Application of derivatives Questions and Answers

The radius of a sphere is increasing at a rate of 5 mm s How fast is the volume increasing in mm s when the diameter is 60 mm Round your answer to two decimal places mm s

The height in meters of a projectile shot vertically upward from a point 3 m above ground level with an initial velocity of 21 5 m s is h 3 21 5t 4 9t after t seconds a Find the velocity in m s after t seconds v t Find the velocity in m s after 2 seconds and after 4 seconds v 2 m s v 4 m s b What is the value in m s of v t when the projectile reaches its maximum height v t m s When in s does the projectile reach its maximum height Round your answer to two decimal places S c What is the maximum height in m Round your answer to two decimal places m d When in s does it hit the ground Round your answer to two decimal places S e With what velocity in m s does it hit the ground Round your answer to two decimal places m s

Use the graph to state the absolute and local maximum and minimum values of the function Assume each point lies on the gridlines Enter your answers as a comma separated list If an answer do not exist enter DNE y 5 4 3 2 1 1 absolute maximum value absolute minimum value local maximum value s local minimum value s A y f x 2 3 4 5 6 e

Finally setting f c 4c 4 equal to 0 and solving for c gives the following result Enter your answers as a comma separated list 4c 4 0 C

y g x W 1 C 0 g is shown 1 i a Which of the following verifies that g satisfies the hypotheses of the Mean Value Theorem on the interval 0 8 Select all that apply Og is continuous on the closed interval 0 8 Og obtains at least one maximum on the closed interval 0 8 g is differentiable on the open interval 0 8 Og obtains at least one minimum on the open interval 0 8 Og is continuous on the open interval 0 8 Og takes only positive values on the closed interval 0 8 b Estimate the value s of c that satisfy the conclusion of the Mean Value Theorem on the interval 0 8 Enter your answers as a comma separated list Round your answers to one decima places If an answer does not exist enter DNE c Estimate the value s of c that satisfy the conclusion of the Mean Value Theorem on the interval 2 6 Enter your answers as a comma separated list Round your answers to one decimal places If an answer does not exist enter DNE

The third hypothesis is satisfied if f a f b To verify that the given function satisfies this hypothesis on the interval 1 3 we must evaluate f 1 and f 3 and compare the re f x f 1 f 3 Since these values Select the same the third hypothesis Select satisfied 2x 4x 9

Tutorial Exercise Verify that the function satisfies the three hypotheses of Rolle s theorem on the given interval Then find all numbers c that satisfy the conclusion of Rolle s theorem f x 2x 4x 9 1 3

Consider the following function g v v 48v 8 Find the derivative of the function g v Find the values of v such that g v 0 Enter your answers as a comma separated list If an answer does not exist enter DNE V Find the values of v in the domain of g such that g v does not exist Enter your answers as a comma separated list If an answer does not exist enter DNE V Find the critical numbers of the function Enter your answers as a comma separated list If an answer does not exist enter DNE V

For each of the numbers a b c d r and s state whether the function whose graph is shown has an absolute maximum or minimum a local maximum or minimum or neither a maximum nor minimum Enter your answers as a comma separated list y absolute maximum local maximum M absolute minimum local minimum a b C d r

If f x 4 sin x In 3x find f x f x Graph fand f in the same viewing window y 8 6 4 2 O 2 4 W X 47 2x 2 f 2 6 2x X 4x P

The number of users on a website has grown exponentially since its launch After 1 month there were 120 users After 4 months there were 960 users Find the exponential function that models the number of users x months after the website was launched Write your answer in the form f x a b Provide your answer below

A radioactive substance has a continuous decay rate of 0 044 per hour How many grams of a 100 gram sample will remain radioactive after 2 hours Round the answer to the nearest tenth of a gram and do not include the unit in your answer If necessary include a leading 0 with any decimals less than 1 For example 0 4 instead of 14

Use the one to one property of logarithms to solve for x Do not include x in your answer log 2x 1 log 3x 5

Solve the following equation for x In 3x 5 6 Round your final answer to 4 decimal places and do not include x in your answer

Solve the following equation for x Provide your answer below log 6x 84 log 3 2

A T shirt manufacturer is planning to expand its workforce It estimates that the number of T shirts produced by hiring x new workers is given by T x 0 25x 3x 0 x 9 When is the rate of change of T shirt production increasing and when is it decreasing What is the point of diminishing returns and the maximum rate of change for T shirt production Graph T and T on the same coordinate system The rate of change of T shirt production is increasing on Type your answer in interval notation

Find the critical numbers the intervals on which f x is increasing the intervals on which f x is decreasing and the local extrema Do not graph f x 3x 147 X Select the correct choice below and if necessary fill in the answer box to complete your choice A The critical number s is are 7 7 Type integers or simplified fractions Use a comma to separate answers as needed OB The function has no critical numbers Select the correct choice below and if necessary fill in the answer box to complete your choice A The function is increasing on 7 x x 7 Type your answer in interval notation Type integers or simplified fractions Use a comma to separate answers as needed

Find the x and y coordinates of all inflection points 9 5 f x 2x 4 What is are the inflection point s Select the correct choice below and if necessary fill in the answer box to complete your choi A The inflection point s is are 72 Type an ordered pair Use comma to separate answers as needed OB There are no inflection points

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 x 9x 13x 7 For what interval s of x is the graph of f concave upward Select the correct choice below and if necessary fill in the answer box to complete your choice A Type your answer in interval notation Type an exact answer Use a comma to separate answers as needed OB The graph is never concave upward

Find the indicated derivative for the function h x for h x 7x7 6x8 h x

6 Determine whether or not the function is continous for the given value of X FCX x 4 X 2 x 0 x 2 x 2 Is the function continous at x 0 select the Correct choice below and if recensing fill in the ander bety OA Yes the point is defined limf x exists and lim F x OB NO the function is not continous at X 0 simptly you Is the function continous cut X 23 OA Yes the point is defined lim Fax exists and lim x XCX 277 B No the function is continous at X 20 not Is the function continous at x 27 watchist xon 950 OA Yes the point is defined lim fox exist and lim F x X 9 2 OB No The Function is not Continuous at X 2

is the knoth which is concave upward or concave d ownward and find the location of a any points of inflection F x x 4x 8x 4 MON correcta select the below and fill in the blanks to complete your choice Type your answer in interval notation use a comma seperale answer answer as needed Use integers or fraction numbers in the expression x to for 57 any JK Dalyvia A The function is concave upward on and concave downund on The function is concave downward on They are no intervals on which the function is concave upward The function is concave upward on There are no b intervals on which the function is concave down word Select the correct choice bellow and if necessary fill in the ansung box A The Function has point s of inflection aut ctype an orded pair Use a Comma to sepeate as needed LAVS a B There is no point of inflection 102

Solve 9x 5 6 Enter an exact answer or round your answer to the nearest tenth Do not include x in your answer

Write In 8t in expanded form Note When entering natural log in your answer enter lowercase LN as In There is no natural log button on the Alta keyboard

27 find all critial for the following function Then use the second derivative test on each critical number to determine whether it leads to a local maximum o minimum F x 2x 3x 12x 2 What is the second deles derivatie of x F x find all critical a local Correct choice bellow numbers and determine whether they lead maximum or minimum select the A The critical number s is a x there is a local maximum at X There is no local in These is The critical number S is are Xz no local maximum There is a local minimumatx The critical number s is are x D There is no local maxima and no local minimam I there are no criticul numbers The critical numbers s is are x These is a local maximum at X There is a local minimum aut X use a comma to separate ansur as needed

Write the expanded form of log x cwor below

13 Determine the location of each local extremum of the function 13 2 FCX x 12x 45x 2 what is are the local minimum minima select the all Correct choice below and if necessary fill in the answerbet A The local minimum minima is are at x 1 use a comma to separate ansuer as needed Type integess or simpired fruction B The function has no local a what is are the local maximm maxima select the correct choice 8 cixs tun wala A The local maximum maxima is are use a comma minimum atx to seperate ansug as needed rype or simplifed fraktion b saktion 779 211 6309900 integers B The Function has has no I local maximum

Evaluate the integral 7 zett utett dt

5 find the equation of the tangent line to the stroph offCX at the given point x 10x 5 at 2 5 equation of the tangent line to the graph of f x the given point is

23 find the followi llowing F u iF F X 7 x X option A t B 17 16 GEM B new a D H B 7

Compute the instantaneous rate of change of the function at at x a 2 F x 9x x a 4 option A 71 1B 72 C 68 Dy 68

18 find the derivative of the given Function y x 3x sophy on 29 Type 5 an exact answers of knowl

Enter the expression which represents In k 2 In 7r as a single logarithm 1 Note When entering natural log in your answer enter lowercase LN as In There is no natural log button on the Alta keyboard Provide your answer below

Solve 7 x 4 49 x 5 Do not include x in your answer Provide your

Solve 4 x 4X 5 Do not include X in your answer Give your answer as a reduced improper fraction if necessary

Solve 744 10 Enter an exact answer or round your answer to the nearest tenth Do not include x in your answer

Enter the expression which represents In 30 2 ln 5 In 15 as a single logarithm and simplify your answer Note When entering natural log in your answer enter lowercase LN as In There is no natural log button on the Alta keyboard

Enter the expression that represents 3 In 2 In 5 In 4 as a single logarithm Note When entering natural log in your answer enter lowercase LN as In There is no natural log button on the Alta keyboard

The vertical asymptote of the curve 8x y x 7 is given by x

Question Use the Quotient Rule of Logarithms to write an expanded expression equivalent to log 2 3 parenthesis around your logarithm functions log x y Note If you are using log you need to type it in and use the subscript button on the keyboard There is no log button provide your answer below Make sure to use

Let f x x 3x 2 cos TX 16 we find f n 2 5

Pulley wheels having diameters of 84 0mm and 226mm are situated with their centres 855mm apart Find the length of the connecting belt between P and Q the points where the belt contacts the wheels

The profit function in millions for a beverage company for the years 2009 through 2017 can be approximated by f x 28x 616x 1025 where x 9 corresponds to the year 2009 a During what year did a local maximum profit occur b What was the maximum profit GEILE a A local extremum occurs at a critical number of f Because the derivative exists for every x the only critical number s occur where the derivative is zero Find the derivative of f x f x A local maximum profit occurred in the year b The maximum profit was million Simplify your answer Round to the nearest integer as needed

estion 4 point s possible Find all critical numbers for the following function Then use the second derivative test on each critical number determine whether it leads to a local maximum or minimum f x 2x 9x 24x 8 What is the second derivative of f x Find all critical numbers and determine whether they lead to a local maximum or minimum Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The critical number s is are x Use a comma to separate answers B The critical number s is are x Use a comma to separate answers C There are no critical numbers D The critical number s is are x There is no local maximum and no local minimum as needed There is a local maximum at x as needed GEILE Use a comma to separate answers as needed There is no local maximum There is a local minimum at x E The critical number s is are x There is a local maximum at x Use a comma to separate answers as needed There is a local minimum at x There is no local minimum

Use the properties of limits to find the indicated limit x 2x 6 lim x x 8x 11 3 2 O A 6 11 OB 1 0 O c OR 8

A Let X X and X3 represent the times necessary to perform three successive repair tasks at a certain service facility Suppose they are independent normal rv s with expected values H and 2 2 and variances 2 and o3 respectively Round your answers to four decimal places USE SALT 2 a If H 3 60 and 6 63 18 calculate P T 198 and P 144 T 198 02 P T 198 P 144 T 198 b Using the s and o s given in part a calculate both P 54 X and P 58 X 62 P 54 X P 58 X 62 c Using the s and oi s given in part a calculate P 12 X 0 5X 0 5X3 6 P 12 X 0 5X 0 5X 6

rite the following logarithmic equation as an exponential equation of the form a b

Evaluate thelog216 Write your answer in simplest form

What is the value of logs 25 Do not include log 25 in your answer

Write the following logarithmic equation as an exponential equation of the form a b log 64 6