Application of derivatives Questions and Answers

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A fence 9 feet tall runs parallel to a tall building at a distance of 3 ft from the building as shown in the diagram e LADDER 9 ft 3 ft We wish to find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building A First find a formula for the length of the ladder in terms of 0 Hint split the ladder into 2 parts Type theta for 0 L 0 B Now find the derivative L 0 Type theta for 0 L 0 C Once you find the value of that makes L 0 0 substitute that into your original function to find the length of the shortest ladder Give your answer

Find the absolute maximum and minimum values of f x x 6 x 5 Give your answers as a single value e g 17 or a comma separated list e g 1 2 3 or NONE The absolute maximum value is The absolute minimum value is f 1 if any over the interval 5 0 and it occurs at z and it occurs at z

Suppose that the monthly cost in dollars of producing x chairs is C x 0 007x3 0 07x 2 19x 700 and currently 40 chairs are produced monthly a What is the current monthly cost b What is the marginal cost when x 40 c Use the result from part b to estimate the monthly cost of increasing production to 42 chairs per month d What would be the actual additional monthly cost of increasing production to 42 chairs monthly a The current monthly cost is Round to the nearest cent as needed b The marginal cost when x 40 is per item Round to the nearest cent as needed c Use the result from part b to estimate the monthly cost of increasing production to 42 chairs per month The monthly cost in increasing production to 42 chairs monthly is Round to the nearest cent as needed d The actual additional monthly cost of increasing production to 42 chairs monthly is Round to the nearest cent as needed

For the demand function q D x 700 X find the following a The elasticity b The elasticity at x 1 stating whether the demand is elastic inelastic or has unit elasticity c The value s of x for which total revenue is a maximum assume that x is in dollars a Find the equation for elasticity E x b Find the elasticity at the given price stating whether the demand is elastic inelastic or has unit elasticity E 1 Simplify your answer Type an integer or a fraction Is the demand at x 1 elastic inelastic or does it have unit elasticity OA unit elasticity OB inelastic OC elastic c Find the value s of x for which total revenue is a maximum assume that x is in dollars OA Round to the nearest cent as needed Use a comma to separate answers as needed OB The total revenue is independent of x

The average cost for a company to produce x units of a product is given by the function A x The change in average cost is approximately dollars 16x 200 X C Use A x to estimate the change in average cost as production goes from 100 units to

Social animals that live in groups can spend less time scanning for predators than solitary individuals However they waste time fighting with the other group members over the available food There is some group size at which the net benefit is greatest because the animals spend least time on these unproductive activities and thus can spend time on feeding mating etc Assume that for a group of size x the fraction of time spent scanning for predators is 1 x 1 S x A and the fraction of time spent fighting with other animals over food is F x B x 1 where A B are constants and A 2B Find the size of the group for which the time wasted on scanning plus time wasted on fighting is smallest

The frame for a kite is to be made from six pieces of wood The four exterior pieces have been cut with the lengths indicated in the figure To maximize the area of the kite how long should the diagonal pieces be List the diagonal pieces lengths in non decreasing order Your answers may depend on a and b and assume a b a a b b

A rectangle is inscribed with its base on the x axis and its upper corners on the parabola y 11 What are the dimensions of such a rectangle with the greatest possible area Width Height

Find the distance from the origin to the closest point s on the graph of 14x 4xy 14y We begin by computing implicitly for points x y on the graph obtaining dy da We need to minimize the following function of a and y f x y There are four critical points located at the following x values listed in increasing order 21 202 003 1 Thus there is are x y and its distance from the origin is EB E 15 point s closest to the origin one of which is

The frame for a kite is to be made from six pieces of wood The four exterior pieces have been cut with the lengths indicated in the figure maximize the area of the kite how long should the diagonal pieces be List the diagonal pieces lengths in non decreasing order Your answers may depend on a and b and assume a b m b b

A construction crew has to dig ditches for a set of pipes that must carry water from point A to both of points B and C see figure Points B and Care 4 km apart and point A is 4 km from the midpoint of B and C They want to dig as little as possible and they believe the best way to do that is to dig from point A to some point D along the midline a distance h from the midpoint of B and C They know that they should choose h satisfying 05h2 but they are not sure exactly what value of h to use 4 km 4 km B D a Formulate an expression h for the total length of ditch b Now calculate the value of which minimizes the amount of ditch to be dug If there is more than one such value give your answer as a comma

Find the absolute maximum and minimum values of f x x 6 5 Give your answers as a single value e g 17 or a comma separated list e g 1 2 3 or NONE The absolute maximum value is The absolute minimum value is if any over the interval 5 0 and it occurs at x and it occurs at x E

Find the dimensions of a right circular cylindrical can with both a top and a bottom that holds 27000 cubic cm and is constructed with the least amount of material possible Radius of can 16 258 Height of can Area of Material 4982 26 cm cm cm

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Let f x x 8x 4 a Find the x value c of the critical point of f x and compute f c The x value of the critical point is c The value of f c b Compute the value of f x at the endpoints of the interval 0 8 f 0 f 8 c Determine the minimum and maximum values of f x on 0 8 Minimum value Maximum value d Find the extreme values of f x on 0 1 Minimum value Maximum value BR

Find the dimensions of a right circular cylindrical can with both a top and a bottom that holds 27000 cubic cm and is constructed with the least amount of material possible Radius of can Height of can Area of Material cm cm cm

A piece of wire 35 m long is cut into two pieces One piece is bent into a square and the other is bent into a circle How much of the wire should go to the square to minimize the total area enclosed by both figures BE

Let f x x 8x 4 a Find the x value c of the critical point of f x and compute f c The x value of the critical point is c The value of f c b Compute the value of f x at the endpoints of the interval 0 8 f 0 f 8 c Determine the minimum and maximum values of f x on 0 8 Minimum value Maximum value d Find the extreme values of f x on 0 1 Minimum value Maximum value

A rectangular photo frame encloses an area of 600 cm The top edge of the frame is constructed out of heavier material than the other three sides The material for the top edge weighs 200 g cm and the other three sides are made from material weighing 100 g cm In this question we will calcula the dimensions of the frame that minimize the total weight a Write an expression in terms of 2 and y for the total weight of the frame Weight E y 9 b Find a constraint that allows you to express y in terms of c The values of x and y that minimize the total weight are

At what value s of a on the curve y 5 90x 3x5 does the tangent line have the largest positive slope If there is more than one value enter your answer as a comma separated list Answer separate by commas x

Consider the function f defined for all real x f x x 4 a Find the x values of the two critical points of f Separate your two answers by a comma b Now use the Second Derivative Test to determine the location and types of the extrema at the critical points you found above The left critical point is a The right critical point is a c Finally using an online graphing tool like Desmos produce the graph of f Notice that f has a local maximum at x 0 Choose the best reason why f has a local maximum at x 0 but no critical point at x 0 f is not continuous at x 0 f is not differentiable at x 0 on this problem

Note If you are asked for value s give your answer as a single value e g 17 or a comma separated list e g 1 2 3 or NONE If you are asked for an interval give your answer in interval notation Type INF for o and INF for 00 Please answer the following questions about the function 8x 24x a Find the locations of the critical points of f where it is increasing and decreasing and its local extrema Note that a function is considered increasing if larger x values give larger y values A function is decreasing if larger x values give smaller y values Critical points are located at x Increasing on the x interval Decreasing on the x interval Local maxima are located at c Local minima are located at x b Find the locations where f is concave up concave down and has inflection points Concave up on the x interval Concave down on the x interval Inflection points are located at c c The function d Sketch a graph dashed lines for hc because for all in the domain of f even odd on f without using a graphing calculator or online tool Plot the y intercept and the x intercepts if they are known Draw neither vertical asymptotes Plot the points where f has local maxima local minima and inflection points Use what you know

The raccoon population on a small island is observed to be given by the function P t 120t0 4t 700 where t is the time in months since observations of the island began The maximum population is attained at The maximum population is Note you can get a larger view of the graph by c months P

Find the point on the line 6x y 9 that is closest to the point 9 7 X y B

Find the extreme values of the function f on the interval 0 and the x value s at which they occur If an extreme value does not exist enter DNE for both the value and location Absolute minimum value Absolute maximum value located at x located at f x 8e cos r

Note If you are asked for value s give your answer as a single value e g 17 or a comma separated list e g 1 2 3 or NONE If you are asked for an interval give your answer in interval notation Type INF for oo and INF for o Consider the function f and its derivatives all defined for x 0 f x x5 log x f x x 5x log x f x 9x 20x log z a Find the x values of the critical points of f you are not asked to enter them here and then use the Second Derivative Test when possible to determine the location and types of the extrema Local maxima at x Local minima at x Separate multiple answers by commas and enter NONE if there are no local maxima Separate multiple answers by commas and enter NONE if there are no local minima

Consider the function f x defined for values of x inside the interval 4 4 Find T a T a and T3 x the Taylor polynomials for f centered at 0 of degree 1 2 and 3 respectively Ty x 5 41F T x

Consider the graph of the function defined for x in the interval 1 13 shown above Identify the marked points as being a global maximum or minimum a local maximum or minimum or none of the above Select all that apply B C A Global aximum B Local maximum C Global minimum D Local minimum E None of the above A Global maximum B Local maximum C Global minimum D Local minimum E None of the above A Global maximum B Local maximum C Global minimum D Local minimum E None of the above D A Global maximum B Local maximum C Global minimum D Local minimum E None of the above E A Global maximum 5 B A B C 7 5 D E F G X 15

Find two numbers differing by 36 whose product is as small as possible Enter your two numbers as a comma separated list e g 2 3 The two numbers are

Consider the function f x x 18x 6 2 x 7 Find the absolute minimum value of this function Answer find the absolute maximum value of this function

Find the absolute maximum and absolute minimum values of the function f x x 6x 63x 11 over each of the indicated intervals a Interval 8 0 1 Absolute maximum 2 Absolute minimum b Interval 5 4 1 Absolute maximum 2 Absolute minimum c Interval 8 4 1 Absolute maximum 2 Absolute minimum

Using a calculator or computer generate a graph like the one shown below by graphing y 2 16 82 8 and y 2x for 1 1 What is the relationship between y 2 16 8a 8 and y 2x Use your graph to estimate to one decimal place the largest magnitude of the error in approximating 2 16 8 8 by 2x for 1 1 error Is the approximation an over or an underestimate when 0 Enter over or under over Note You can earn partial credit on this problem Preview My Answers Submit Answers EEE You have attempted this problem 3 times Your overall recorded score is 50 10

The hyperbolic cosine function cosh is defined as follows e ez cosh x 2 Find the sixth degree Taylor Polynomial Te x for cosh 4x about x 0 To x BE 0 20 0

The function f x is defined for a T x f x Find the sixth degree Taylor Polynomial To x for f x about x 0 1 5 as follows x 25 e 2 1 53 x 24 26

Suppose that the second order Taylor polynomial of some function f about x 0 is given by x x 7 24 Now consider the function Find S x the second order Taylor Polynomial for g about x 0 S x T x 3 g x f x e x E

Consider the function f x sin 2x cos x Find the third degree Taylor polynomial T3 x of f about the point x 0 Hint you could find this manually using differentiation via the product rule or you could use the expansions of sin x and cos x directly T3 x B

and its derivatives a Identify the graph that displays f in blue and f in red A 6m 2 2 D f is concave up on M and 24 2 18m 12 2 b Indicate the x intervals where f is concave up and concave down Give your answer in interval notation Type INF for co and INF for co

W8 Problem 3 1 point You are given the following graph of the function f x 5 18 5 0 1 0 Find the x value of the point where the second derivative changes sign from negative to positive B 5 0 10

Find the linear approximation L x of y e5 log x at a 1 L x

The figure below shows f x and its linear approximation at x a y 2x 3 The linear approximation is shown in blue a What is the value of a a b What is the value of f a f a c Use the linear approximation to approximate the value of f 2 2 f 2 2 d Is the approximation an under or overestimate The approximation is an Click on the ima estimate 2 X

Suppose that f x is a function with f 130 64 and f 130 5 Estimate f 128 f 128

Find the linear approximation L x of f x log x at x 1 and use it to estimate log 1 15 L x log 1 15 Notice that the linear approximation of f x at x 1 is the Talor polynomial of degree 1 for f about x 1

Given that f 2 6 1 2 and f 2 5 9 4 approximate f 2 6 f 2 6

1 point You are given the following graph of the function f 5 P 18 5 0 18 1 5 10 5 0 10 4 0 Find the x value of the point where the second derivative changes sign from negative to positive

Claire needs to borrow 12 000 from a local bank She compares the monthly payments for a 5 1 loan for three different periods of time What is the monthly payment for a one year loan A two year loan A five year loan

Evaluate the following indefinite integrals 4 cos x sin x sin x 2 csc x dx sin x a b dx

Match each of the following graphs with the correct function by placing the letter of the graph on the line provided Each letter can only be used once Do not use a calculator or desmos to draw the graph 3 f x x 1 2 2 E 4 f x x 1 F 5 f x x 2x 1 q 1 x 1 x 1 b 2 C I C 6 f x x x 1 9 1 b a X da C 3 G 7 f x x 2x 2 a a 1 b 1 CHO H 9 f x x 2x a 1 6 2 x a a 1 1 1 2 1 2 1 cia B 8 f x x 2x 9 1 f 1 0 X x 12 0 1 2 1 0 x f 1 1 2 1 1 1 X 1 0 1 0 15 1 f 1 1 1 2 1 0 1 0 1 1 x 1 1 2 1 1 f 1 1 1 1 1 olla x 1 9 1 D 2 COO D 10 f x x 2x 2 x 1 1 x 1 y 1 x 1 0 1 1 X 1 0 25 0 5 To 1 1 1 1 F 1 2 1 1 1 1 1 0 10 1 11 2 y 1 0 y 1 3 1 1

Find the quadrant in which the terminal side of 8 10 radians is located None of the other answers x One Correct Answer Four Three Three

10 973 Use the following problem to answer the questions below A man flies a kite and lets out 200 feet of string The angle of elevation of the string is 30 Make a figure and answer the following questions A To find the height off the ground the kite is you will use the trigonometric function x 50 feet Correct Answer 100 feet X cot 30 Correct Answer sin 30 B How high off the ground is the kite