Application of derivatives Questions and Answers

a Find the inverse of the function below on the given interval and write it in the form y f x b Verify the relationships f f x x and f f x x f x x 36 for x 0 a f x

Find the following derivative x 1 Inx d dx dx 6 x 1 Inx

Find the inverse of the function f x x 5 The inverse function of f x x 5 is f x 165465 2010 2018 ALDE LE BUR CONSER DEPRESS Il es CERVE Gestal PAR 1000 S MEREN SELINS PRYLAR

Find all local and global max min and any inflection points for the function g x 2 cos x sin 2x for t Interval 0 27

A chemist has a 100 gram sample of a radioactive material He records the amount of radioactive material every week for 6 weeks and obtains the following data Complete parts a f Week x 0 1 2 3 4 5 6 Weight y in grams 100 0 86 0 72 0 64 1 53 1 45 4 38 8 a Treating week x as the independent variat use a graphing utility to fit an exponential funct to the data y ab Round to four decimal places as needed

The size P of a certain insect population at time t in days obeys the function P t 700e0 04t a Determine the number of insects at t 0 days b What is the growth rate of the insect population c Graph the function using a graphing utility d What is the population after 10 days e When will the insect population reach 840 f When will the insect population double b The growth rate of the insect population is 4 c Graph P t 700e0 04t Choose the correct graph below The graphing window is 0 10 1 by 0 1400 100 OA B d The population after 10 days is approximately 1044 insect s Do not round until the final answer Then round to the nearest whole number as needed e It will take about 4 6 day s for the insect population to reach 840 Do not round until the final answer Then round to the nearest tenth as needed f It will take about day s for the insect population to double Do not round until the final answer Then round to the nearest tenth as needed c D o 57

The Hubble Space Telescope HST has excellent resolving power because there is no atmospheric distortion of the light Its 2 40 m diameter primary mirror can collect light from distant galaxies that formed early in the history of the universe How far apart can two star clusters be from each other if they are 14 0 billion lightyears away from Earth and are barely resolved by the HST using visible light with a wavelength of 400 nm

The population of a colony of mosquitoes obeys the law of uninhibited growth Use this information to answer parts a through c a If N is the population of the colony and t is the time in days express N as a function of t Consider No is the original amount at t 0 and k 0 is a constant that represents the growth rate N t Noekt Type an expression using t as the variable and in terms of e b The population of a colony of mosquitoes obeys the law of uninhibited growth If there are 1000 mosquitoes initially and there are 1800 after 1 day what is the size of the colony after 4 days Approximately mosquitoes Do not round until the final answer Then round to the nearest whole number as needed

After the release of radioactive material into the atmosphere from a nuclear power plant in a country in 1990 the hay that country was contaminated by a radioactive isotope half life 8 days If it is safe to feed the hay to cows when 9 the radioactive isotope remains how long did the farmers need to wait to use this hay The farmers needed to wait approximately Round to one decimal place as needed www days for it to be safe to feed the hay to the cows

20 The equation 35n t 50 0 relates t the total cost in dollars of boarding a dog in a kennel with the number of nights n the dog is boarded and the cost of medical insurance By a Write the equation in the form y mx b b Use the grid to graph the total cost versus the number of nights of boarding c What is the total cost to board a dog for 7 nights at the kennel 235 P d What is the slope of the line 35 e What does the slope represent Per night to board the dog f What is the cost of medical insurance

Write the expression as a sum and or difference of logarithms Express powers as factors log2 4x ww log2 4x Type an exact answer in simplified form

K x f x f x Use the following table to determine the indicated derivatives or state that the derivative cannot be determined 6 0 1 5 2 a OA 3 4 1 3 2 b 1 c 3 d f 3 3 3 7 3 4 Type an integer or a fraction O B 4 cannot be determined Part 1 of 4 6 8 2 a Choose the correct answer below and if necessary fill in the answer box within your choice ECCE O Points 0 of 1

Suppose that G x log2 2x 2 3 a What is the domain of G b What is G 7 What point is on the graph of G c If G x 1 what is x What point is on the graph of G d What is the zero of G a The domain of G is 1 00 Type your answer in interval notation b G 7 1 The point 7 1 is on the graph of G Type an ordered pair c If G x 1 then x 1 The point is on the graph of G Type an ordered pair

Find four intervals on which f is one to one making each interval as large as possible Four intervals on which f is one to one are a 6 6 3 3 0 0 Type your answer in interval notation Use a comma to separate answers as needed CHECKED 8 7 6 5 Ay 10 V FIAB 3 2 10 2

Use the given graph of f x to find the intervals on which f x is increasing the intervals on which f x is decreasing and the local extrema Sketch a possible graph of y f x Af x 5

K a Find the inverse of the function below on the given interval and write it in the form y f x x and f f x x b Verify the relationships f f x f x x 6 a f x x 5 9 A B 6 b Verify the relationship f f x x Select the correct choice below and fill in the answer box es to complete your choice x 6 x GEEXED x x 6 X Type integers or simplified fractions

Complete the sentence below If the graph of a logarithmic function f x log ax a 0 a 1 is increasing then its base must be larger than www If the graph of a logarithmic function f x log ax a 0 a 1 is increasing then its base must be larger than 1 5 1

Complete the sentence below The domain of the logarithmic function f x log ax is The domain of the logarithmic function f x log ax is xx 1 or 1 00 x 0 x 1 or 0 1 x x 0 or 0 00 x 0 x 1 or 0 00

Question Solve the given equation for a 762 2 41

Question 1 A vehicle purchased for 27 500 depreciates at a constant rate of 5 Determine the approximate value of the vehicle 13 years after purchase Round to the nearest whole dollar Question Help Video Message instructor

Score 5 14 5 14 answered Question 8 Solve for x 48x 3 832 10 x

Question 2 Which of the following are equivalent to 5 log u 1 3 Assume u 0 1 VTS log D log u 3 1 log u5 log 5 15 5 log u5 3 Question Help Message instructor

Consider the following function Find the inverse function express it as a function of x and find the derivative of the inverse function f x f x x x 6 6x x 1 GREE

nsider the following function Find the inverse function express it as a function of x and find the derivative of the inverse function X x 6 f x 6x9

Find the inverse function and graph both f x and f x on the same set of axes Check your work by looking for the required symmetry in the graphs f x 9 8x The inverse function is f x SIER

Find the inverse function on the given interval if specified and graph both f and f on the same set of axes Check by looking for the required symmetry in the graphs f x x 8 for x 8 Find the inverse function f x for x

1 Simplify the expression completely cos sec 3 csc

Let 1 5 and 6 5 0 Compute the dot product 6 b

Given p 5 7 and 4 4 find p 4u p 4u

Given p 6 2 and 6 10 5 find p b 7 6

Given 1 8 and 3 1 find b a b g

Perform the vector addition 12 5 7 4

Use Newton s method to find all roots of the equation correct to eight decimal places Start by drawing a graph to find initial approximations Enter your answers as a comma separated list earctan 3x 27x 1 X

A graphing calculator is recommended A company generates a continuous stream of income of 22t million dollars per year where t is the number of years that the company has been in operation a Find the integral for the present value of this stream for the first 6 years at 6 interest compounded continuously Do not use a graphing calculator 6 S PV t dt b Evaluate your answer to part a using FnInt on a graphing calculator Enter your answer in millions of dollars Round your answer to the nearest whole number million

A graphing calculator is recommended Radioactive waste is leaking out of cement storage vessels at the rate of te0 19t hundred gallons per month where it is the number of months since the leak began Find the total leakage in gallons during the first 2 months Verify your answer using FnInt on a graphing calculator Round your answer to the nearest whole number gallons

Given that cos 0 and sin 0 0 determine the values of the sine and cosine functions for 20 41 sin 20 Type a simplified fraction

K expres 2 sin 55 cos 99 2 sin 55 cos 99 Do not include the degree symbol in your answer

sin sin 5x ba 12 CID Simplify your answer including any radicals Use integers or fractions for any numbers in the expression Type an exact answer using radicals as needed

Consider n independent trials of an experiment in which each trial has two possible outcomes success or failure The probability of a success on each trial is p and the probability of a failure is q p In this context the term nCk pkqnk in the expansion of p q gives the probability of k successes in the n trials of the experiment A fair coin is tossed eleven times To find the probability of obtaining six heads evaluate the term shown below in the expansion of Round your answer to three decimal places 11 11 6 5

Simplify the factorial expression 10 5 3

Use sigma notation to write the sum 1 1 1 2 4 32 n 0 1 1 1 8 Use a graphing utility to find the sum Round your answer to three decimal place

Perform the indicated operation below and simplify the result sin x CSC X COS X sec x COS X sec x sin x CSC X

Watch the video that describes rewriting an expression in terms of sine and cosi Click here to watch the video Write the expression in terms of sines and or cosines and then simplify cot x sin x cotx sin x

Use the Binomial Theorem to expand and simplify the expression 2 x 3 4 4 x 3

A trough has a trapezoidal cross section with a height of 7 m and horizontal sides of width 7 2 m and 7 m Assume the length of the trough is 14 m See the figure to the right Complete parts a and b below MICKERS 7 m 7 m 3 5 m 14 m a How much work is required to pump the water out of the trough to the level of the top of the trough when it is full Use 1000 kg m for the density of water and 9 8 m s for the acceleration due to gravity Draw a y axis in the vertical direction parallel to gravity and use the midpoint of the bottom of one edge of the trough as the location of the origin For Osy 7 find the cross sectional area A y in terms of y A y Simplify your answer Use integers or fractions for any numbers in the expression

A swimming pool has the shape of a box with a base that measures 26 m by 19 m and a uniform depth of 2 9 m How much work is required to pump the water out of the pool when it 3 is full Use 1000 kg m for the density of water and 9 8 m s for the acceleration due to gravity Points 0 of 1 Draw a y axis in the vertical direction parallel to gravity and choose one corner of the bottom of the pool as the origin For 0 y 2 9 find the cross sectional area A y A y Simplify your answer Save

A deposit of 100 is made at the beginning of each month in an account that pays 2 interest compounded monthly The balance A in the account at the end of 6 years is given by A 100 1 0 02 100 1 0 02 Find A Round your answer to the nearest cent A

Find the sequence of the first five partial sums S S S3 S4 and Ss of the geometric sequence by adding terms 8 12 18 27 81 2 S S S3 S4 S5

Find a formula for the nth term of the geometric sequence Then find the indicated nth term of the geometric sequence 7th term 2 14 98 an

Determine whether or not the sequence is geometric If it is find the common ratio r If an answer does not exist enter DNE 11 26 41 56 O Yes the sequence is geometric O No the sequence is not geometric