Limits & Continuity Questions and Answers

A zoo supplier is building a glass walled terrarium whose interior volume is to be 128 ft Material costs per square foot are estimated as shown below 1 00 Walls Floor Ceiling What dimensions of the terrarium will minimize the total cost What is the minimum cost X ft y Z ft 2 00 2 00 ft

Use the Comparison Test to determine if the series converges or diverges 3 En 13 n 4 n

An athletic field consists of a rectangle with a semicircle at each end as shown below A 880 meter track runs along the perimeter of the field Determine the dimensions of the inner rectangle of greatest area D Enter the solutions using a comma separated list

An 11 foot tall fence runs parallel to a building which is situated 10 feet away from the fence Determine the minimum length of a ladder that can reach the house from the ground on the opposite side of the fence while clearing the fence Round the solution to the nearest ten thousandth

Solve for the variable P to two decimal places 69 100 P 0 074 6 P Round to two decimal places as needed

Johan would like to determine the largest rectangle that can be inscribed in a semicircle of radius 525 Determine the dimensions of such a rectangle Width y r x T T

1 point If tan 0 1 0 0 2 then sin 0 equals cos 0 equals sec 0 equals

0 Let f x y 1 1 x y for x y 00 00 and let Da x y x y a for a 0 Which of the following statement is truc a SD f x y d x y ln 2 b SD f x y d x y ln 4 c JD f x y d x y r ln 6 d Sp f x y d x y ln 8 e None of the above statements is true

3 Define a sequence an as follows Let a Which of the following statement is true a limnoo an 1 b limnan 0 c limno an n 0 d 1 nan 0 c None of the above statements is truc 1 and an 1 0 5an 1 n for n 2

12 30 points Let f x F x and h a be defined as the following 2 e 2 1 TX f f t dt f x F x h x a 15 points Find limo h x b 15 points Find limr o h x f x 1 F x

Each unit of a product can be made on either machine A or machine B The nature of the machines makes their cost functions differ Machine A Machine B C x 20 C y 220 9 Total cost is given by C x y C x C y How many units should be made on each machine in order to minimize total costs if x y 25 760 units are required The minimum total cost is achieved when units are produced on machine A and Simplify your answer units are produced on machine B

A lamina occupies the part of the disk x y 9 in the first quadrant and the density at each point is given by the function p x y 4 x y A What is the total mass 81pi 2 B What is the moment about the y axis My ff xp x y dxdy C What is the moment about the x axis Mx ff yp x y dxdy D Where is the center of mass 1 527887 E What is the moment of inertia about the origin ff x y p x y dxdy

6 Let f x y sin x y x y for z y 00 00 Which of the following statement is true f x y lim 1 3 0 0 x x y f x y dy x cos x a x cos x ar 1 b c f x y 2x cos x y d The tagent plane to the surface z f x y at the point 0 0 0 is z 2x y e None of the above statements is truc cos x y sin x y 1 for a 0

A lamina occupies the part of the disk x y 9 in the 2 first quadrant and the density at each point is given by the function p x y 4 x y A What is the total mass 81pi 2 B What is the moment about the y axis My ff xp x y dxdy C What is the moment about the x axis Mx ff yp x y dxdy D Where is the center of mass 1 527887 E What is the moment of inertia about the origin f x y p x y dxdy

Use the graph of the third degree polynomial and the factor x2 2x 4 to write the factored form of the polynomial suggested by the grap The leading coefficient is 1 y 9 6 3 f x SKADEN 4 3 2 1 0 1 2 3 3 6 9 x 1 x 1 N MASEL BANTAY X 4 X

3 Suppose that is a differentiable function such that f x f x and f x 0 for all IE 00 00 Which of the followig statement is false a f x f 0 e for a 00 00 b f x f x for x E 00 00 c f is strictly increasing on 0 d in f x for x 00 00 Then g is a constant function on 0 00 d Let g x c lim f x 0

Let f x 2x x 1 x 2 Determine the zeros multiplicities of the zeros and behavior of the graph at zero List the zeros from least to greatest Zero Multiplicity The behavior of Graph at Zero If it crosses the x axis type cross in the answer box If it turns around and doesn t cross the x axis type turn in the answer box

The graph of a degree 4 polynomial function is given below f x by Find the function in the factored form Hint List the zeros from least to greatest 0 12 x x

The graph of a polynomial of degree 5 is given t 5 Zeros Select 10 10 O 6 10 5 Factored form of the polynomial Select

Determine whether the following function is a power or polynomial function If it is a power function identify the power If it is a polynomial function identify the degree of the polynomial Function f x 3 f x dre f x 2 3x 5 4 f x 5x4 action 3x x Power or Polynomial 2 Select Select Select Select V V 5 Power or degree Select Select Select Select

Find the directional derivative of the function at the given point in the direction of the vector v 0 5 v 10 24 f x y 7e sin y Dur 0

Use the demand equation to find the revenue function Graph the revenue function and indicate the regions of inelastic and elastic demand on the graph x f p 35 p 18 or The revenue function is R p Graph the revenue function and indicate the regions of inelastic and elastic growth Choose the correct graph below O A AR P 312stic 0 0 Inelastic 18 Q Q 5 OB AR p 312plastic 0 6 Elastic 18 Q E O C AR p 3020 Inelastic 0 0 1 9 Elastic 18 Q Q 5 OD AR p Elastic 3020 0 0 9 Inelastic 18 Q 5

Use the given equation which expresses price p as a function of demand x to find a function f p that expresses demand x as a function of price p Give the doma of f p p 30 0 5x 0 x 60 x f p ps

Give the domain of f the domain of g and the domain of m where m x f g x 2 f u In u g x 9 x The domain of f is Type your answer in interval notation The domain of g is Type your answer in interval notation The domain of m is Type your answer in interval notation

Find the absolute maximum and minimum values of fon the set D f x y 2x y 2 D x y x y 1 absolute maximum value absolute minimum value

The one to one function h is defined below h x 7x 3 5x 2 1 Find h x where h is the inverse of h 1 Also state the domain and range of h in interval notation n x 0 Domain of h 1 010 0 0 0 0 OUD 0 0 0 0 8 8 C

Find all solutions that satisfy the equation sin t 36 Assume 0 t 2

Find all solutions that satisfy the equation cos t 91 Assume 0 t 2

Match each statement with the expression that completes it for the given graph Assume the graph extends infinitely in each direction Drag and drop options on the right hand side and submit For keyboard navigation SHOW MORE Asx x y Asx 00 Y 2 2

Use the graph to determine where the function is discontinuous JIL O At 1 O At 2 5 7 777 On the interval 0 1

Suppose 2y sin 8 t 6 2 Answer the following questions What is the period of the function What is the horizontal shift of the function What is the phase shift of the function

Evaluate the limit lim x 9 x 9 T 3 x

Find the limit lim I 1 03 Does Not Exist 02 I 1 2 if it exists

You are given that lim f x 3 lim g x 4 and Z a za lim h x x g x lim ha I a 022 0 17 0 8 2 Find the limit

Question 5 1 point Listen Find the limit lim 2 1 O 0 3 Does Not Exist 02 2 1 if it exists

Find the derivative of the function f x 8x 3 40x 8x 3 Ost 8x 3 40 8x 3 5 8x 3

Find the limit lim 2 4 3 O 17 2 O 04 102 4x 1 3x 2

Find the limit lim h 0 Does not exist 0 O 75 2 5 h 2 5 5 h 75 h if it exists

Govan f 0 2 1 Find lim 32 2 ifz 1 3 f ifr z 1 Oo 0 3 01 Does Not Exist

Consider the function f x 5 x 8 for the domain 5 Find f x where f 1 Also state the domain of f in interval notation 1 f x for the domain is the inverse of f 2 C

Use the given graph to find the indicated quantities If limit does not exist say DNE a lim fr I 3 b mx 2 c f 3 P

Solve for a in the following triangle Note Figure is not drawn to scale B A 5 I x 1 C

The one to one functions g and h are defined as follows g 1 2 1 9 2 3 3 9 7 8 h x X 3 11 Find the following g 2 0 h x 0 0 D hh 010 X S 2

Find an exponential function that fits the experimental data collected over time t 0 1 2 3 4 1000 900 00 810 00 729 00 656 10 y

sin 0 cos 0 tan 0 csc 0 sec 0 cot 0 X X X X ometric ratios of the angle 0 in the triangle 25 24 7

Use traces to sketch the surface If an answer does not exist enter DNE Select Update Graph to see your response plotted on the screen Select the Submit button to grade your response y 2z 2x Write an equation for the cross section at z 1 using x and y Write an equation for the cross section at z 0 using x and y Write an equation for the cross section at z 1 using x and y Write an equation for the cross section at y 1 using x and z Write an equation for the cross section at y 0 using x and z Write an equation for the cross section at y 1 using x and z Write an equation for the cross section at x 1 using y and z Write an equation for the cross section at x 0 using y and z Write an equation for the cross section at x 1 using y and z

limt 00 e 4t 2 1 2 1 Find the limit lim

3 Evaluate the following limits You must state the indeterminate form before applying L Hopital s Rule I a 1 1 2 a lim b lim z ln x I 40 Hint x In x c lim COS I d lim where a is a non zero real number In r In cos r x Hint You will need to apply L Hopital s rule twice Recall that tan 0 0 and sec 0 1 e lim 1 x Hint on next page I 0 1

Solve the optimal maxu t Sit 3 Sce ent control problem 1 2x t 2u t dt x t 4u t X101 1 x 2 free

A water sprinkler sends water out in a circular pattern Determine how large an area is watered The radius of watering is 18 ft Use 3 14 for CICIELI The area of the circle is ft Type a whole number or decimal rounded to the nearest hundredth as needed