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mathematics
june 2018
15

Mathematics Questions and Answers June 15, 2018

Expert-verified solutions to help you succeed

Watch the video Extending the Power Rule Further. Then compute each of the following derivatives. d d x ( x 2 ) = ( x e ) ' = d d y ( 4 y 5 ) =

find the slope of ( - 1 , 2 ) and ( 2 , 4 )

For Problems 3 0 - 3 3 , use the technique derived i n the previ - ous problem t o solve the given differential equation. x y ' - y = x 2 l n x .

QUESTION 2 0 Assume that the sequence { a n } defined below, converges. Find the limit of the sequence { a n } . a 1 = 3 , a n 1 = a n 3 for n = 1 , 2 , 3 dotsdots 3 1 0

Suppose that a rectangle has an area of 7 8 square meters. Express the perimeter as a function of the length of one of the sides. =

Determine h ( x 2 ) for the following function. h ( x ) = - 3 x x 2 - 7 x - 1 2

Global supply chains that connect organizations in different countries are possible because of demand planning pull - based models the Internet push - based approaches

Question 4 of 8 Calculate the derivative. ( Use symbolic notation and fractions where needed. ) d d x 0 x ( 5 t 5 - 3 t 3 ) d t =

Add. StartFraction 8 Over x EndFraction plus StartFraction 4 Over x minus 7 EndFraction 8 x + 4 x 7

Integral of x sqrt ( 1 - 8 x squared ) dx

Integrate. x s i n ( 9 - 2 x 2 ) d x = C

For the function y = f ( x ) , find f ' ( a ) using m t a n = lim x a f ( x ) - f ( a ) x - a . f ( x ) = 1 x - 4 , a = - 2 f ' ( a ) = - 1 ( x - 4 ) 2

Consider the function f ( x ) = 8 x 8 2 x 7 - 9 x 2 - 8 . Enter the antiderivative of f ( x ) C

( 3 points ) Determine the surface area o f the portion o f z = 6 y + 2 x 2 that i s above the triangle i n the x y - plane with vertices ( 0 , 0 ) , ( 8 , 0 ) and ( 8 , 2 ) .

The average births per woman in a certain developing country can be modeled according to the equation g ( x ) = 0 . 0 0 7 x 2 - 0 . 4 x + 7 . 4 3 where x is the number of years since 1 9 5 8 . Step 1 of 2 : What was the average births per woman in the country in 2 0 0 0 ? Answer Keyboard Shortcuts

7 . The velocity of a moving vehicle is given by the equation = ( 2 + 3 ) 4 The task is to: a ) Use the Chain Rule to determine an equation for the acceleration when = .

Write the integral in terms of u and d u . Then evaluate. 6 x e - x 2 d x , u = - x 2 ( Use symbolic notation and fractions where needed. Use C for the arbitrary constant. Absorb into C as much as possible. ) 6 x e - x 2 d x =

Question 1 8 Find the derivative o f l n ( ( x 2 - 1 2 x ) 3 x 3 - 9 2 )

Consider the following vectors: vec ( a ) = 2 vec ( ) + vec ( ) - 3 vec ( k ) , vec ( b ) = - 2 vec ( ) + 3 vec ( ) , and vec ( c ) = 2 vec ( ) - 4 vec ( k ) . a ( 1 mark ) v e c ( a ) * v e c ( b ) bvec ( b ) vec ( c ) cvec ( c ) onto vec ( a ) - vec ( b ) c o s = ( v e c ( a ) ) * ( v e c ( b ) ) | ( v e c ( a ) ) | | ( v e c ( b ) ) | S p r o j

Calculate the integral by interchanging the order of integration. 1 1 2 2 ( 3 x + 5 y + 9 ) dx dy

Consider the following vector equation. r ( t ) = ( : 3 t - 4 , t 2 + 5 : ) ( a ) Find r ' ( t ) . r ' ( t ) =

The oxygen content t days after organic waste was dumped into a pond is given by f ( t ) = 1 0 0 t 2 + 1 0 t + 1 0 0 t 2 + 2 0 t + 1 0 0 ( a ) Use the intermediate value theorem ( read and understand it first ) to conclude that the oxygen content of the pond must have been at a level of 9 0 % at some time. ( b ) At what time ( s ) is the oxygen

3 ( 3 . 1 , 3 . 2 ) 1 of 5 This quit This que Solve for t to two decimal places. 4 = e 0 . 0 1 t t ~~ ( Do not round until the final answer. Then round to the neares

Solve the given differential equation by undetermined coefficients. y 8 y + 2 0 y = 2 0 0 x 2 9 1 xex

A ball is thrown vertically upward. After t seconds, its height h ( in feet ) is given by the function h ( f ) = 1 0 4 t - 1 6 7 . After how long will it reach its maximun Do not round your answer.

Question 1 0 Evaluate using trigonometric substitution: d x x 3 - 2 5 2 - x 2 - 2 5 2 x C l n | x x 2 - 2 5 2 5 | C s i n - 1 ( x 5 ) C l n | x - x 2 - 2 5 2 5 | C

Find the general indefinite integral. ( Remember the constant of integration. ) ( 4 x 2 5 6 x 2 1 ) d x

Evaluate the iterated integral. 2 5 1 x 2 2 y e - x d y d x

Problem 1 . Wheat production, W , i n Bushel s A cre i s a function o f temperature T i n C and annual rainfall, R i n centimeters. Suppose that scientists observe that average temperature i s increasing b y 0 . 1 5 C ? year and rainfall i s decreasing b y 0 . 1 0 c m ? year . Further, i t i s estimated that d e l W d e l T = - 3 , and , d e l W d

If a ball is thrown in the air, its height in feet after t seconds is given by y = 4 0 t 1 6 t 2 . Find the average velocity for the time period beginning when t = 1 and lasting for 0 . 2 5 seconds.

The standard equation of a circle with center at \ ( ( 3 , - 4 ) \ ) and radius 3 is The standard equation of the given circle is ( Type an equation. )

Differentiate the following function: f ( t ) = t 6 - 1 t 6 f ' ( t ) =

Between 2 0 0 6 and 2 0 1 6 , the number of applications for patents, N , grew by about 3 2 % per year. The a ) Find the function that satisfies this equation. Assume that t = 0 corresponds to 2 0 0 6 , when app b ) Estimate the number of patent applications in 2 0 2 0 . c ) Estimate the rate of change in the number of patent applications in 2 0 2

Find the equation of a line with slope - 5 through ( 4 , 7 )

Which exponential function has a horizontal asypmptote at y = 1 and is decreasing from ( , ) ?

Calculate the following derivative. ( Use symbolic notation and fractions where needed. ) d d x ( 1 0 e 1 1 x + 7 e - 1 1 x ) 1 5 =

Question 3 5 pts Evaluate the following integral - c o s ( ( x - 1 ) ) ( x 1 ) d x - 1 0 1 None of the above

Given g ( x ) = 5 sin ( x ) - 5 cos ( x ) / ( sin ( x ) + cos ( x ) ) , find g ' ( x )

( 2 0 p t s ) Use the Divergence ( G a u s s ' ) Theorem t o calculate the flux o f F across S , where F ( x , y , z ) = 3 x y 2 i + x e z j + z 3 k and S i s the surface o f the solid bounded b y the cylinder x 2 + y 2 = 1 and the planes z = - 1 and z = 2

Evaluate the integral. 1 6 t 3 - 5 t 2 2 7 t - 5 ( t 2 1 ) ( t 2 2 ) d x Select the correct answer. a 2 l n ( t 2 1 ) 5 2 l n ( t 2 2 ) - ( 5 2 2 ) 2 t a n - 1 ( t 2 2 ) c 1 1 2 l n ( t 2 1 ) 1 6 2 l n ( t 2 2 ) - ( 5 2 2 ) t a n - 1 ( t 2 2 ) c 3 2 l n ( t 2 1 ) 1 5 2 l n ( t 2 2 ) - ( 5 2 2 ) t a n - 1 ( t 2 2 ) c

Find the sum o f the first 1 1 terms i n the geometric series 1 , - 4 , + 1 6 dots.. . 4 5 1 7 2 4 1 9 4 2 7 8 8 3 8 8 6 1 - 8 5 9 1 4 2

Find f ' ( x ) using the rules for finding derivatives f ( x ) = 3 x 2 - 5 x 1 5 x 8 f ' ( x ) =

Evaluate the integral. 0 4 2 + 7 c o s 2 c o s 2 d

1 . Explain how a cooling reaction that releases heat affects the equilibrium. ( 2 marks )

Section 3 . 4 - Ay = ptotes Asymptote littical - Set downinates - 0 , wolve for x . The VA is the lies x - A . Baticoral - dre ( NUM ) sadre ( DEN ) dog ( NUM ) - des ( DEN ) dog ( NUM ) f ( x ) = 1 x f ( x ) = 3 x x 2 - 4 f ( x ) = 6 - 2 x 2 x 2 - 9 f ( x ) = x 3 x 5 x 1 5 y = 0 Etemple 1 . Find all anymptotes. Graph ( a ) ass ( b ) . ( a ) f ( x

Find the derivatives and simplify the answers: f ' ( x ) if f ( x ) = sin ^ - 1 ( 1 - 2 x ) + ln ( 1 + 2 x )

A ) For what x - values does p ( x ) = x 4 + 4 x + 4 x 2 + B ) What is the equation of the tangent line at x = - 1 ? ( write in slope / intercept form y = m x + b ) A ) For what x - values does p ( x ) = x 4 + 4 x + 4 x 2 + B ) What is the equation of the tangent line at x = - 1 ? ( write in slope / intercept form y = m x + b )

Problem 4 . Find and classify the local maximum and local minimum values and saddle points o f the function f ( x , y ) = x 2 + x y + y 2 + y

Hes within the sphere x 2 y 2 z 2 - 8 1 , above the x y - plane, and below the cone z = x 2 y 2 2 .

Use the Ratio Test to determine whether the series is convergent k = 1 9 k ! Identify a k Evaluate the following limit . lim k | a k 1 a k | Since lim k | a k 1 a k | ? 1 , - - Select - - -

in a murder investigation the temperature of the corpse was 3 2 . 3 \ deg c at 1 : 3 0 pm and 3 0 . 6 \ deg c an hour later. normal body temperature is 3 7 . 0 \ deg c and the ambient temperature was 2 0 \ deg . how long ago did the murder take place

t x = 1 - 3 s i n ( 4 t ) , y = 2 + 3 c o s ( 4 t ) , 0 t 1 2

Differentiate. g ( x ) = 3 x - 2 4 x 5 x 3 g ' ( x ) =

What is the angle between the given vector and the positive direction of the x - axis? ( Round your answer to the nearest degree. ) 1 6 i 1 2 1 Need Help?

f ( x ) = e x l n ( x 2 2 ) f ' ( x ) = Show My Work ( Optional )

ning furstom - Wherefount nace washington 6 6 2 4 ? 5 1 1 4 0 9 PM sing and Decreasing ( Ountican 4 , 4 . 4 . 3 Mat Stown and 4 at 1 5 para Prisar 1 0 1 1 5 Determene whither the statement is true or have, and equate uts If the statement the ar fabe? negative q , pouldre and negative. negative tructor Search Determine whether the statement is true

In a murder investigation, the temperature of the corpse was 3 2 . 9 C at 1 : 3 0 p . m . and 3 0 . 4 C an hour later. Normal body temperature is 3 7 . 0 C , and the ambient temperature was 2 0 . 0 C . How long ago ( in minutes before 1 : 3 0 p . m . ) did the murder take place? ( Round your answer to the nearest minute. ) minutes before 1 : 3 0 p

find the general solution to the following differential equation p ' ( x ) = 1 0 x / x ^ 6 - 5 + 1 8 ^ 5

Solve the initial value problem. t - 1 d y d t = 3 s i n 2 y , y ( 1 ) = 5 4

Evaluate the integral. ( Remember the constant of integration. ) 1 5 d x 3 x x x 2

Content HW 5 . 1 Antiderivative 1 9 . [ 0 / 1 Points ] LARAPCALO Find the cost function for the marginal cost and fixed cost. \ table [ [ Marginal Cost,Fixed Cost

[ - / 1 Points ] Evaluate the integral. ( Remember to use absolute values where appropriate. Rerr 5 ( x - 1 ) ( x 2 4 ) d x Need Help?

Consider the series. 2 0 3 2 0 3 2 2 0 3 3 2 0 3 4 cdots This can be written as a geometric series in the form ( Give an exact answer. Use symbolic notation and frac c = r =

Using Euler's method to solve the second order differential equation over the range x = - 1 to 2 using a ST EP size of 1 with y ( - 1 ) = 3 and y ' ( - 1 ) = 7 3 x y ' ' - 5 y ' ' x 2 ( x 2 y ) = y ' x

First, compute the gradient o f the function f ( x , y ) = 3 + 3 x 2 - 7 y 2 . Then evaluate i t a t the point ( - 1 , - 2 ) . The gradient i s .

Approximate the critical numbers of the function shown in the graph. Determine whether the function has a relative maximum, a relative minimum, an absolute maximum, an absolute minimum, or none of these at each critical number on the interval shown. Approximate the critical numbers. ( Enter your answers as a comma - separated list. ) x = List the

Completed: 2 2 of 3 2 Save score: 2 1 / 3 2 pts ( 6 5 . 6 3 % ) Use logarithmic differentiation to evaluate f ' ( x ) . f ( x ) = ( x 3 ) 9 ( 2 x - 8 ) 1 1 f ' ( x ) =

Find the Taylor series for f ( x ) centered at the given val R n ( x ) 0 . 1 f ( x ) = l n ( x ) , a = 2 f ) Find the Issociated radius of convergence, R . R =

Evaluate the integral. ( Remember the constant of integration. ) root x ^ 2 6 4 / x ^ 3 dx

a drug response curve describes the level of medication in the bloodstream acter a drug is administered

Use geometric series t o write the expanding decimal fraction 0 . 2 6 2 6 2 6 2 6 dots

Find unit tangent vector vec ( T ) ( t ) for the curve vec ( r ) = ( : e - t c o s t , e t s i n t : ) For the acceleration vec ( a ) ( t ) = - 3 2 hat ( j ) and the given initial velocity vec ( v ) ( 0 ) = 6 0 0 3 i 2 6 0 0 hat ( j ) and the initial position vec ( r ) 0 = ( : 0 , 0 : ) find the position of the object vec ( r ) ( t ) . Show all

Find the domain of the function. Upper F left parenthesis x right parenthesis equals 3 x Superscript 5 Baseline minus 3 x squared plus 1 F ( x ) = 3 x 5 3 x 2 + 1

V = 0 2 2 x ( s i n x - 4 2 x 2 ) d x

I f F ( x ) = 0 x t 3 + 1 2 d t , then F ' ( 2 ) = - 2 2 3 1 8

Find the derivative of the following hyperbolic and inverse hyperbolic functions. i ) y = s i n h ( 1 - 3 x ) c o s h ( x 2 )

I n your answer, all vectors should b e given a s a component form. The motion o f a point i s given b y the equations: x = s i n t - t , and , y = t 3 where x and y are i n meters and t i s i n seconds. Find the acceleration vector o f the point a t t = 2 s e c . [ 1 0 p t s ]

An animal population currently has 7 2 0 0 members and is reproducing at a rate of R ( t ) = 2 2 5 0 6 0 t members per yeac. The proportion of members that survive after tyears is given by S ( t ) = 1 t 1 . ( a ) How many of the original members survive three years? animals ( b ) How many new members are added during the next three years? animals

[ 1 / 1 Points ] SCALCET 9 7 . 5 . 0 3 5 . ASK YOUR TEACHER Evaluate the integral. ( Remember the constant of integration. ) 1 4 d x 1 4 e x Fantastic!

Question 6 Calculate lim x - 4 x 2 + 9 2 - 5 x + 4 . Round your answer to three decimal places, if needed. Question 6 Calculate lim x - 4 x 2 + 9 2 - 5 x + 4 . Round your answer to three decimal places, if needed.

lim x 2 x 2 - 2 x - 8 x 2

Evaluate the integral. Integral from nothing to nothing 4 5 x e Superscript negative 9 x Baseline dx 4 5 xe 9 x dx

Find d y d x by applying the exponential and logarithm rules. i ) , y = ( x 2 1 ) 3 ( x 3 - 1 ) 5 ( x 4 2 ) 2

from 2 to 2 of [ 1 / ( t ^ 3 ( t ^ 2 1 ) ) ] dt Evaluate the integral Select the correct answer.

f 1 ( u ) = g 1 ( x ) = 2 x f 1 ' ( u ) = 3 s e c ( u ) t a n ( u ) g 1 ' ( x ) = 2 Determine the following. f ' ( x ) = f ' ( 5 ) =

An object is moving along the number line; its velocity at each time t 0 is given by: v ( t ) = 2 ( t - 2 ) ( t - 1 0 ) . Determine the intervals ( if any ) during which the object is moving to the left.

Question 1 Find the derivative of f ( x ) = 2 3 x f ' ( x ) = 2 3 x ( l n 2 ) ( 3 ) f ' ( x ) = 3 x 2 3 x - 1 f ' ( x ) = 2 3 x - 1 f ' ( x ) = 2 3 x ( l n 3 x )

Evaluate the integral. ( Remember the constant of integration. ) x x 2 5 dx

Let f ( x , y ) = l n ( 2 x ) + y - 1 2 - 7 with x ( r , s ) = r t a n - 1 ( s ) and y ( r , s ) = 3 s 2 . By using the chain rule, find d e l f d e l r .

A linear function f ( x , y ) = a x + b y + c has n o critical points. Therefore, the global minimum and maximum values o f f ( x , y ) o n a closed and bounded domain must occur o n the boundary o f the domain. Furthermore, i t can b e easily seen that i f the domain i s a polygon, then the global minimum and maximum values o f f must occur a t a

MAC 2 3 1 1 Learning Activity # 9 Related Rates Name: q , Group #: A 2 5 - ft ladder is leaning against a house when its base is being pushed towards the wall. By the time the base is 1 2 feet from the house, the base is moving at a rate of 1 f t s e c . ( a ) At what rate is the ladder sliding up the wall at this moment? ( b ) At what rate is the

Let f ( x ) = 3 x 2 + 2 x - 1 x 2 + 3 x + 2 . Find ( A ) lim x - 3 f ( x ) ( B ) lim x - 1 f ( x ) ( C ) lim x 2 f ( x )

Differentiate the function. y = 5 x 2 - 4 7 x 3 2 Which of the following shows how to find the derivative of f ( x ) ? A . y ' = ( 5 x 2 - 4 ) ( d d x ( 5 x 2 - 4 ) ) - ( 7 x 3 2 ) ( d d x ( 7 x 3 2 ) ) ( 7 x 3 2 ) 2 B . y ' = ( 7 x 3 2 ) ( d d x ( 5 x 2 - 4 ) ) - ( 5 x 2 - 4 ) ( d d x ( 7 x 3 2 ) ) ( 7 x 3 2 ) c . y ' = ( 7 x 3 2 ) ( d d x ( 5 x

SCalcet 9 3 7 0 0 9 The height ( in meters ) of a projectile shot vertically upward from a point 3 m above ground level with an initial velocity of 2 2 . 5 m s is h = 3 2 2 . 5 t = 4 . 9 t 2 after 6 seconds. ( a ) Find the velocity ( in m s ) after 2 seconds and after 4 seconds. v ( 2 ) = , m s v ( 4 ) = , m s ( b ) When does the projectile reach

Evaluate the following integral. t a n 3 1 7 x d x t a n 3 1 7 x d x = ( U s e parentheses t o clearly denote the argument o f each function. )

Junker Renovation completely overhauls junked or abandoned cars. Data shows their 1 9 7 0 ' s models hold their value quite well. The value F ( x ) of one of these cars is given by F ( x ) = 8 5 - 1 5 x x 3 , where x is the number of years since repurchase and F is in hundreds of dollars. Step 1 of 3 : What is the initial resale price of the car?

uiz ( 1 . 7 - 1 . 8 ) Differentiate the function. y = ( 2 x 2 - 5 ) - 1 0 d y d x =

Use the limit definition to find the derivative of the function. f ( x ) = x 2 3 x - 3 First, find f ( x h ) - f ( x ) h . x 2 2 h x h 2 - 3 x - 3 h - 3 - x 2 Next, simplify the numerator. Divide out the h . So now, find the limit lim h 0 f ( x h ) - f ( x ) h =

code class = "asciimath" > 3 . 2 y ( t ) = { ( 0 , x ( t ) 0 , ) , ( x ( t ) x ( t - 2 ) , x ( t ) > = 0 ) : }
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