, usinv.v,v), where u Second, let f(x. y. z) be the distance function from the point (x. y, z) to the z - axi% Integrate the function f over S. Pick 2 of the following and solve. Compute fc(:2 + yeI)dx+ (ez + yz)dy + (2xz + y2/2 )dz along the curve C: r(t)= 5 cost, sint, cost, sint). > Consider the curve coordinates this is just the wobbly circle r = 3 + cos St). Compute where F - (-y,x)/(x2 + y2). State and prove Clairaut's theorem. State and prove the chain rule. State and prow Green s theorem for simply-connected regions. Bonus if you can prove Green's theorem for non-simply-connected regions