Application of derivatives Questions and Answers

Match each equation with its graph 5 A A 7 N 2 y Yx 2 B 2 y 7 ay

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

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

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

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

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

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

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

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

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

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

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

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

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

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

In 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

graph 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

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

Use 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

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

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

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

What is the hybridization of the central atom in BrCl3 sp sp sp3 sp d

aluate the following integral 2 812 25 2 812 25 dz dz tan 9z 5

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

Question The rate of increase of rain in Austin can be modeled with the formula R the amount of total rainfall in inches where t is time in days t using the formula R 3 log 8 O 29 O 100 nearest whole number what is the approximate time of which there is a total of 30 inches of rain O28 10 101 To the

The pH scale for acidity is defined by pH log 0 H where H is the concentration of hydrogen ions measured in moles per liter M A substance has a hydrogen ion concentration of H 3 8 x 10 M Calculate the pH of the substance

until 11 A certain radioactive isotope has a half life of 2 27 minutes If we initially have 5632 grams of this substance how many half lives will pass remain Your answer should be a whole number help numbers Part 2 What length of time is needed for this decay process to occur Round your answer to the nearest tenth holn numbers minutes grams

Use the graph of y f x assuming f x 0 if x b or f to answer parts A through F AY a b ww 561 e 9 h Q F Identify intervals on which f x is decreasing Choose the correct answer below O A a c c e g h OC a c c d e g OE b c c e g h OB a b e g O D e g OF b c c d e g ffy is never docrossing on to h

Show that f x logax and g x log x where a 1 and b 1 grow at a comparable rate as x 00 log x What rule of logarithms can assist with simplifying the limit function X 8 STER Change of base formula Power rule Product rule O Quotient rule Use the change of base formula to rewrite each given logorithm in terms of the natural logarithm and im lim

K Use limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates 8 MITTED Set up the functions as an expression of a limit to determine which grows faster lim X 00 tom

Express as a sum or difference of logarithms without exponents 9 log c 2 X 92 y z 927 x 2 The answer is log Simplify your answer Use integers or fractions for any numbers in the expression COD

K Evaluate the following limit Use H pital s Rule when it is convenient and applicable e 1 lim x 0 3x 9x Use l H pital s Rule to rewrite the given limit so that it is not an indeterminate form lim x 0 3x 9x lim x 0

Evaluate the following limit Use T H pital s Rule when it is convenient and applicable log 8X log 6x lim X 00 lim X 00 log 8x log 6x Type an exact answer COO a

In 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 3x P x 1 x 0 x 1 COLL 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 concave upward on and concave downward on OB P x is concave upward on C P x is concave downward on

Use the graph of y f x assuming f x 0 if x b or f to answer parts A through F Q C d ww 9 h D Identify intervals on which f x 0 Choose the correct answer below O A a b e g OC d e g h OE a b d e g h OG ac c h OB b c c e g h O D a c c d e g OF e g OH fx 0 over a h

O Points 0 of 1 se limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates 27 In x Inx Set up the functions as an expression of a limit to determine which grows faster lim X10 W

Use the graph of y f x assuming f x 0 if x b or f to answer parts A through F a b C d www Ay O A a c c d e g OC b c c d e g OE d e g h OG b c c e g h 9 h B Identity intervals on which the graph of t is concave downward Choose the correct answer below O B e g OD a b e g OF a c c e g h OH The graph is never concave downward

Points 0 of 1 limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates Set up the functions as an expression of a limit to determine which grows faster lim X 00

Prove that lim x 00 OE Write the limit in the form L H pital s Rule can be used to evaluate the limit but it must be put in the form A L lim xin 818 OB L lim x In x 00 e for a 0 Oc L lim xin X 00 OD L lim x In x x lim X40 lim In 1 a x 1 x X 8 In 1 lim In 1 a x 840 lim xx X In 1 a x 1 In 1 ax olo or 88 How should the limit be rewritten Apply l H pital s Rule by finding the derivative of the numerator and the denominator separately In 1 a x lim

Let R be the region bounded by the following curves Let S be the solid generated when R is revolved about the given axis If possible find the volume of S by both the disk washer and shell methods Check that your results agree y x 2 1 x 0 y 26 revolved about the y axis A 26 S H G 1 2 2 dy Type exact answers OB S dy S dy Type exact answers Set up the integral s that gives the volume of the solid as a single integral if possible using the shell method Use increasing limits of integration Select the correct choice below and fill in any answer boxes within your choice OA SO x dx s dx Type exact answers dx Type exact answers ok answer

Use the graph of y f x assuming f x 0 if x b or f to answer parts A through F www 9 h Q Q A Identify intervals on which the graph of f is concave upward Choose the correct answer below OA The graph is never concave upward OC a c c d e g OE e g OG a c c e a h OB d e g h OD a b e g OF a b d e g h OH b c c e a h

Find the intervals on which the graph of f is concave upward the intervals on which the graph of f is concave downward and the inflection points f x In x 4x 20 A w 4 10 07 Type your answer in interval notation Type an exact answer Use a comma to separate answers as neede OB The graph is never concave downward Determine the x coordinates of any inflection points of the graph of f x Select the correct choice below and if necessary fill in the answer box to complete your choice Type an exact answer Use a comma to separate answers as needed OA X OR There are no inflection points

Identify the relationship between Za and Zb Show your work here a O Corresponding angles O Alternate exterior angles Same side exterior angles b O Alternate interior angles O Same side interior angles

An 8 ft tall fence runs parallel to the wall of a house at a distance of 6 ft Find the length of the shortest ladder that extends from the ground to the house without touching the fence Assume the vertical wall of the house is 20 ft high and the horizontal ground extends 25 ft from the fence The length of the shortest ladder is 36 89 ft Round the final answer to the nearest tenth as needed Round all intermediate values to the nearest thousandth as needed Correct 4

Given the following information evaluate the other five functions 5 sec 8 and 0 8 3 cos 8

OA K Use a right triangle to write the following expression as an algebraic expression Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x Hint Use sin 20 2 sin 0 cos 8 sin 2 cos1 17x Identify the right triangle that can be used to simplify the given expression Choose the correct answer below 17x a Part 2 of 2 1 17x a O C HW Score 45 65 10 5 of 23 points O Points 0 of 1 sin 2 cos 17x 3ux 1 289x Simplify your answer including any radicals Use integers or fractions for any numbers in the expression O D 4 4 17x 17x 1 a Save

Find the value of a Show your work here a 97 4a 7

Angles Q and P are a linear pair Angle Q is 49 What is the value of angle P Show your work here

If Zx and 4k are supplementary and 4k 142 solve for x Write your answer in degrees k