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

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Integrate. Integral left parenthesis sine left parenthesis StartFraction x Over 9 EndFraction right parenthesis plus StartFraction 1 Over 8 x EndFraction plus RootIndex 7 StartRoot x cubed EndRoot right parenthesis dx sinx 9 + 1 8 x + 7 x 3 dx = enter your response here + C

answer for part b and part d

Determine the integral that can be used to find the arc length of the curve defined by the parametric equations x ( t ) = 6 e - 4 t and y ( t ) = 9 e - 7 t from 0 . 9 t 1 . 7 . Provide your answer below: s = 2 , d t
Determine the integral that can be used to find

Find the equation of the line with m = 6 that passes through the point ( 4 , 1 ) .

code class = "asciimath" > Evaluate the indefinite integral. re ( r ) / ( 4 ) dr

find derivative of 1 / ( 8 + 8 x ) ^ 2

write an equation of a perpondicular line to y = - ( 2 ) / ( 3 ) x throough point ( 1 4 , 3 0 ) in slope intercept form

Find the Flux of F across S F ( x , y , z ) = x i + 2 y j + 3 z k , S is the cube with vertices ( + - 1 , + - 1 , + - 1 )
Find the Flux of F across S F ( x , y , z ) = x i

by writing equivalent expressions to given values. You are substituting the x an Distributive 2 ( 3 x + 2 y ) 2 ( 3 ( - 1 ) + 2 ( 3 ) ) Substitute 2 ( - 3 + 6 ) finish

The volume of a cantaloupe is approximated by V = 4 3 r 3 . The radius is growing at the rate of 0 . 5 c m ? week, at a time when the radius is 7 . 7 cm . How fast is the volume changing at that moment? The volume is changing at a rate of about c m 3 ? week ( Round to one decimal place as needed. )
The volume of a cantaloupe is approximated by V =

Given the equation y = xx 5 , determine the differential dy for x = 6 and dx = 0 . 0 1 . Round your answer to four decimal places if necessary.

If vec ( F ) = ( : - x , 0 , z : ) , which of the following must be correct? P . The flux of vec ( F ) across the plane z = 1 is 0 . Q . The flux of vec ( F ) across the plane x = 1 is 0 . R . The flux of vec ( F ) across a unit sphere is 0 . a . P only b . Q only c . R only d . P and Q only e . P , Q , and R
If vec ( F ) = ( : - x , 0 , z : ) , which of the

integral x ^ e + e ^ x

Consider the following. 3 6 - x 2 2 x d x Find the indefinite integral using the substitution x = 6 s i n ( ) . ( Remember to use absolute values where appropriate. Remember the constant of integration. )
Consider the following. 3 6 - x 2 2 x d x Find

integral pi 0 x ^ 2 cos ( 4 x ) dx

David invested $ 2 9 , 1 8 7 in a retirement account with continuous compounding. If the annual interest rate is 2 . 1 % , how much will be in the account in 1 4 years?

In Problems 1 4 3 9 , find an antiderivative. 1 7 . f ( x ) = 5

Integrate ( 2 5 - x ^ 2 ) ^ ( 1 / 2 )

Use the limit definition to find f ( x ) if f ( x ) = 1 x + 1 through plugging in without lhopital walk me through process

Consider the following. cos 3 ( x ) sin 4 ( x ) dx Which of the following are true? The power of cosine is odd and positive. The power of sine is odd and positive. The powers of both sine and cosine are even and nonnegative. Which of the following guidelines can be applied to find this indefinite integral? Save one sine factor and convert the

The area of the region bounded by the curves y = 9 - x 2 and y = 0 is equal to a . 1 6 3 b . 3 6 c . 1 4 3 d . 1 8 e . 4 3
The area of the region bounded by the curves y =

a . If F ( x ) = 3 x 1 t d t , then F ' ( x ) = q , b . If F ( x ) = x 7 1 t d t , then F ' ( x ) = q , c . If F ( x ) = 1 8 x 3 1 t d t , then F ' ( x ) = q , d . If F ( x ) = 2 + c o s x x 2 + 1 1 t d t , then F ' ( x ) = q , Note: You can earn partial credit on this problem.
a . If F ( x ) = 3 x 1 t d t , then F ' ( x ) = q

Let y ( t ) represent your bank account balance, in dollars, after t years. Suppose you start with $ 6 0 0 0 0 in the account. Each year the account earns 4 % interest, and you deposit $ 7 0 0 0 into the account. This can be modeled with the differential equation: d y d t = 0 . 0 4 y + 7 0 0 0 y ( 0 ) = 6 0 0 0 0 Solve this differential equation
Let y ( t ) represent your bank account balance,

Find the surface area of revolution about the x - axis of R . Surface area = 2 . 6 4 . 8 f ( x ) d x where f ( x ) = Now integrate to find surface area = 2 . 6 4 . 8 f ( x ) d x =
Find the surface area of revolution about the x -

Suppose y varies jointlly as x and z . Find y when x = 9 and z = - 5 , if y = - 9 0 when z = 1 5 and x = - 6 .

Parametrize the line segment beginning at ( 1 , 3 ) and ending at ( 4 , 9 ) . Use t as the parameter. ( Express numbers in exact form. Use symbolic notation and fractions where needed. ) ( x ( t ) , y ( t ) ) Determine the range of values for parameter t . ( Give your answer as an interval in the form ( * , * ) . Use the symbol for infinity, for
Parametrize the line segment beginning at ( 1 , 3

g ) Evaluate S F * n d s if F = ( x + 2 y ) i - 3 z j + x k and S is the surface of 2 x + y + 2 z = 6 bounded by x = 0 , x = 1 , y = 0 and y = 2 .
g ) Evaluate S F * n d s if F = ( x + 2 y ) i - 3

( 8 marks ) Solve the IVP y ' = x + 5 y , y ( 0 ) = - 1 2 5 by finding Integrating Factor ( IF ) using the formula I F = e P ( x ) d x
( 8 marks ) Solve the IVP y ' = x + 5 y , y ( 0 )

Use Stoke's Theorem to evaluate C F * d r , where F = x 2 y i + x 3 j + e z t a n - 1 z k , where C is the curve with parametric equations r ( t ) = ( c o s ( t ) , s i n ( t ) , s i n ( t ) ) , 0 t 2
Use Stoke's Theorem to evaluate C F * d r , where

A cylindrical tank, lying on its side, has a radius of 1 0 ft 2 and length 4 0 ft . Assume that the tank is half - filled with water and that water leaks through a hole in the bottom of area B = 4 in 2 . Determine the water level y ( t ) and the time te when the tank is empty.

tric Counes Part II: Write your solution to the following problems. Both your answer and justification will be graded. An answer without proper justification will be give 7 . ( 3 5 points ) Consider the parametric equations x = 2 t 3 - 6 t y = t 2 - 3
tric Counes Part II: Write your solution to the

Integral 3 x ^ 3

Question: Given the function f ( x ) = x 4 8 x 2 4 f ( x ) = x ^ 4 - 8 x ^ 2 - 4 : a ) Find the local extreme values of the function using the first derivative test. b ) Find the inflection points, if any, and determine the intervals of concavity. c ) Sketch the graph of the function using the results from parts ( a ) and ( b ) .

The length of a rectangle is 3 m less than twice the width, and the area of the rectangle is 4 4 m ^ ( 2 ) . Find the dimensions of the rectangle Length : m Width : m

If f ( x ) = x 2 4 + x , find f ( 4 ) . f ( 4 ) =

Find the derivative of the function f ( x ) = 4 x - 8 using the limit definition of the derivative. Provide your answer below: f ' ( x ) =
Find the derivative of the function f ( x ) = 4 x

Approximate the area under the curve y = x 2 - 4 x + 3 on the interval 0 , 5 by using i . Right endpoints with n = 5 . ii . Left endpoints with n = 5 . iii. Right endpoints with n = 8 . iv . Left endpoints with n = 8 . Provide plots of the curves with rectangles representing each approximation. Round to four decimal places as necessay.
Approximate the area under the curve y = x 2 - 4

Let f ( x ) = 0 x ^ 0 - 1 , 0 < 0

Calculate the limit algebraically. ( If an answer does not exist, enter DNE. ) lim x 2 x 4 + 1 0 x 3 9 , 0 0 0 x 6 + 5

Question 2 . 2 . 0 . 4 . 9 0 . The value of lim x e 2 x - e - x 2 e x - e - x A . is 1 2 . B . is . C . is - . D . exists.
Question 2 . 2 . 0 . 4 . 9 0 . The value of lim x

Question 1 of 1 View Policies Current Attempt in Progress Compute the flux of the vector field, vec ( F ) = xvec ( i ) + yvec ( j ) , through the parameterized surface, S , which is oriented toward the origin and given, for 0 s 1 , 0 t 1 by x = 6 s , y = s + t , z = 2 + s - t Enter an exact answer. S vec ( F ) * d v e c ( A ) =
Question 1 of 1 View Policies Current Attempt in

Write an equation for a line perpendicular to y = 2 x - 3 and passing through the point ( - 4 , 3 )

Find the solution to the differential equation below, subject to the given initial condition. dydx + y 5 = 0 , y ( 0 ) = 1 2

The automobile has a mass of 1 . 5 Mg and center of mass at G . Take s = 0 . 3 .

Factor out the greatest common factor from the following polynomial. 7 x ^ ( 7 ) - 2 8 x ^ ( 6 ) + 3 5 x ^ ( 5 )

Evaluate _ ( - 5 ) ^ 5 ( 4 2 ) by interpreting it in terms of areas

y ' ' ' + 4 y ' ' - y ' = 2 - 3 cos 3 x + e ^ x

Approximate the open intervals where the given function is increasing and decreasing. Round any decimals to the nearest tenth. f ( x ) = x 3 - 3 x 2 + 1
Approximate the open intervals where the given

A biologist who is studying blue jays finds that adult blue jays eat an average of 3 0 insects each day. One particular blue jay eats 2 7 insects each day. This particular blue jay eats q , % of the average number of insects each day. 2 7 3 0 3 9 0
A biologist who is studying blue jays finds that

Find the greatest common factor of the terms of the polynor 1 6 a ^ ( 3 ) - 1 6 a ^ ( 2 )

9 x ^ 2 - 6 4 y ^ 2 - 8 1 = 0 . Find the vertices and asymptotes of the hyperbola.

Which expression is equivalent to ( 1 0 y ^ ( 7 ) - 8 y ^ ( 6 ) + 3 y ^ ( 4 ) ) / ( y ^ ( 3 ) ) ? Assume that the denominator does not equal zero.

r ( t ) = { t 2 - 2 t - 1 i f t 3 a t + b i f t > 3 Find a and b such that the given function, r ( t ) , is differentiable everywhere. a = b = Make a plot that verifies your result.
r ( t ) = { t 2 - 2 t - 1 i f t 3 a t + b i f t style=

Is x - 2 a factor of f ( x ) = 3 x ^ ( 2 ) + 7 x - 2 0 ? Do not use division and explain your answer.

[ 0 / 2 Points ] LARCALCET 7 5 . 4 . 0 9 4 . The velocity function, in feet per second, is given for a particle moving along a straight line. v ( t ) = t 2 - t - 1 8 2 , 1 t 1 5 ( a ) Find the displacement. ( b ) Find the total distance that the particle travels over the given interval. Need Help?
[ 0 / 2 Points ] LARCALCET 7 5 . 4 . 0 9 4 . The

Question 4 ( 1 point ) Describe the vector using Cartesian notation Tail at ( 6 , - 1 ) and head at ( 1 , - 4 ) Question 4 options: [ - 1 , - 4 ] [ 1 , 4 ] [ - 5 , - 3 ] [ 5 , 3 ]

f ( x ) = x ^ 4 - 8 x ^ 2 + 7 find the antiderivative

Find the most general antiderivative of f ( u ) = 4 u 4 5 uu 2 . Note: Any arbitrary constants used must be an upper - case " C " . F ( u ) =

Calculate the following values. If answer does not exist, explain lim x 3 f ( x ) = lim x - 3 + f f ( x ) = f ( - 3 ) = lim x 1 - f ( x ) = lim x 1 + f ( x ) f ( x ) = lim x 5 5 f ( x ) = f ( 5 ) = lim x 8 - f ( x ) =
Calculate the following values. If answer does

derivitive of cos ( 3 x )

a ) evaluate the following limit lim x - 1 x 2 + 2 x + 1 x + 1 b ) based on your answer does the function x 2 + 2 x + 1 x + 1 have a vertical asymptote at x + 1 = - 1 . explain if not and what the graph looks like at x = - 1 . c ) evaluate the following limit lim x x 3 + 2 x 2 + 1 3 x 3 - 4 x d ) does the function f ( x ) = x 3 + 2 x 2 + 1 3 x 3 -
a ) evaluate the following limit lim x - 1 x 2 +

[ 1 0 points ] The function v ( t ) , is a continuous and differentiable function giving the velocity, of a car , in f t s e c , traveling on a straight road, for 0 t 5 0 . Values of the function are given for several values of t at 5 - second intervals as shown in the table below. \ table [ [ t ( s e c ) , 0 , 5 , 1 0 , 1 5 , 2 0 , 2 5 , 3 0 , 3
[ 1 0 points ] The function v ( t ) , is a

A city has a population of 3 4 0 , 0 0 0 people. Suppose that each year the population grows by 8 . 2 5 % . what will the population be after 1 4 years

Evaluate: from - 2 to 5 ( 5 - | x | ) dx by interpreting in terms of areas.

Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [ 0 , 2 ] . lim n goes to infinity, summation I = 1 to n , ( ( 8 i ^ 2 / ( n ^ 3 ) - ( ( 4 i / n ^ 2 ) )

Question: The function A = 2 0 0 e ^ { - 0 . 1 t } measures the amount ( A ) in kg of a radioactive substance that is left in tt years. How much time will need to pass until there are 7 5 kg left?

Use the Fundamental Theroam to evaluate the definite intergral exactly S 6 1 1 / x dx =

Your Answer Correct Answer A population of bacteria is introduced into a nutrient solution in a dish. The nutrient solution is replenished by the exact same amount each day. At first the bacteria population grows very quickly. Soon, population growth must slow down, as the nutrients become more scarce given the number of bacteria. Eventually the
Your Answer Correct Answer A population of

2 0 Solve the following system of first order differential equations x ' = 2 x + 3 y , y ' = 2 x + y , using: ( a ) Substitution, ( b ) Method of Laplace Transform, ( c ) Eigenvalues and eigenvectors. ( please solve all parts and show steps and all work. This is the differential equation class. )
2 0 Solve the following system of first order

QUESTION 1 4 The price and demand for the manufacture of tissue boxes is related by the function p = 2 8 - 0 . 0 7 x , where p is the retail price per tissue box at a demand of x tissue boxes for 0 x 4 0 0 . A ) Find the revenue as a function of x . B ) Find the marginal revenue function. C ) Find R ( 1 0 0 ) and R ' ( 1 0 0 ) and interpret the
QUESTION 1 4 The price and demand for the

Evaluate the integral. 7 y 3 3 - 7 y 4 2 d y 7 y 3 3 - 7 y 4 2 d y =
Evaluate the integral. 7 y 3 3 - 7 y 4 2 d y 7 y

Suppose functions f ( x ) and g ( x ) are continuous on the finite interval a , b , a b , and a b [ f ( x ) + g ( x ) ] d x = 0 . Then ( Choose all the correct statements below ) ( A ) b a [ f ( x ) + g ( x ) ] d x = 0 . ( B ) If f ( x ) = ( x 2 - 1 ) and g ( x ) = ( 2 - x 2 ) , then a b [ f ( x ) + g ( x ) ] d x = 0 ( C ) a b [ f ( x ) - g ( x )
Suppose functions f ( x ) and g ( x ) are

Find the derivatives of the following functions. a ) 0 x 2 ( t 3 + 2 t ) d t , 2 x ( x 2 ) 3 + 2 ( x 2 ) = 2 x ( x 6 + 2 x 2 ) b ) g ( t ) = a r c c o s ( t 2 ) = - 1 2 t 2 1 - t 2 c ) f ( x ) = 2 x s e c h ( x 2 ) d ) f ( x ) = 5 x 2 + 1 e ) y = e x - e - x e x + e - x f ) f ( x ) = l n ( 3 x 2 - 3 3 ) g ) f ( x ) = 2 ( x 3 ) c o s h 4 x
Find the derivatives of the following functions.

A calculator will NOT be available for this question. x y = 1 x 2 + y 2 = 5 The point ( x , y ) lies in the third quadrant of the x y - plane and satisfies the equations above. What is the value of y

Consider the following velocity components: u = 2 y + 3 z , v = 5 x + 2 y , w = 2 x + 3 y . Estimate the velocity potential?Question 1 Answera. ( x , y ) = 7 x - 7 y + 3 zb . ( x , y ) = 5 x - 2 y + zc . ( x , y ) = 8 x - 4 y + zd . Does not exist

Evaluate using integration by parts or substitution. 1 5 x e - 3 x dx

Given: f ( x ) = ( 1 - x ) e x Complete the following: I have completed 1 - 3 but need help with number 4 please 4 . State all asymptotes
Given: f ( x ) = ( 1 - x ) e x Complete the

Classify the given function as even, odd, or neither. f ( x ) = 7 x 2 + 9 A function f is called an even function of x if The function f is called an odd function of x if
Classify the given function as even, odd, or

Find the product using the difference of two squares formula. ( 2 x + 5 ) ( 2 x - 5 )

The curves vec ( r ) 1 ( t ) = ( : 5 t , t 6 , - 3 t 2 : ) and vec ( r ) 2 ( t ) = ( : s i n ( - 3 t ) , s i n ( - 4 t ) , t - : ) intersect at the origin. Find the acute angle of intersection ( in radians ) on the domain 0 2 , to at least two decimal places.
The curves vec ( r ) 1 ( t ) = ( : 5 t , t 6 , -

Evaluate the integral. 3 xe - x 2 dx

Use the Fundamental Theorem to evaluate the definite integral exactly. 0 to 2 ( 1 2 x ^ 2 + 2 ) dx Enter the exact answer. 0 to 2 ( 1 2 x ^ 2 + 2 ) dx =

Find f ' ( x ) and f ' ' ( x ) . ( x 3 + 8
Find f ' ( x ) and f ' ' ( x ) . ( x 3 + 8

Calculate the iterated integral. 9 0 2 0 t 2 sin 3 ( ) d dt

Find E y d V , where E is the solid bounded by the parabolic cylinder z = x 2 and the planes y = 0 and z = 1 1 - 3 y
Find E y d V , where E is the solid bounded by

calculate dz / dx of the following function z = ( ( x - y ) ^ 2 - y ) ^ 3 where y = x + e ^ x

Question 2 . ( 5 + 5 Points ) Determine the most general antiderivative of the following functions. ( a ) f ( x ) = x 2 + x 2 + 2 x . ( b ) f ( x ) = c o s ( 2 x ) .
Question 2 . ( 5 + 5 Points ) Determine the most

\ ( \ qquad \ ) 2 3 . A rectangular area cilong A tobechered inclowed w / 2 5 0 ft it fencing What are the dinienoines that wille produre the greatest inclosed area?
\ ( \ qquad \ ) 2 3 . A rectangular area cilong A

1 2 : 6 pts The masses m i are located at the points p i Find the center of mass of the system. m 1 = 1 , m 2 = 2 , m 3 = 9 P 1 = ( - 3 , - 7 ) , P 2 = ( 6 , 1 ) , P 3 = ( 6 , - 1 ) a ) Find x - bar. b ) Find y - bar
1 2 : 6 pts The masses m i are located at the

Evaluate the indefinite integral: 8 dxxln ( 6 x ) =

Raggs, Ltd . a clothing firm, determines that in order to sell x suits, the price per suit must be p = 1 8 0 - 0 . 7 5 x . It also determines that the total cost of producing x suits is given by C ( x ) = 5 0 0 0 + 0 . 7 5 x ^ 2 . a ) Find the total revenue, R ( x ) . b ) Find the total profit, P ( x ) . c ) How many suits must the company produce

Find the equation of the tangent line to x 2 - x y + 3 y 2 = 3 at the point ( - 1 , - 1 ) .
Find the equation of the tangent line to x 2 - x

Consider the differential equation f ' ' ( x ) + 2 f ' ( x ) + f ( x ) = e - x . What is the approximate value of the particular solution at x = 9 ? , r 2 + 2 r + 1 = e - x Hint - 1 : This is a differential equation with constant coefficients! Hint - 1 : This is a differential equation with constant coefficients! Hint - 2 : You can take e 9 9 0 2
Consider the differential equation f ' ' ( x ) +

Step 2 To determine our indices of integration, we need to identify where the point of intersection occurs. In other words, we need to find the \ ( x \ ) - coordinate for the point of intersection. To do this we set \ ( 2 \ cos ( 4 x ) \ ) equal to \ ( 2 - 2 \ cos ( 4 x ) \ ) , and solve for \ ( x \ ) . What is the \ ( x \ ) - coordinate of the
Step 2 To determine our indices of integration,

noticed that in the solutions provided for the problems involving differential equations ( like in problems 7 - 1 0 ) , the integrating factor was not explicitly found before solving the equations. However, the question specifically asks to find the integrating factor. Could you please clarify how the solutions were derived without this step, or

1 6 0 0 = ( 3 h ^ 2 ) / 4 ; find h

What is the work done in foot pounds when 5 pounds of water is released at 1 5 0 0 feet

A rain gutter is made from a sheet of metal 9 in wide. The gutters have a 3 in base and two 3 in sides folded up at an angle of \ theta . What angle \ theta maximizes the cross - sectional area of the gutter?
A rain gutter is made from a sheet of metal 9 in

True / False . If f is integrable on a , b , then there exists a number x in the open interval ( a , b ) such that a x f = x b f If your answer is true, then explain your reasoning. If you chose false, then give a counterexample. ( A ) True ( B ) False ( C ) not enough information ( D ) none of the above
True / False . If f is integrable on a , b , then

3 x - 1 2 x 2 - 8 x - 1 0 Find the antiderivative

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