Calculus Questions

WW7 Problem 17 1 point Consider the following graph of a derivative f a At what x values do the points of inflection of f occur 3 b On which interval s is f concave down f is concave down for in Click on the graph to enlarge Enter your answer as a value or a comma separated list 1 0 Enter your answer using interval notation 1 1

Find two positive integers such that their sum is 64 and a maximize and b minimize the sum of their squares The two positive integers that maximize the sum of their squares are help numbers The two positive integers that minimize the sum of their squares are help numbers

You are building five identical pens adjacent to each other with a total area of 1400m as shown in the figure below What dimensions should you use to minimize the amount of fencing

You are constructing an open top box for your cat to sleep in The plush material for the square bottom of the box costs 4 ft and the material for the sides costs 4 ft You need a box with volume 4 ft Find the dimensions of the box that will minimize the cost The dimensions of the box that will minimize the cost are feet enter a comma separated list help numbers

You have 790 ft of fencing to make a pen for hogs You have a river on one side of your property and will use the river as one side of the pen You wa to maximize the area of the pen using all of the available fencing Use the picture below for this problem W The area of the pen as a function of its length 1 is A 1 ft For what value of I will A l 0 The area of the pen as a function of its width w is A w ft For what value of w will A w 0 W

You need to construct fence around an area of 2601 ft What are the dimensions of the rectangular pen that minimize the amount of material needed ft by

A company plans to manufacture a rectangular box with a square base an open top and a volume of 108 cm Determine the dimensions of the box that will minimize the surface area What is the minimum surface area Enter only the minimum surface area and do not include units in your answer

Question 4 Find all zeros of the function f x 3x 24x 51x30 Enter the zeros separated by commas Question Help Video Message instructor

A box with a square base and open top must have a volume of 119164 cm We wish to find the dimensions of the box that minimize the amount of material used First find a formula for the surface area of the box in terms of only x the length of one side of the square base Hint use the volume formula to express the height of the box in terms of x Simplify your formula as much as possible A x Next find the derivative A x A x Now calculate when the derivative equals zero that is when A x 0 Hint multiply both sides by x A x 0 when

A cylinder shaped can needs to be constructed to hold 200 cubic centimeters of soup The material for the sides of the can costs 0 04 cents per square centimeter The material for the top and bottom of the can need to be thicker and costs 0 06 cents per square centimeter Find the dimensions for the can that will minimize production cost Helpful information h height of can r radius of can Volume of a cylinder V h Area of the sides A 2 rh Area of the top bottom A To minimize the cost of the can Radius of the can Height of the can Minimum cost cents

A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle Find the area of the largest possible Norman window with a perimeter of 20 feet Round your answer to one place after the decimal square feet

A 22 g bullet traveling 255 m s penetrates a 2 0 kg block of wood and emerges going 150 m s Part A If the block is stationary on a frictionless surface when hit how fast does it move after the bullet emerges Express your answer to two significant figures and include the appropriate units

An object at rest is suddenly broken apart into fragments A and B by an explosion The fragment A acquires six times the kinetic energy of the fragment B Part A What is the ratio of their masses Express your answer using two significant figures VAX

r 30 O A 20 20 20 OB 20 20 20 O C 20 20 20 O D 20 20 20

75057 20 sos Choose the correct graph below O A 5 AY 5 5 OB 5 5 O C O D 5 5 5

Identify the symmetries of the curve below Then sketch the curve r sin 2 Identify the symmetries of the curve Select all that apply A The curve is symmetric about the y axis B The curve is symmetric about the origin C The curve is symmetric about the x axis

r 5 cos 50 O A 10 10 OB 10 10 10 O C 10 10 10 OD 10 10 10

Find the slope of the curve below at the given points Sketch the curve along with its tangent lines at these points r 9 9 cos 0 0 The slope at the point 0 Simplify your answer B TU RIN 2 is

5 A table of selected values is given for a one to one function g At z 5 g has a vertical asymptote x g x 5 undefined 2 14 0 11 1 0 3 6 a What is the y intercept of g 9 b Write the equation for the horizo

The figure below gives the behavior of the derivative of a function g x on 2 x 2 Notice that no scale is given on the y axis on the graph a At what x values does the graph of g x have inflection points X minimum at x b What x values give the global maxima and minima of g on 2 2 maximum at x Graph of g x Click on the graph to get a larger version Enter your answer as a comma separated list of values or enter none if there are none c If g 2 5 what are possible values for g 0 653 g 0 is in Enter your answer as an interval or union of intervals giving the possible values Thus if you know 3 g 0 1 enter 3 1 Enter infinity for co the interval 3 3 to indicate a single point d How is the value of g 2 related to the value of g 0

Let f x Find the domain and range of f 1 Is it possible to do this without finding the expression for f Explain 1 x 5

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 Let which has the following derivatives f x 2xe x 4 e f x 2e 2xe 2xe x 4 e a Find the locations the x values of the points of inflection b Determine the x intervals on which f is concave up or down The left most interval is The middle interval is The right most interval is f x x 4 e and on this interval f is and on this interval f is I and on this interval f Concave Down

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 x e 2 4 a List the location the x values of all the critical points of f x b Use interval notation to indicate where f x is increasing Increasing for in c Use interval notation to indicate where f x is decreasing Decreasing for a in d List the x values of all local maxima of f x values of local maxima e List the x values of all local mimima of f x values of local minima f List the x values of all the inflection points of f values of inflection points g Use interval notation to indicate where f x is concave up Concave up for a in h Use interval notation to indicate where f x is concave down BAY

3 Find the inverse of y x 3 and state its domain and range

1 Consider the function B t that gives the temperature of Boston Massachusetts on a particular day where t is measured in hours after midnight Is B t a one to one function How do you know

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 pe INF for oo and INF for 0 Consider the function f x x 4x 18x 15 a List the locations the x values of the critical points of f x b List the x values of all local maxima of f x values of local maxima c List the x values of all local mimima of f x values of local minima f List the x values of all the inflection points of f values of inflection points g Use interval notation to indicate where f x is concave up Concave up for a in h Use interval notation to indicate where f x is concave down Concave down for in

2 The graph of y f x is shown below on the ay plane for 6 6 Is fa one to one function on its domain Explain 10

Owners of a bike rental company that charges customers between 5 and 25 per day have determined that the number of bikes rented per day n can be modeled by the linear function n p 200 8p where p is the daily rental charge How much should the company charge each customer per day to maximize revenue Do not include units or a dollar sign in your answer

Consider the following graph of a derivative f a At what a values do the points of inflection of f occur Click on the graph to enlarge it Enter your answer as a value or a comma separated list of values or NONE b On which interval s is f concave down Enter your answer using interval notation

Consider the function f x 2 sin x 3 2 This function has two inflection points which occur at the x values A and B where 0 A B 2T A and B In the interval 0 A the function fis Concave up Concave down In the interval A B the function fis Concave up Concave down In the interval B 2T the function fis Concave up Concave down x

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 x x 4 5x 12x 1 and its derivatives f x 3x 9x 12 f x 6x 9 a Determine the x intervals on which f is concave up and concave down f is concave up on f is concave down on b What are the x y coordinates of the inflection points of f Enter your answer as an x y coordinate e g 91 13 and separate multiple answers by commas

Suppose that f x x 3x a List the x values of all critical points of f x 3 Use interval notation to indicate where f x is increasing A function is increasing if larger x values give larger y values Don t include points where the function changes from increasing to decreasing or from decreasing to increasing Increasing for a in C Use interval notation to indicate where f x is decreasing A function is decreasing if larger x values give smaller y values Don t include points where the function changes from increasing to decreasing or from decreasing to increasing Decreasing for a in D List the x values of all local maxima of f x values of local maxima E List the x values of all local minima of f x x values of local minima F Use interval notation to indicate where f x is concave up Concave up for a in G Use interval notation to indicate where f x is concave down Concave down for a in H List the revalues of all the inflection points of f

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 co and INF for 00 Let A Use interval notation to indicate where f x is increasing Increasing for a in B Use interval notation to indicate where f x is decreasing Decreasing for a in C List the values of all local maxima of f values of local maxima D List the values of all local minima of f values of local minima 4 f x 4x

A company plans to manufacture a rectangular bin with a square base an open top and a volume of 800 cm The cost of the material for the base is 0 1 cents per square centimeter and the cost of the material for the sides is 0 5 cents per square centimeter Determine the dimensions of the bin that will minimize the cost of manufacturing it What is the minimum cost Round any calculations and the final answer to the nearest hundredth if necessary

8 The profit a company makes by manufacturing and selling a thromdimbulators is given in dollars by p x 0 1 100a 2000 Current production levels are a 300 units a Find the linearization L a to p x at 300 b Use the linearization to estimate the new profit obtained if production can be raised from 800 to 10 units

1 QUESTION 6 2 points Let F be an antiderivataive of f x 2 e such that F 0 3 Then find F 4

QUESTION 9 3pts Consider f x 1 x 3 a USE Only The Definition of the DERIVATIVE to find f 1 b Find the equation of the tangent line to the graph of f at the point 1 f 1

QUESTION 16 3 points Find cos x sin x 6 da

818 2 7 0002 9

QUESTION 3 2 points Find the derivative of f x In cosa 2 1 x 2 4e

QUESTION 5 2 points Consider the function f x 23 24x 2 defined on the closed bounded interval 3 3 Find its Global absolute maximum and find its Global absolute minimum

QUESTION 6 2 points Let F be an antiderivataive of f x 1 antiderivataive of f x e find F 4 e 1 such that F 0 3 Then

QUESTION 2 2 points Information on functions f z g z f z and g x for some z is given in the table Let h r gof x I g 1 Find A 2 and h 2 1 2 4 3 3 2 4 3 1 2 3 5 2 4 3 4 3 1 1 4

Find the slopes of the curve 4 y y x at the two points shown here 43 4 3 Ay 3 4 The slope of the curve y y x at the point 3 4 is 2 Type an exact answer using radicals as needed

oblem 5 a Find the value of c that satisfies the conclusion of the Mean Value Theorem for the function f x ln x on the interval 2 16 b If f 1 4 and f x 9 for 1 x 9 then according to the Mean Value Theorem what is the smallest possible value for f 9 c Let f be the function whose graph is shown below y 3 2 0 0 N 3 4 1 2 5 6 7 8 X Estimate the two values of c that satisfy the conclusion of the Mean Value Theorem on the interval 2 7

blem 4 Consider the function f given below x 1 if 1 x 2 x 3 4 if 2 x 5 Which of the following statements is true f x A f has both an absolute maximum and an absolute minimum B f has an absolute maximum but no absolute minimum C f has an absolute minimum but no absolute maximum D f has no absolute maximum and no absolute minimum

An inverted cylindrical cone 24 m deep and 12 m in diameter across the top is being filled with water at a constant rate of 16 m min At what rate is the water rising in the tank when the depth of the water is 1 meter Answer 10 meters Answer 23 meters Answer

5 0 5 5 Shown above is the graph of two functions Both are defined for a 0 a The red line shows the function 3 10 or f x 10g 2 b Find the three values of a at which f x g x Give your answers in order from smallest to largest and g z log z or c Find the two values of a at which f x and g x have parallel tangent lines Give your answers in order from smallest to la

roblem 7 Let f x x In x On what interval is f concave down A e72 17 0 B 0 el7 72 C 0 e72 17 D 0 E e 17 72 0 F 0 e 17 72 G H e 72 17 00 7 72

At noon ship A is 50km west of ship B Ship A is sailing south at 25km h and ship B is sailing north at 30km h How fast is the distance between the ships changing at 4 00pm The distance between the ships is changing at a rate of kilometers per hour