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  5. january 2018
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Mathematics Questions and Answers January 5, 2018

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Evaluate the limit . lim x 1 x 5 1 x 3 1

integral of ln ( 2 x / pi )

Find the center of mass of a thin triangular plate bounded by the y - axis and the lines y = x and y = 2 - x if its density is ( x , y ) = 6 x + 3 y + 3 .
Find the center of mass of a thin triangular

Can the first order reaction come to an end? If it can then at what time?

If the density of the solid region E at any point ( x , y , z ) is given by ( x , y , z ) = x y , find the total mass of the solid below.
If the density of the solid region E at any point

x equals fraction numerator negative 5 plus - or - minus 3 square root of 5 over denominator 5 end fraction

Find F ( s ) . { te ^ 8 t }

Calculate the derivative ddx x 1 ( 3 t 5 t ) 2 1 dt

Find the unique solution to the function below, if you know f ( 1 ) = 6 : f ( x ) = ( x 4 + x 2 - 6 ) d x f ( x ) =
Find the unique solution to the function below,

True or false a b f ' ( x ) [ f ( x ) ] 4 d x = 1 5 [ f ( x ) ] 5 + C
True or false a b f ' ( x ) [ f ( x ) ] 4 d x = 1

sum i = 0 to n - 1 ( ( - 1 ) ^ i ) / ( i + 1 )

p ( t ) = 1 / 1 + ae ^ kt When would half the population have heard the rumor?

For each of the following series determine if the series converges or diverges. If the series converges, give its value. a ) n = 1 ( 1 - s i n ( n ) ) ( 1 + s i n ( n ) ) n 2 + 8 n + 1 b ) n = 4 4 1 + 2 n 3 2 - 3 n
For each of the following series determine if the

Problem 4 . ( 1 point ) The volume of the solid obtained by rotating the region bounded by y = x 2 , y = 4 x about the line x = 4 can be computed using the method of washers via an integral V = a b ? with limits of integration a = and b = The volume of this solid can also be computed using cylindrical shells via an integral V = with limits of
Problem 4 . ( 1 point ) The volume of the solid

The temperature at a point ( x , y , z ) is given by T ( x , y , z ) = 2 0 0 e x 2 3 y 2 7 z 2 where T is measured in C and x , y , z in meters. ( a ) Find the rate of change of temperature at the point P ( 2 , 1 , 2 ) in the direction towards the point ( 6 , 3 , 3 ) . C / m ( b ) In which direction does the temperature increase fastest at P ? ( 8

Find the future value of the income ( in dollars ) given by f ( t ) over t 1 years at annual interest rate r . If the function f represents a continuous investment over a period of t 1 years at an annual interest rate r ( in decimal form ) compounded continuously, then the future value of the investment is given by Future value = ert 1 t 1 0 f ( t

Let S be the upwards - oriented surface of the square pyramid with vertices ( 0 , 0 , 1 ) , ( 1 , 1 , 0 ) , ( 1 , - 1 , 0 ) , ( - 1 , 1 , 0 ) , and ( - 1 , - 1 , 0 ) which is above the x y - plane, not including the base. Given that ( y 2 + z 2 , x 2 + z 2 , x 2 + y 2 : ) = curl vec ( G ) for some vector field vec ( G ) , calculate S ( : y 2 + z 2
Let S be the upwards - oriented surface of the

Calculate the integral: 4 x 4 3 x 3 + 3 x 2 + 2 x 2 ( x 1 ) dx

Rewrite the inequality 3 x - 2 y > = 1 2 in standard form.

Use Stokes' theorem to evaluate C F * d r where C is oriented counterclockwise as viewed from above. F ( x , y , z ) = xyi + 5 z j + 7 y k , C is the curve of intersection of the plane x + z = 5 and the cylinder x 2 + y 2 = 3 6
Use Stokes' theorem to evaluate C F * d r where C

Use part I of the Fundamental Theorem of Calculus to find the derivative of f ( x ) = int - 5 , x Square root of t ^ 3 + 5 ^ 3 dt

Compute the surface integral over the given oriented surface: F = y 4 i + 4 j - x k , portion of the plane x + y + z = 1 in the octant x , y , z 0 , downward - pointing normal S F * d S =
Compute the surface integral over the given

Simplify the rational expression. If the rational expression cannot be simplified, so state. ( y ^ ( 2 ) - 1 0 y + 1 6 ) / ( y ^ ( 2 ) - 1 7 y + 7 2 )

Use a suitable change of variables to determine the indefinite integral. ( Use C for the constant of integration. ) ( 1 6 x - 9 ) - 3 d x
Use a suitable change of variables to determine

Find the Fourier series expansion fot the function f F ( x ) = n = 1 2 n ( 1 + ( - 1 ) n + 1 ) s i n ( n x L ) f F ( x ) = n = 1 ( - 1 ) n n s i n ( n x L )
Find the Fourier series expansion fot the

( 4 pts ) If lim n a n = 0 , then
( 4 pts ) If lim n a n = 0 , then

Use the component form to generate an equation for the plane through P 1 ( 4 , 1 , 5 ) normal to n 1 = i - 2 j + k . Then generate another equation for the same plane using the point P 2 ( 3 , - 2 , 0 ) and the normal vector n 2 = - 2 2 i + 2 2 2 j - 2 2 k .
Use the component form to generate an equation

Question content area top Part 1 Evaluate the following limit . Use l ' H pital ' s Rule when it is convenient and applicable. ModifyingBelow lim With x right arrow 0 limx 0 StartFraction 1 minus cosine 5 x Over 1 1 x squared EndFraction

( 9 points ) Consider the integral I = 0 1 x a r c t a n ( x ) d x . ( a ) Write explicitly an approximation for I using the right - hand sum with n - slices. You can use the sigma notation or the . . . notation. If you use notation such as x or x k , be sure to explain exactly what it means. ( b ) Find a number of slices for which your
( 9 points ) Consider the integral I = 0 1 x a r

Verify that the given differential equation is exact, then solve it: ( x + t a n - 1 y ) d x + x + y 1 + y 2 d y = 0
Verify that the given differential equation is

Rational functions v and w both have a point of discontinuity at x = 7 . Which equation could represent function w ? A . w ( x ) = v ( x 7 ) B . w ( x ) = v ( x + 7 ) C . w ( x ) = v ( x 7 ) + 7 D . w ( x ) = v ( x ) + 7

If a > 0 , then show that lim ( 1 / 1 + na ) = 0

Solve this Differential equation
Solve this Differential equation

lim ( x , y ) ( 1 , - 1 ) s i n ( x + y ) + e x + y x 2 + y 2 + 2 2 = A . 1 2 B . 1 C . 0 D . 1 2 2
lim ( x , y ) ( 1 , - 1 ) s i n ( x + y ) + e x +

lim n 0 n + n + n + . . . 2 2 2 = ? ( A ) 0 ( B ) 1 ( C ) e ( D )
lim n 0 n + n + n + . . . 2 2 2 = ? ( A ) 0 ( B )

Factor completely if possible. Remember that you can check by multiplying. s ^ ( 2 ) + 7 st + 7 t ^ ( 2 )

Evaluate the double integral I = A 5 + x 2 1 + y 2 d x d y when A = { ( x , y ) : 0 x 1 , 0 y 1 } .
Evaluate the double integral I = A 5 + x 2 1 + y

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. y = 5 u 3 1 + u 2 du 1 3 x

Consider the indefinite integral x + 4 ( 4 x + 5 ) 2 d x . The substitution u = 4 x + 5 transforms the integral into: d u ( This answer must be a function of u . ) Note: You are not asked to evaluate the integral.
Consider the indefinite integral x + 4 ( 4 x + 5

f ( x ) = arcsec ( x ) - 6 x find f ' ( x )

Locate and classify any critical points. ( Round your numerical answers to three decimal places. Order your answers from smallest to largest w . ) h ( w , z ) = 0 . 8 w 2 + 1 . 2 z 3 4 . 6 wz ( w , z , h ( w , z ) ) = Select Classification relative maximum relative minimum saddle point ( w , z , h ( w , z ) ) =

( a ) Approximate f by a Taylor polynomial with degree n at the number a . ( b ) Use Taylor's Inequality to estimate the accuracy of the approximation f ( x ) ~~ T n ( x ) when x lies in the given interval. ( c ) Check your result in part ( b ) by graphing | R n ( x ) | . 1 3 . f ( x ) = 1 x , a = 1 , n = 2 , 0 . 7 x 1 . 3 1 4 . f ( x ) = x - 1 2
( a ) Approximate f by a Taylor polynomial with

1 I Write down the equation in slope intercept form if slope m = 4 and y - intercept is 2 . 5

The answer above is NOT correct. ( 1 point ) If f ( x ) = 6 t a n - 1 ( 3 s i n ( 4 x ) ) , find f ' ( x ) .
The answer above is NOT correct. ( 1 point ) If f

Which of the following are Taylor series representations of f ( x ) = 1 x 3 , a = 2 Check ALL that apply n = 0 ( - 1 ) n 3 * 4 * c d o t s * ( n + 2 ) n ! 2 n + 3 ( x - 2 ) n n = 0 ( - 1 ) n 2 * 3 * 4 * d o t s * ( n + 2 ) n ! 2 n + 4 ( x - 2 ) n n = 0 ( - 1 ) n ( n + 2 ) n ! 2 n + 3 ( x - 2 ) n n = 0 ( - 1 ) n ( n + 2 ) ( n + 1 ) 2 n + 4 ( x - 2
Which of the following are Taylor series

derivative of integral sqrt ( 2 + ( cos ( t ) ) ^ 3 ) dt ) with lower limit 2 and upper limit x

( 1 0 points ) Find a power series representation for the following function and determine the interval of convergence f ( x ) = 5 x + 3 4 x 2 + 3 x - 1
( 1 0 points ) Find a power series representation

The graph of f ' , the derivative of f , is shown to the right. Identify the intervals when f is increasing and decreasing. Include a justification statement. Increasing: Decreasing:
The graph of f ' , the derivative of f , is shown

Evaluate the line integral in Stokes' Theorem to evaluate the surface integral S ( g r a d F ) * n d S . Assume that n points in an upward direction. F = ( : x + y , y + z , z + x : ) ; S is the tilted disk enclosed by r ( t ) = ( : 5 c o s t , 6 s i n t , 1 1 2 c o s t : ) . Find the value of the surface integral. S ( g r a d F ) * n d S = ( Type
Evaluate the line integral in Stokes' Theorem to

The equation 4 x 2 + 9 y 2 - 3 6 = 0 represents an ellipse in standard position. Find the coordinates of the foci and the equations of the directrices. Give your answers exactly. The foci are ( + - f , 0 ) where f > 0 and f = The directrices are x = + - d where d > 0 and d =
The equation 4 x 2 + 9 y 2 - 3 6 = 0 represents

Evaluate the definite integral below. 1 0 3 x 3 4 x 4 + 2 dx

Let f ( t ) = { 3 t 0 t 2 sin ( 4 t ) 2 < t 4 e 6 ( t 4 ) t > 4 ( a ) f ( t ) can be written in the form g 1 ( t ) + g 2 ( t ) U ( t 2 ) + g 3 ( t ) U ( t 4 ) where U ( t ) is the Heaviside function. Enter the functions g 1 ( t ) , g 2 ( t ) , and g 3 ( t ) , into the answer box below ( in that order ) , separated with commas. ( b ) Compute the

Use logarithmic differentiation to find d / dx ( y ) Express your answer in terms of just x . y = ( ( x + 3 ) ^ 3 * ( x + 4 ) ^ 8 ) / ( ( x - 7 ) ^ 9 )

Solve the differential equation. y 4 d y d x = 3 x 2 y 5 - 6 x 2 The solution is ( Type an equation. )
Solve the differential equation. y 4 d y d x = 3

Given l n y = - a x + b x + c , what is d y d x ? A , b , and c are constants. Select one or more: A . d y d x = - a - b x 2 B . d y d x = - a x 2 - b x 2 C . d y d x = - a x - b x D . d y d x = - a x + b x
Given l n y = - a x + b x + c , what is d y d x ?

# 5 - Find the volume of the solid when the area is rotated around the x axis :
# 5 - Find the volume of the solid when the area

A very large tank initially contains 1 0 0 L of pure water. Starting at time t = 0 a solution with a salt concentration of 0 . 7 k g L is added at a rate of 6 L m i n . The solution is kept thoroughly mixed and is drained from the tank at a rate of 4 L m i n . Answer the following questions. Let y ( t ) be the amount of salt ( in kilograms ) in
A very large tank initially contains 1 0 0 L of

Evaluate lim 4 2 1 6 2 9 + 2 0 .

lim x 0 + l n ( s i n x ) l n ( t a n x ) Use L'hospitals rule
lim x 0 + l n ( s i n x ) l n ( t a n x ) Use

( 1 ) A culture initially has P 0 number of bacteria. At t = 1 hour the number of bacteria is measured to be 3 2 P 0 . If the rate of growth is proportional to the number of bacteria P ( t ) present at time t , determine the time necessary for the number of bacteria to triple.
( 1 ) A culture initially has P 0 number of

Problem 5 . Prove that the limits do not exist: ( b ) lim x 0 s i n ( 1 x ) .
Problem 5 . Prove that the limits do not exist: (

Perform the integration. ( If necessary, use C to represent a constant. ) t a n 2 ( x ) c s c 3 ( x ) s e c 3 ( x ) d x =
Perform the integration. ( If necessary, use C to

Find the integral. ( c o s ( t ) ) 5 s i n ( t ) 2 d t
Find the integral. ( c o s ( t ) ) 5 s i n ( t )

How to find the axis of symetry

find the indicated derivative dy / dx if y = 2 x ^ ( ( 3 ) / ( 2 ) )

Consider the series n = 0 x n n ! a . Find the radius of convergence. b . Find S 6 ( x ) . c . Evaluate the polynomial you found in ( b ) at x = 1 d . Find lim n S n ( 1 ) accurate to 6 decimal places. ( Hint: Keep find sums until the 1 s t six decimal places stabilize )
Consider the series n = 0 x n n ! a . Find the

What yourre solving for r . What's given in the problem ( 6 ) / ( r ^ ( 2 ) ) - ( 1 ) / ( 7 ) = ( 1 ) / ( r ^ ( 2 ) ) How to solve

2 x 2 + 1 2

A city has 3 5 0 0 0 residents and is adding new residents steadily at the rate of 1 5 0 0 per year. If the proportion of residents who remain after t years is given by S ( t ) = 1 / ( t + 1 ) , what is the population of the city six years from now?

Write out the first few terms of the sequence given by a 1 an 3 ; 2 an 1 + 4 . Then find a recursive definition for the sequence 1 0 2 4 5 2 1 0 8

Find the average value, ? b a r ( I ) , of the function f ( x , y ) = 3 ( x + y ) l n x over the rectangle A = { ( x , y ) : 1 x 3 , 0 y ' 3 } .
Find the average value, ? b a r ( I ) , of the

- 2 4 minat The arower above is NOI correce. ( 6 , 0 , 6 ) to ( 6 , 4 , 6 ) back to ( 6 , 0 , 0 ) . Circulation = C vec ( F ) * d v e c ( r ) =
- 2 4 minat The arower above is NOI correce. ( 6

Identify the asymptotes, domain and range. y = ( 3 ) / ( x ) + 2 HA = q , VA = q , Domain = q , Range = q ,

Solve each equation by factoring. Simplity all irrakional and solu 2 x ^ ( 2 ) = 5 0

Question 2 : ( 1 0 points ) The decay of radioactive materials can be used to date archaeological remains. If the amount present at time \ ( t \ ) ( in years ) is given by \ ( \ mathrm { N } ( \ mathrm { t } ) \ ) ( in grammes ) , then the decay satisfies the equation \ [ \ frac { \ mathrm { d } } { \ mathrm { ~d } t } N ( t ) = - a N ( t ) \ ]
Question 2 : ( 1 0 points ) The decay of

Write a formula for both. Please show your work. 1 4 . 2 0 . Show all images Show all images Show all images done loading
Write a formula for both. Please show your work.

Derivative of - 4 / x

Which of the following represent a linear inequality in open - sentence form? ( Select all that apply. ) 8 + x > 5 2 x + 8 = 1 0 y 3 < 5 2 x - 7 y > 1 0

Cathys clock company determined where x represents "quality units" cost function C ( x ) = 2 x + 2 / ( x + 6 )

Find the limit . lim x - 1 - ( 5 / x ^ 2 - 1

Solve the equation e 3 x = 8 e ^ { 3 x } = 8 e 3 x = 8 for xxx .

Lim ( x ^ 2 - 2 x - 8 ) / ( x - 4 )

( a ) If you walk due south, will you start to ascend or descend?ascenddescend At what rate? vertical meters per horizontal meter \ ( x \ ) ( b ) If you walk northwest, will you start to ascend or descend?ascenddescend At what rate? ( Round your answer to two decimal places. ) vertical meters per horizontal meter ( c ) In which direction is the
( a ) If you walk due south, will you start to

Problem 7 . For the initial value problem y ' ' - x y - 1 = 0 y ( 0 ) = y ' ( 0 ) = 0 Find the first 3 nonzero terms in the power series expansion of y around x = 0 .
Problem 7 . For the initial value problem y ' ' -

Find a vector equation with parameter t for the line through the point ( 7 , 0 , 2 ) and perpendicular to the plane

Find an antiderivative of the function f ( x ) = 7 e 6 x . e ^ 7 x 1 / 7 e ^ 7 x 7 e ^ x 1 / 7 e ^ x

Tutorial Exercise A particle moves in a velocity field v ( x , y ) = ( : x 2 , x + y 2 : ) . If it is at position ( x , y ) = ( 4 , 1 ) at time t = 6 , estimate its location at time t = 6 . 0 1 . Step 1 Submit Skip _ ( you cannot come back ) Need Help?
Tutorial Exercise A particle moves in a velocity

An equation of a line through ( 5 , 1 2 ) which is parallel to the line y = 4 x + 4 has slope:

Evaluate the integral. [ HINT: Use a shortcut. ] 1 1 8 ( 3 x + 1 ) 5 dx 0

8 Multiple Choice 1 point The function f is defined by f ( x ) = 1 6 x ^ ( 4 ) + 3 If f ( a ) = c , which of the following equations gives the value of a in terms of c ? Select one answer.

prove that lim n - > infty 1 / sqrtn = 0

8 Find the exact length of the curve x = e t c o s ( 2 t ) , y = e t s i n ( 2 t ) , 0 t
8 Find the exact length of the curve x = e t c o

Evaluate. 2 cubed 2 3 minus left parenthesis 3 plus 8 right parenthesis ( 3 + 8 ) plus + 5 5 equals = 2 cubed 2 3 minus enter your response here plus

( 1 point ) According to a simple physiological model, an athletic adult male needs 2 0 calories per day per pound of body weight to maintain his weight. If he consumes more or fewer calories than those required to maintain his weight, his weight changes at a rate proportional to the difference between the number of calories consumed and the

Find the net outward flux of the field F = + y j + z k 3 , = x 2 + y 2 + z 2 2 , across a surface S , where S is a closed surface with the topmost point ( 0 , 0 , 1 0 ) which is joined with the bottommost point ( 0 , 0 , - 1 0 ) through straight lines via ten vertices in the x y plane. Each vertex in the x y plane must be at least 5 unit distance
Find the net outward flux of the field F = + y j

A ladder of length L leans on a wall perpendicular to the ground. The ladder starts sliding on the wall and the ground. Define appropriate parameter ( s ) and find the acceleration of the center of the ladder with respect to an observer fixed to the ground. Solve the problem WITHOUT using Magic Formula.
A ladder of length L leans on a wall

Solve the following differential equations with prescribed initial conditions. d y d x = 4 s i n 2 x , y ( 0 ) = 2 d x d t = t + t x 2 , x ( 0 ) = 1
Solve the following differential equations with

find an equation for the line that passes though the points ( 1 , - 6 ) and ( - 5 , 3 )

The population of a country from 1 9 5 0 to 2 0 0 0 has been modeled with the function P ( t ) = 9 4 e 0 . 1 3 t where P is measured in millions and t is measured in years with t = 0 in the year 1 9 5 0 . What was the average population of the country in the second half of the 2 0 th century? ( Round your answer to the nearest whole number. )
The population of a country from 1 9 5 0 to 2 0 0

The graph of the function f ( x ) is given below. On the same axes, sketch the graph of f ^ ( ' ) ( x ) . Then, explain why you drew the graph of f ^ ( ' ) ( x ) the way you did. Explanation:
The graph of the function f ( x ) is given below.

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