# Application of derivatives Questions and Answers

Calculus

Application of derivatives54 57 Applying reduction formulas Use the reduction formulas Exercises 50 53 to evaluate the following integrals 54 55 x cos 5x dx 56 x sin x dx 57 In x dx 58 Two methods Evaluate sin x In cos x dx in the following two ways a Use integration by parts 59 Two methods a Evaluate f b Evaluate X x 1 X b Use substitution dx using integration by parts dx using substitution c Verify that your answers to parts a and b are consistent 60 Two methods a Evaluate fx ln x dx using the substitution u x and evaluating In u du b Evaluate xln x dx using integration by parts c Verify that your answers to parts a and b are consistent

Calculus

Application of derivativesHelium is pumped into a spherical balloon at a constant rate of 2 cubic feet per second How fast is the radius increasing after 3 minutes At what time if any is the radius increasing at a rate of 140 feet per second The radius is increasing at a rate of 0 0082 ft sec Type an integer or a decimal Do not round until the final answer Then round to four decimal places as needed Select the correct choice below and if necessary fill in the answer box to complete your choice A The time at which the radius increases at a rate of 140 feet per second is sec Type an integer or a decimal Do not round until the final answer Then round to five decimal places as needed There is no solution

Calculus

Application of derivativesProblem 7 A chain lying on the ground is 14 ft long and weighs 90 lbs How much work is required to raise one end of the chain to a height of 8ft 4s

Calculus

Application of derivativesProblem 5 Find an integral which represents the volume of the solid obtained by rotating the region enclosed by the curves y x 2 and y 2x 8 about the x axis A E 7 x H x 2x 8 dx B 7 2x 8 x2 2 dx C Problem 5 Select 2x 8 dx F 2x 8 dx F x 2x x4 2x 8 dix 4 2 2 2x 8 x D 10 S x 2x 8 x G 2x 8 x dx 2x 8 x dx

Calculus

Application of derivativesProblem 3 Which of the following represents the area of the region enclosed by the curves y x and y 2x 8 for 3 x 4 A 3 x 2x 8 dx C 2 3 x2 2x 8 dx H F x 2x 8 dx E x2 2x 8 dx x 2x x 2x 8 dx dx 2 x 2x 8 B x 2x 8 2 2 2 1 x 2x 8 dx D 2 x 2x 8 dx 3 x 2x 8 dx x 2x 8 dx G G x 2x 8 dx 3 x x 2x 8 dx 2 2x 8 dx x 2x 8 dx

Calculus

Application of derivativesProblem 8 Suppose that the temperature of a cup of coffee t minutes after it is poured is given by T t 24 77e kt where k 4 Find the average temperature of the coffee during the first 37 minutes after it is poured Enter YOUT

Calculus

Application of derivativesProblem 1 Evaluate the following integrals 1 S 5x 177 2 dbx a b je Sesved 4

Calculus

Application of derivativesWrite an equation for the function graphed here 5 4 3 2 f x 3 2 1 2 3 3

Calculus

Application of derivativesThe top and bottom margins of a poster are 2 cm and the side margins are each 4 cm If the area of printed material on the poster is fixed at 390 square centimeters find the dimensions of the poster with the smallest area Width Height

Calculus

Application of derivativescos arcsin u CHE cos arcsin u Simplify your answer including any radicals Use integers or fractions for any numbers in the expression Rationa all denominators

Calculus

Application of derivativessin sin 1 sin sin 1 Type an exact answer using radicals as needed Rationalize all denominators

Calculus

Application of derivativesy sec 2 HECH y Type an exact answer using as needed Use integers or fractions for any numbers in the expression

Calculus

Application of derivativesxact value of the expression sin 2 sin 2 Type an exact answer using as needed

Calculus

Application of derivativesIf a function f has an inverse and f x 1 then f 1 If a function f has an inverse and f x 1 then f 1

Calculus

Application of derivativesComplete the statement y cos 1 x means that x y cos x means that x for 0 y t for 0 syst

Calculus

Application of derivativesWatch the video that describes finding inverse function values Click here to watch the video Find the exact value in radians of the expression cot 1 3 3 cot 1 Type an exact answer using as n

Calculus

Application of derivativesK Watch the video that describes finding inverse cosine values Click here to watch the video Find the exact value of the expression arccos arccos Type an exact answer using a

Calculus

Application of derivativesA boat sails directly toward a 150 m skyscraper that stands on the edge of a harbor The angular size of the building is the angle formed by lines from the top and bottom of the building to the observer see figure Complete parts a and b below Round to five decimal places as needed de b Graph as a function of x and determine the point at which the angular size changes most rapidly Choose the correct graph below dx OA Q OB 0 1000 by 0 0 008 REISSE 0 1000 by 0 0 008 For what value of x is the angular size changing most rapidly x Round to the nearest integer as needed M 0 1000 by 0 008 01 OD 150 m S 0 1000 by 0 008 01

Calculus

Application of derivativesFind two numbers A and B with AB whose difference is 44 and whose product is minimizec A B

Calculus

Application of derivativeswy oc cos 0 Yo Find the derivative of the inverse cosine using the theorem of the derivative of the inverse function The theorem states let f be differentiable and have an inverse on an interval 1 If x a point of I at which f x 0 then f isdifferentiable at yo Rx and Y x sin xo 1 sin xo Part 2 of 6 R 00 cos 0 How can the expression on the right side be rewritten in terms of f x OA Substitute using xocos Yo OB Substitute using Yo cos xo OC Rewrite the expression using cac Xosin xo OD Substitute using the identity sinx cos 1 sin xo HW Score 60 58 13 93 of 23 points O Points 0 of 1 CITES Save

Calculus

Application of derivativesA parcel delivery service will deliver a package only if the length plus the girth distance around does not exceed 120 inches Find the maximum volume of a rectangular box with square ends that satisfies the delivery company s requirements Maximum Volume

Calculus

Application of derivativesGiven the following information evaluate the other five functions RECU and 0 so cos sin 0 35 tan 8 45 4 3 csc 8

Calculus

Application of derivativesFind the standard form of the equation of the parabola with the given graph 6 5 4 2 6 3 2 1 T y 11 9 7 5 3 1 3 5 7 9 11 0 0 1 2

Calculus

Application of derivativesA woman pulls a sled which together with its load has a mass of m kg If her arm makes an angle of with her body assumed vertical and the coefficient of friction a positive constant is u the least force F she must exert to move the sled is given by mgu sin cos 0 If 0 15 find the maximum and minimum values of F for 0 0 T 2 Your answers will involve the unspecified constants m and 9 maximum force minimum force Newtons Newtons F

Calculus

Application of derivativesA rectangular storage container with an open top is to have a volume of 60 cubic meters The length of its base is twice the width Material for the base costs 60 dollars per square meter Material for the sides costs 6 dollars per square meter Find the cost of materials for the cheapest such container Minimum cost

Calculus

Application of derivativesA Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle What is the area of the largest possibe Norman window with a perimeter of 44 feet

Calculus

Application of derivatives2200cm of material is available to make a box with a square base and an open top find the largest possible volume of the box Volume cm

Calculus

Application of derivativesFind the length L and width W with W L of the rectangle with perimeter 44 that has maximum area and then find the maximum area L W Maximum area

Calculus

Application of derivativesHow would you divide a 16 inch line into two parts of length A and B so that A B 16 and the product AB is maximized Assume that A B A B

Calculus

Application of derivativesse the graph of f x to discuss the graph of y f x Organize your conclusions in a table and sketch a possible graph of y f x Select the correct choice below and if necessary fill in the answer box to complete your choice OA The function f has inflection points at x Use a comma to separate answers as needed OB The function f has no inflection points Choose the correct graph of y f x below OA M OB 1 GEXER G OC A Q OD MD

Calculus

Application of derivativesAntua 9 h D Identify intervals on which f x 0 Choose the correct answer below OA a c c e g h OC d e g h OE b c c e g h OG a b d e g h E Identify intervals on which f x is increasing Choose the correct answer below OA d e g h OC a b e g OE e g O B a b e g OD e g OF a c c d e g OH f x 0 over a h OB a c c e g h OD b c c e g h OF a b d e g h

Calculus

Application of derivativesUse the graph of y f x assuming f x 0 if x b ord and f x 0 if x 1 to answer parts A through F ANTA E Identify intervals on which f x is increasing Choose the correct answer below OA d e g h C a b e g CHOKES OE e g OG a c c d e g F Identify intervals on which f x is decreasing Choose the correct answer below OA d e g h OC b c c d e g OE a c c d e g OG b c c e g h OB a c c e g h OD b c c e g h OF a b d e g h OH f x is never increasing on a h OB a b e g OD a c c e g h OF e g OH f x is never decreasing on a h

Calculus

Application of derivativesUse the graph of y f x assuming f x 0 if x b ord and f x 0 if x f to answer parts A through F ANTA AM a E Identify intervals on which f x is increasing Choose the correct answer below OA d e g h OC a b e g OE e g OG a c c d e g F Identify intervals on which f x is decreasing Choose the correct answer below OA d e g h OC b c c d e g OE a c c d e g OB a c c e g h OD b c c e g h OF a b d e g h OH f x is never increasing on a h OB a b e g OD a c c e g h OF e g

Calculus

Application of derivativesUse the graph of y f x assuming f x 0 if x b or d and f x 0 if x f to answer parts A through F AVINA GULB D Identify intervals on which f x 0 Choose the correct answer below OA a c c e g h OC d e g h OE b c c e g h OG a b d e g h E Identify intervals sol which f x is increasing Choose the correct answer below OA d e g h OC a b e g OE e g OG a c c d e g OB a b e g OD e g OF a c c d e g OH f x 0 over a h OB a c c e g h OD b c c e g h OF a b d e g h OH f x is never increasing on a h

Calculus

Application of derivativesUse the graph of y f x assuming f x 0 if x b ord and f x 0 if x f to answer parts A through F AVD A D Identify intervals on which f x 0 Choose the correct answer below OA a c c e g h OC d e g h OE b c c e g h OG a b d e g h E Identify intervals on which f x is increasing Choose the correct answer below OA d e g h OC a b e g OE e g OG a c c d e g SLILA food in de Lalain OB a b e g OD e g OF a c c d e g OH f x 0 over a h OB a c c e g h OD b c c e g h OF a b d e g h OH f x is never increasing on a h

Calculus

Application of derivativesgraph of y f x Organize your conclusions in a table and sketch a possible graph of y f x OB The function f is never concave upward Select the correct choice below and if necessary fill in the answer box to complete your choice OA The function f has inflection points at x k Use a comma to separate answers as needed OB The function f has no inflection points Choose the correct graph of y f x below OA M OB SAL A x O C Ax O D

Calculus

Application of derivativesIn Silicon Valley a number of computer firms were found to be contaminating underground water supples with toxic che containers A water quality control agency ordered the companies to take immediate corrective action and contribute to a monetary pool for the testing and cleanup of the underground contamination Suppose that the required monetary pool in millions of dollars is given by the following function where x is the percentage expressed as a decimal fraction of the total contaminant removed Complete parts A through E P x 3x 1 x 0 x 1 GRZEN OC P x is concave downward on C Find any horizontal and vertical asymptote Find the vertical asymptote s if any Select the correct choice below and if necessary fill in the answer box to complete your choice OA The vertical asymptote s is are x Simplify your answer Use a comma to separate answers as needed OB There are no vertical asymptotes Find the horizontal asymptote s if any Select the correct choice below and if necessary fill in the answer box to complete your choice OA The horizontal asymptote s is are y Simplify your answer Use a comma to separate answers as needed OB There are no horizontal asymptotes

Calculus

Application of derivativesIn Silicon Valley a number of computer firms were found to be contaminating underground water supplies with toxic chemicals stored in leaking underground containers A water quality control agency ordered the companies to take immediate corrective action and contribute to a monetary pool for the testing and cleanup of the underground contamination Suppose that the required monetary pool in millions of dollars is given by the following function where x is the percentage expressed as a decimal fraction of the total contaminant removed Complete parts A through E P x 3x 1 x 0 x 1 A Where is P x increasing Decreasing Select the correct choice below and fill in the answer box es to complete your choice Type your answer in interval notation Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed OA P x is increasing on and decreasing on OB P x is increasing on OC P x is decreasing on B Where is the graph of P concave upward Downward Select the correct choice below and fill in the answer box es to complete your choice Type your answer in interval notation Use integers or fractions for any numbers in the expression Use comma to separate answers as needed OA P x is concave upward on and concave downward on OB P x is concave upward on OC P x is concave downward on C Find any horizontal and vertical asymptote Find the vertical asymptote s if any Select the correct choice below and if necessary fill in the answer box to complete your choice A The vertical asymptote s is are x

Calculus

Application of derivativesgraph of y f x Organize your conclusions in a table and sketch a possible graph of y f x 12 the juncUUI111ds d Uld III al X Use a comma to separate answers as needed OB The function f has no local minimum Select the correct choice below and if necessary fill in the answer box to complete your choice OA The graph of f is concave downward on the subinterval s Type your answer in interval notation Use a comma to separate answers as needed OB The function f is never concave downward Select the correct choice below and if necessary fill in the answer box to complete your choice OA The graph of f is concave upward on the subinterval s Type your answer using interval notation Use a comma to separate answers as needed OB The function f is never concave upward

Calculus

Application of derivativesUse the graph of y f x assuming f x 0 if x b ord and f x 0 if x f to answer parts A through F AVT 9 h 2 C Identify intervals on which f x 0 Choose the correct answer below OA a c c e g h OC d e g h OE b c c e g h G a b d e g h OA e g OC a c c d e g OE b c c e g h OG b c c d e g D Identify intervals on which f x 0 Choose the correct answer below OB d e g h OD a c c e g h OF a b e g OH f x 20 over a h OB a b e g D e g OF a c c d e g OH f x 0 over a h

Calculus

Application of derivativesUse the graph of f x to discuss the graph of y f x Organize your conclusions in a table and sketch a possible graph of y f x 300 Af x Use a comma to separate answers as needed Q OA The function f is decreasing on the subinterval s Type your answer in interval notation Use a comma to separate answers as needed OB The function f is never decreasing Select the correct choice below and if necessary fill in the answer box to complete your choice OA The function f is increasing on the subinterval s Type your answer using interval notation Use a comma to separate answers as needed OB The function f is never increasing Select the correct choice below and if necessary fill in the answer box to complete your choice OA The function f has a local maximum at x

Calculus

Application of derivativesSummarize the pertinent information obtained by applying the graphing strategy and sketch the graph of f x X 4 your choice OA There is an inflection point at x Type an integer or a decimal Use a comma to separate answers as needed OB There are no inflection points Which graph below shows f x OA OB Q O C Q O D Q

Calculus

Application of derivativesUse the given information to sketch the graph off Domain All real x lim f x 3 X 1 x 0 3 f x 6 S 3 0 0 1 x 0 ND 0 6 3 X 00 D 3 6 3 0 3 0 ND 6 CID Choose the correct graph below O A O C OB O D

Calculus

Application of derivativesPart 1 of 6 Find the derivative of the inverse cosine using the theorem of the derivative of the inverse function The theorem states let f be differentiable and have an inverse a point of I at which f x 0 then f is differentiable at y f x and 7 Ye x Let yo cos xo Using the given formula what is the resulting equation 1 OA cos yo OB cos x O C OB cos yo COS xo sin xo sin Xo sin Xo 1 sin xo HW Score 30 43 7 of 23 points O Points 0 of 1

Calculus

Application of derivativesWhat is the hybridization of the central atom in BrCl3 sp sp sp3 sp d

Calculus

Application of derivativesGiven the following information evaluate the other five functions 5 sec 8 3 and 0 0 cos sin A 35 Part 2 of 5 NI 2 ACCO