Application of derivatives Questions and Answers

Use differentials to estimate the amount of paint needed in m to apply a coat of paint 0 06 cm thick to a hemispherical don with diameter 54 m Round your answer to two decimal places m

Find the differential of each function a y x sin 4x dy b y In 2 1 dy

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m highe than the bow of the boat If the rope is pulled in at a rate of 1 m s how fast in m s is the boat approaching the dock when it is 7 m from the dock Round your answer to two decimal places m s

Two sides of a triangle have lengths 11 m and 17 m The angle between them is increasing at a rate of 2 min How fast in m min is the length of the third side increasing when the angle between the sides of fixed length is 60 Hint Use the Law of Cosines Round your answer to three decimal places m min

Boyle s Law states that when a sample of gas is compressed at a constant temperature the pressure P and volume V satisfy the equation PV C where C is a constant Suppose that at a certain instant the volume is 300 cm the pressure is 120 kPa and the pressure is increasing at a rate of 20 kPa min At what rate in cm min is the volume decreasing at this instant cm min

Find the linearization L x of the function at a f x x a 4 L x

The altitude of a triangle is increasing at a rate of 1 cm min while the area of the triangle is increasing at a rate of 2 cm2 min At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 190 cm cm min

The length of a rectangle is increasing at a rate of 3 cm s and its width is increasing at a rate of 8 cm s When the length is 7 cm and the width is 5 cm how fast is the area of the rectangle increasing in cm s cm s

Two sides of a triangle are 8 m and 9 m in length and the angle between them is increasing at a rate of 0 06 rad s Find the rate in m s at which the area of the triangle is increasing when the angle between the sides of fixed length is rad 3 m s

Each side of a square is increasing at a rate of 5 cm s At what rate in cm s is the area of the square increasing when the area of the square is 81 cm cm s

A kite 100 ft above the ground moves horizontally at a speed of 4 ft s At what rate in rad s is the angle in radians between the string and the horizontal decreasing when 200 ft of string have been let out rad s

Determine if the series converges or diverges Give a reason for your answer 2n 3n 7 n 1 Select the correct choice below and if necessary fill in the answer box to complete your choice V The series converges because the limit found in the nth Term Test is OA Type an integer or a fraction The series diverges because the limit found in the nth Term Test is B O B Type an integer or a fraction The Direct Comparison Test with En shows that the series diverges O C D The Direct Comparison Test with shows that the series converges n

Drag the function equation on top of the matching graph given the transformation 6 YA 8 LL f x 2 f x 2 6 x

Consider the following parametric equation What is the length of the curve for 0 0 to 0 x 13 cos 0 0 sin 0 y 13 sin 0 0 cos 0

Consider the parametric curve given by the equations x t t 30t 26 y t 30t 37 How many units of distance are covered by the point P t x t y t between t 0 and t 10

Consider the curve given by the parametric equations a Determine the point on the curve where the tangent is horizontal t x t t 147 y 3 t 147 b Determine the points t1 t2 where the tangent is vertical and t t t t Graph the curve on a calculator Turn in a sketch of the curve

If x et and y t 8 2 find an equation of the tangent in terms of x and y to the curve at 1 64 Tangent line Graph the curve on a calculator Turn in a sketch of the curve with the tangent line

show and and that seventh term 1 243 find them geometric sequence with first term 3 has two possible sums to infinity

a Find dy da dy da expressed as a function of t for the given the parametric equations d y dx2 b Find d y dx2 c Except for at the points where Concave expressed as a function of t 8 S cos t 7 sin t dy is undefined is the curve concave up or concave down Enter up or down da

g x x 27x 61 a Select the correct choice below and if necessary fill in the answer box within your choice OA The critical value s of the function is are Use a comma to separate answers as needed OB The function has no critical values b Select the correct choice below and if necessary fill in the answer box es within your choice OA The relative maximum point s is are and there are no relative minimum points Simplify your answer Type an ordered OB The relative minimum point s is are Simplify your answer Type an ordered pair using integers fractions Use a comma to separate answers as needed and there are no relative maximum points pair using integers or fractions Use a comma to separate answers as needed OC The relative minimum point s is are and the relative maximum point s is are Simplify your answers Type ordered pairs using integers or fractions Use a comma to separate answers as needed D There are no relative minimum points and there are no relative maximum points

f x 5 x x Describe any relative extrema Select the correct choice below and if necessary fill in the answer box es to within your choice and there are no relative maximum points OA The relative minimum point s is are Simplify your answer Type an ordered OB The relative minimum point s is are Simplify your answers Type ordered pair using integers or fractions Use a comma to separate answers as needed and the relative maximum point s is are pairs using integers or fractions Use a comma to separate answers as needed and there are no relative minimum points OC The relative maximum point s is are Simplify your answer Type an ordered pair using integers or fractions Use a comma to separate answers as needed OD There are no relative minimum points and there are no relative maximum points Identify any intervals over which the function is increasing or decreasing Select the correct choice below and fill in the answer box es within your choice and is not increasing anywhere or fractions for any numbers in the expression Use a comma to separate answers as needed and decreasing over the interval s OA The function f x is decreasing over the interval s Type your answer in interval notation Use integers OB The function f x is increasing over the interval s Type your answers in interval notation Use integers or fractions for any numbers in the expressions Use a comma to separate answers as needed and is not decreasing anywhere OC The function f x is increasing over the interval s Type your answer in interval notation Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed Choose the correct graph below O A Q Q G OB Ay Q O C Ay Q O D OU

let x tan find f s

Answer the following questions for the function x f x b The formula for the first derivative f x can be factored Set f x 0 to find the two critical numbers Hint You can factor out 18 from all terms in the formula for f x The critical values are x Use a comma to separate answers as needed c At the negative critical value listed in part b what is the value of the second derivative Type Integers or simplified fractions Consequently what can be concluded about the graph of f Select the correct choice below and if necessary fill in the answer boxes within your choice OA The graph of f is concave up and f has a relative minimum at OB The graph of f is concave down and f has a relative minimum at OC The graph of f is concave up and f has a relative maximum at OD The graph of f is concave down and f has a relative maximum at OE No conclusion can be made d At the positive critical value listed in part b what is the value of the second derivative Type Integers or simplified fractions Consequently what can be concluded about the graph of f Select the correct choice below and if necessary fill in the answer boxes within your choice OA The graph of f is concave down and f has a relative minimum at OB The graph of f is concave up and f has a relative maximum at 0 OC The graph of f is concave up and f has a relative minimum at OD The graph of f is concave down and f has a relative maximum at E No conclusion can be made e Set the formula for the second derivative f x 0 to find any possible inflection points f x 0 atx What can be concluded about the graph of f at this value of x Select the correct choice below and if necessary fill in the answer boxes within your choice OA There is a point of inflection at where the graph of f changes from concave down to concave up OB There is a point of inflection at where the graph off changes from concave up to concave down OC No conclusion can be made f Choose the graph of the function f x OA Q OB 1500 O C 4400

9 To convert from a degrees Celsius to y degrees Fahrenheit we use the formula f x x 32 Find the inverse function if it exists and be sure you could explain its meaning f x Submit answer help formulas

Find the largest domain for f x x 2 2 with upper a bound 6 such that the function f is one to one and non decreasing One to one domain help intervals Your answers must be in interval notation Find the inverse of f restricted to that domain f x help formulas

Let the graph of f x be given below Find the following 5 f 0 f 0 Tiles tmp quick 2023 Click on graph to enlarge help numbers help numbers

Find the inverse of f x 3 3 f x

8 The Sum given by a Find b Show geometric the of 5 1 2 1 first term of that three terms the progression terms of certain of the the series is series 4 3 4 3 series Ug are in

A patient needs to keep their blood sugar levels within 10 milligrams per deciliter of 93 milligrams per deciliter What blood sugar levels are okay for that patient Enter solution as an interval or union of intervals Note this is a hypothetical situation help intervals In absolute value notation where xa represents the score on the test this is OA x 10 93 OB x 93 10 OC x 10 93 OD x 93 10 OE None of the above

Starting graph a shifting f x 3 units upward y 9x 3 b shifting f x 2 units to the right y 9x 2 c reflecting f x about the x axis y 9x d reflecting f x about the line y 2 y 9x 2 gid

2x 22 2 3x

Consider the parabola given by the equation f x 4x 10x 10 Find the following for this parabola A The vertex B The vertical intercept is the point C Find the coordinates of the two a intercepts of the parabola and write them as a list separated by commas It is OK to round your value s to to two decimal places

Question 3 b What is the y intercept The graph of f x 4 is reflected about the y axis and shifted upward 4 units a What is the equation of the new function g x g x c What is the domain d What is the range Question Help VIDEO Post to forum

A table of values for f g f and g is given below a If F x f f x find F 2 b If G x g g x find G 3 a F 2 b G 3 E x f x g x x g x 6 7 9 123 2 317 7 282 57 7

Consider the function r x f g h x Given that h 1 4 h 1 5 g 5 3 Find the value of r 1 or enter NA if there is not enough information to compute this derivative r 1 g 5 6 and 6 2

Find the derivative of Assume that a and b are constants h x 5 b h

4 The graph of a function g is shown on the zy plane on the interval 7 7 A table of selected values is given for a different function h which has only one a real zero H h z 5 2 2 7 3 0 13 7 20 0 2 4 Let f x 2g x 3h x Find f 4 b Let j x g x h a How many a intercepts does j have on the interval 7 7 c Let k x M g x h x Which values are restricted from the domain of k Explain X g x y

Show all the necessary steps for full credit 1 10 pts Use the graph to find the following limits and function value Ay 175 10 15 15 X INO a b C d e f 2 f 7 lim f x X 7 lim x x 2 lim f x X 2

4 8 points Find the 2nd derivative of the following function Write the final answer in exponential notation using only positive exponents f x 5x 1

9 points Use the information below to sketch the graph for f x on the interval 2 x 2 Label any points on the x axis or y axis which are important Note Precision in drawing is not required as long as it is clear from the graph what you intended to do be careful that your graph is visibly curved where it is supposed to be curved f x 0 for x 1 f x 0 for x 0 f 1 0 f x 0 for 1 x 1 f 1 0 f x 0 for x 1 f 0 0 f x 0 for 0 x 1 f x 0 for x 1 2

During the winter if the low temperature outside is a C the daily cost to heat a building can be determined using the function f x 6 1 3 Find and interpret the given function values and determine an appropriate domain for the function Round all function values to the nearest hundredth Answer Attempt 1 out of 2 f 6 meaning when the low temperature outside is interpretation f 9 interpretation f 4 5 interpretation in the context of the problem meaning when the low temperature outside is V in the context of the problem meaning when the low temperature outside is C it would cost C it would cost in the context of the problem C it would cost Based on the observations above it is clear that an appropriate domain for the function is to heat the building This to heat the building This to heat the building This

When an airplane leaves the runway its angle of climb is 170 and its speed is 275 feet per second Find the plane s altitude after 1 minute Round your answer to the nearest whole number X ft

Solve the right triangle shown in the figure Round your answers to two decimal places A B 82 2 C 90 a b C 186 15 B a 7 8 C 25 5 X b 0 O

Given the parametric equations Write the derivative dy dx as a function of t x 2t 4 y sin 3t

A sign on the roadway at the top of a mountain indicates that for the next 4 miles the grade is 9 5 see figure Find the change in elevation for a car descending the 4 mile stretch Round your answer to two decimal places mi mi Not drawn we

4 of 10 4 5 points Starting from a stop a mouse runs to the right first then comes to a stop for a few seconds and then runs to the left Consider the graphs below Graph A Graph C Graph B a 2 points Which of the above graphs could describe the position of the mouse as a function of time b 2 points Which of the above graphs could describe the velocity of the mouse as a function of time c 1 point Based on the graphs that you have chosen in the previous parts Is the mouse accelerating decelerating or neither at the beginning

An observer in a lighthouse 350 feet above sea level observes two ships directly offshore The angles of depression to the ships are 6 and 0 9 5 see figure How far apart are the ships Round your answer t decimal place 350 ft

When an airplane leaves the runway its angle of climb is 17 and its speed is 275 feet per second Find the plane s altitude after 1 minute Round your answer to the nearest whole number ft

From a point x 110 feet in front of a public library the angles of elevation to the base of the flagpole and the top of the flagpole are 0 28 roof Find the height of the flagpole aft STEP 1 Let F be the height of the flagpole and L be the height of the public library Convert the angle 39 45 into decimal form Use this value and an appropriate trigonometric function to find F L the total height of the library and flagpole feet F L STEP 2 Find the height of the library L LI feet STEP 3 Subtract your result from Step 2 from your latter result in Step 1 to determine the height of the flagpole F F feet Need Help Read

When an airplane leaves the runway its angle of climb is 14 and its speed is 275 feet per second How long will it take the plane to climb to an altitude of 10 000 feet 16 000 feet Round your answers to one decima place 10 000 ft 16 000 ft S S