Please answer all questions on the images. Ignore the writing on it and start from scratch. Please follow all directions, show all work and provide explanations, and circle/box the answers. Please write neatly. If you have already answered this question please do not resubmit your work, it was very difficult to read and there were some errors. Thank you!

Show all your work, unsupported answers will receive not credit. Use correct mathematical notation at all times, points will be deducted for incorrect notation. Include units in your answers to word problems. Box your answers. 1. (10 pts. Each) Find each derivative. Use Product, Quotient and/or Chain rules when needed. Do not simplify your answer. a) (5x3 )e)+2x "Neck to b) f, ( x ) if f ( x, y ) = * ty x + y ( 10 x ) ey 2 +2x TIX + ( 5 x 2 ) (er +2x ) ( 2 ) Fy LX,MY = Xtyz = 10 xey" + 2 x + 10/2 prix xty = ( 2y) (xty ) = 1(x+yz ) ( X + y ) 2 ( x 4 = 1+2 - 1 - 12 - 2-42 2. (10 pts. each). Evaluate the following limits, use co or - co when appropriate. Use L'Hospital's rule when needed (be . X sure to justify why you can use LHR) a) lim - x -De x-+l' In x why 1 form: 0? ? b) lim e"* +1 X-+00 Vx why - 1 form : 00 72 In x c) lim -2 _1 -lm V/ = / 2 form: 0/0 Xol 2 X 23. (12 pts.) Suppose that a company's sales (in thousands of dollars) are related to its television adverting by S(x, /) = 20,000 + 10xt + 20x , where x is the number of commercials per day and i is the length of the commercials in seconds. a) Find S, (x, 1) and S, (x,1) Sx(x, + ) = lut tyox Start ) = lux b) Evaluate S, (x,1) and S, (x, 1) if the company is currently running ten commercials that are 30-seconds long and interpret your answer (be specific about all quantities involved and use correct units). Use that information to determine if the company should increase the number of commercials or the length of each commercial to have a higher increase in sales. Fx ( 10, 30) = 10(30) + 40(10) = 300+400-$ 700 thousand at a level of I commercials At 30 seconds long, each additional Commercial will increase sales by $ 700,000. SE (W, 30 ) = loCio) =$100 thousand at a level of to commenials at 30 seconds long each, anaddition second of the commercial would raise sales boy 4 100, Ow. . The company should increase the of commercials to have ahigher increase in 4. (8pts) Use the information in the charts to answer the following questions and to sketch the graph of the sales function f(x) a) List all the critical points (both coordinates) and classify them as max, min, or neither 72 b) List all the inflection points (, 3 ) 20 4 - 3 x -4 -2 f(x) 3 0 2 + + ND - - - O -4 -2 + + + + + -3 This study source was downloaded by 100000790750407 from CourseHero.com on 10-36-2023 2177:30 GM/10360 +725. (15 pts.) The function f(x) = xe" has first and second derivatives f '(x) = (1-x)e- and f(x)" = (x-2)e"* (you don't have to check this result). a) Determine the domain of the function and find all the horizontal asymptotes, vertical asymptotes, and holes (if any) (You must show all the limits to get full credit for this part) b) Find all the critical points, determine the intervals where the function is increasing and where it is decreasing, and classify the critical points as local maxima, minima, or neither c) Find the inflection points and determine the intervals where fis concave up or concave down d) Find the x- and y-intercepts e) Graph the function. Your graph must clearly show ALL the information found above YOU MUST SHOW ALL YOUR WORK TO GET CREDIT, Calculator graphs will receive no credit. WRITE YOUR ANSWER TO EACH PART IN THE BOX PROVIDED BELOW D: I VA : None HA: lim Xex = OV y = Q y-int : 4. ( ' Ly) - ( 1 x ) e- x = 0 .37 X-inti ( ( x ) + Vy= : 367 7C.P X = 0 1 inc: ( - 0, ! ) Dec. : (1, 20 ) f" ( x ) = ( x- 2 ) e * = 0 D: RV ANSWERS vainone X= Z y = . 27 > inflection HA: 1 = 0 point (" ( x ) - +7/ CIP= (1:37) - local max cont. 2 inc : (-09 , 1 ) dec: ( los) ( - , 2 ) : conc down / ( 21 127 ) zinflection puma ( 2, cs ) : conv, up ( 2, co) 7 cone . up y- int: 0 X- infio