Calculus Questions

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

E 6 1 91

7 2 9 90 16 1 3 112X 7

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

Consider the function Rewrite f in the form b a T a 1 bx 2 What are a a and b f x Find T4 a the fourth degree Taylor polynomial for f centered at 0 4 5 202

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

Recall that for 1 x 1 the Taylor expansion for f x a lim 1 1 c which is also known as a geometric series Use this result to calculate the following limits b 11 400 1 1 1 3 3 3 3 4 8 3 16 32 64 9 27 81 1 1 z 1 3 n centred at z 0 is given by 1 x x x x Note Both of these objects are generally referred to as infinite sums or series The first expression explicitly gives the nth term and asks you to calculate the limit as noo whereas the second implicitly suggests via the that what you are calculating is an infinite sum of terms Althoug they look different they are asking for the same thing for you to calculate the sum of all their infinite terms

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

Read the following lines and answer the question that follows I can t wait to hear from you Which of the following is used to convey the same in the formal style OI look forward hearing from you O I am eager to hear from you O I think you will reply soon O I cannot wait to hear from you

Read the following dialogue and answer the question that follows Huck have you ever told anybody about that Bout what You know what Oh course I haven t Which of the following would you consider the dialogue to be O Quick paced O Lacks emotion O Full of information

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

Which of the following is are NOT TRUE O Narrative writing tells a made up story O Narrative nonfiction is also called biographical writing First person pronouns are used in biographies O Personal narratives talk about real events and people

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

Suppose that the second order Taylor polynomial of some function f about x 0 is given by x x 7 24 How consider the function 5 x 3 T x 3 Find S x the second order Taylor Polynomial for g about x 0 g x f x e x 2347 168

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 graph of f not f is given below At which of the marked values of a is A f x greatest x B f x least x c f x greatest x D f x least x E f x greatest F f x least x x1 x2 x3 x4 x5 x6 X

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

The hyperbolic cosine function cosh x is defined as follows cosh x Find the sixth degree Taylor Polynomial To x for cosh 4x about x 0 To x x 2 x 1 x6 e ex 2 X x5

Given that the Taylor Polynomial for f x e of degree 4 about x 0 is x 2 3 2 3 4 e of degree 15 about x 0 Enter the terms in order of increasing degree find the Taylor Polynomial for g x g x 22 B 2 T 89 T C f x 1 x Hint You could calculate this manually but is there an easier way How do we know the polynomial only has five terms Now find each of the following

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 degree 3 Taylor polynomial Ts a of the function about 2 T3 x f x 3x 2 2

Find the Taylor polynomial for the function f x 10x x about the point x 5 that has three nonzero terms Enter the polynomial terms in order of increasing degree 10x x x 5

Let T3 a be the Taylor Polynomial of degree three for f z centered at 0 If f 0 15 f 0 14 f 0 11 T3 x f 0 5

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

6 7 2965 14 3 2 14 1 24 3 4 5 6 What is the period for the graph show above What is the vertical stretch for the graph shown above Write an equation for the function graphed above Use y for the output

Rewrite the following equation in logarithmic form using a base 10 logarithm Use log for the base 10 logarithm b 10

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

A particular computing company finds that its weekly profit in dollars from the production and sale of x laptop computers is P x 0 007x 0 2x 700x 900 Currently the company builds and sells 11 laptops weekly a What is the current weekly profit b How much profit would be lost if production and sales dropped to 10 laptops weekly c What is the marginal profit when x 11 d Use the answer from part a and c to estimate the profit resulting from the production and sale of 12 laptops weekly

Consider the function f x 11 x Find the Taylor polynomial of degree 1 for f about x 14 y Again note that this is just the linear approximation of fat x a

Using a calculator or computer generate a graph like the one shown below by graphing y 2 16 8x 8 and y 2x for 1 x 1 What is the relationship between y 2 16 8x 8 and y 2x Use your graph to estimate to one decimal place the largest magnitude of the error in approximating 2 16 8x 8 by 2x for 1 x 1 error Is the approximation an over or an underestimate when x 0 Enter over or under BEB

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

The graph of f not f is given below At which of the marked values of is A f x greatest x B f x least x c f x greatest D f x least x E f x greatest F f x least x x1 x2 x3 x4 x5 x6 X