# Application of derivatives Questions and Answers Calculus
Application of derivatives
54 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 derivatives
Helium 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 derivatives
Problem 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 derivatives
Problem 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 derivatives
Problem 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 derivatives
Problem 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 derivatives
Problem 1 Evaluate the following integrals 1 S 5x 177 2 dbx a b je Sesved 4 Calculus
Application of derivatives
Write an equation for the function graphed here 5 4 3 2 f x 3 2 1 2 3 3 Calculus
Application of derivatives
The 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 derivatives
cos 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 derivatives
sin sin 1 sin sin 1 Type an exact answer using radicals as needed Rationalize all denominators Calculus
Application of derivatives
y sec 2 HECH y Type an exact answer using as needed Use integers or fractions for any numbers in the expression Calculus
Application of derivatives
xact value of the expression sin 2 sin 2 Type an exact answer using as needed Calculus
Application of derivatives
expression cos m2 2 Type an exact answer using x as Calculus
Application of derivatives
If 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 derivatives
Complete 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 derivatives
Watch 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 derivatives
K 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 derivatives
A 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 derivatives
Find two numbers A and B with AB whose difference is 44 and whose product is minimizec A B Calculus
Application of derivatives
Find the eccentricity of the ellipse 2 1 x 25 36 Calculus
Application of derivatives
wy 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 derivatives
A 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 derivatives
Given 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 derivatives
Find 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 derivatives
Match each equation with its graph 5 A A 7 N 2 y Yx 2 B 2 y 7 ay Calculus
Application of derivatives
A 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 derivatives
A 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 derivatives
A 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 derivatives
2200cm 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 derivatives
Find 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 derivatives
How 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 derivatives
se 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 derivatives
Antua 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 derivatives
Use 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 derivatives
Use 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 derivatives
Use 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 derivatives
Use 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 derivatives
graph 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 derivatives
In 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 derivatives Calculus
Application of derivatives Calculus
Application of derivatives
Use 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 derivatives Calculus
Application of derivatives
Summarize 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 derivatives
Use 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 derivatives
Part 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 derivatives
What is the hybridization of the central atom in BrCl3 sp sp sp3 sp d Calculus
Application of derivatives
aluate the following integral 2 812 25 2 812 25 dz dz tan 9z 5 Calculus
Application of derivatives
Given 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