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mathematics
june 2018
3

Mathematics Questions and Answers June 3, 2018

Expert-verified solutions to help you succeed

Evaluate the integral. 4 x 2 ( 3 + 4 x 2 ) 7 d x

Calculate the integral, i f i t converges. I f i t diverges, enter diverges for your answer. 4 9 5 0 1 y 2 - 4 9 d y =

Prove the trigonometric identity. s e c x s i n x t a n x + c o t x = s i n 2 x Drag a n expression t o each box t o correctly complete the proof. s e c x s i n x t a n x + c o t x = s i n 2 x = s i n 2 x = s i n 2 x = s i n 2 x = s i n 2 x = s i n 2 x s i n 2 x = s i n 2 x s i n x c o s x 1 s i n x c o s x c o s x s i n x s i n 2 x + c o s 2 x s

find the integral ( 5 / x + 6 / x ^ 3 ) dx , use parentheses to clearly denote the argument of each function

Fourier's law of heat transfer states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is , F = k T , which means that heat energy flows from hot regions to cold regions. The constant k > 0 is called the conductivity, which has metric units of joules per meter per second - kelvin or watts per

Determine the Fourner Series for f ( t ) = 2 s i n ( 3 t ) + 2 s i n ( 4 t ) . ( Hint use trigonomerric representation ) deliucrel

sinear given that x > 0 and y > 0 , is the following statement true or false? x 2 y 2 2 = x y

Question 1 4 2 pts If ( i = 1 N x i ) 2 N = 5 and N = 5 , then the sum of x = q , 5 of - 5 2 5 or . 2 5 1 2 5 or - 1 2 5 carit be determined - not enough information

A juice company has found that the marginal cost of producing x pints of fresh - squeezed orarge juice is given by the function below, shere C ( p ) is in dollers Approderafe fre totil coat of producing 2 7 3 pt of juice, using 3 subintervals over [ 0 , 2 7 3 ] and the left endpoint of each subinterval. C ' ( x ) = 0 . 0 0 0 0 0 8 x 2 - 0 . 0 0 4

use integration b y parts t o evaluate the integral: I n ( t ) t 1 5 2 d t

Approximate the sum of the series correct to four decimal places. \ infty ( 1 ) n 1 n 2 6 n n = 1

Find f ' ( x ) for the following function. Then find f ' ( 6 ) , f ' ( 0 ) , and f ' ( - 5 ) . f ( x ) = 9 x - 1 f ' ( x ) =

6 ( 1 0 p t ) . Use separable equation method t o solve the equations. ( a ) y ' = 4 x y 2 . ( b ) y ' = 1 + x + y + x y .

Evaluate the integral 0 s i n 2 ( t ) c o s 2 ( t ) d t

Find the volume under the surface z = f ( x , y ) and above the rectangle with the given boundaries. z = 6 x 2 y ; R = { ( x , y ) : 0 x 4 , 0 y 3 }

A cosmetics company is planning the introduction and promotion of a new lipstick line. The marketing research department has found that the deman in a particular city is given approximately by p = 1 2 e , 0 x 2 where a thousand lipsticks were sold per week at a price of p dollars each. At what price will the weekly revenue be masimized? Price =

Find the integration norm o f the given polynomial o n C [ 0 , 1 ] . f ( x ) = 3 x - 4

If $ 7 4 5 0 is invested at a rate of 8 % compounded continuously, what will be the balance after 5 years?

Question 2 lim x 4 2 x 2 - 3 x - 2 0 5 x - 4 x - 4 =

This i s Section 2 . 6 , problem 2 1 . The general ( n o n z e r o ) solution o f the Bernoulli differential equation d y d x + y x = x 2 y 2 can b e written f ( x , y ) = C f ( x , y ) =

Find the interval of convergence of x = 2 n ( x - 9 ) n ( Use symbolic notation and fractions where needed. Give your atsurs as intervals in the form ( * * , * * ) . Use the symbol ~~ for infinity, of for combining intervals, and an appropriate type of parsethesis " C , or ' 1 depending on whether the interval is open or closed. ) xin

Find the surface area of the surface cut from the bottom of the paraboloid x 2 y 2 - z = 0 by the plane z = 4 . [ 8 pts ] Find the flux of the velocity vector field vec ( F ) = 0 , y z , z 2 > outward through the surface 5 cut from the cylinder y 2 z 2 = 1 , z 0 , by the planes x = 0 and x = 1 . [ 8 pts ] Verify Stokes' theorem for vec ( F ) = ( :

SCalcCCS 5 . 4 . 0 2 2 . Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. g ( x ) = s i n ( x ) 1 4 t 2 2 d t c o s ( x ) 4 s i n 2 ( x )

Calculate the integral, i f i t converges. I f i t diverges, enter diverges for your answer. 0 2 1 x 2 e - x 2 d x =

Give the partial fraction decomposition for the following function. 2 8 x - 1 5 x 2 - 3 x 2 8 x - 1 5 x 2 - 3 x =
$Give the partial fraction decomposition for the$

Evaluate the integral using any appropriate algebraic method o r trigonometric identity. 4 7 3 x 7 x 4 - 4 d x

4 . 6 1 ) Find equations for the tangent plane and normal line to the surface = 2 + 2 at the point ( 2 , 1 , 5 ) .

Find the exact solution for ln ( 4 ) + ln ( 4 x ^ 2 - 7 ) = ln ( 9 3 )

Find f ' ( - 4 ) i f f ( x ) = x 4 6 - 6 x f ' ( - 4 ) = ( S i m p l i f y your answer. Type a n integer o r a fraction. )

a lim r 0 9 r 5 s i n ( + x )

Evaluate the integral: x 3 3 6 - x 2 2 d x + C

] ) : } - x 9 4 x 2 - 9 2 C 2 x 4 x 2 - 9 2 C [ - 2 x 4 x 2 - 9 2 C

Evaluate the integral: 2 5 2 - x 2 1 4 x 2 d x ( A ) Which trig substitution is correct for this integral? x = 2 5 s i n ( ) x = 5 s e c ( ) x = 2 5 t a n ( ) x = 5 t a n ( ) x = 2 5 s e c ( ) x = 5 s i n ( ) ( B ) Which integral do you obtain after substituting for x and simplifying? Note: to enter , type the word theta. d ( C ) What is the value

A particle moves according to a law of motion s = f ( 0 ) , t 0 , where t is measured in seconds and s in feet. F ( t ) = 0 . 0 1 t 4 - 0 . 0 5 t 3 ( a ) Find the velocity at time t ( n n ? s ) . v ( t ) = ( b ) What is the velocity after 2 second ( s ) ? v ( 2 ) = t s ( c ) When is the particle at rest? t = s ( smaller value ) t = s ( larger

Find the potential function f for the field F = 6 + 7 y j + 4 z k . A general expression for the infinitely many potential functions f for the field F = 6 + 7 y j + 4 z k i s f ( x , y , z ) =

convert the equation f ( t ) = 2 0 0 ( 1 . 3 1 ) t t o form f ( t ) = a e k t a = k =

Find the imit. Use THaspital's Rule where appropriate. If there is a more elementary method, consider using it . lim x 2 5 x 2 - 5 x - 2 5

Evaluate the indefinite integral. ( Use symbolic notation and fractions where needed. Use C for the arbierry constant. Absorb into C as much as possible. ) ( 2 7 s i n ( x ) - 2 8 c o s ( x ) ) d x =

Approximate the change in the volume of a right clrcular cylinder of fixed radius r = 2 2 c m when its haight decreases from h = 1 1 c m to ) = ( a 2 ( h ) Use a linear approximation.

evaluate i f the l i m i t exists. otherwise enter dne. lim x 0 s i n ( 1 1 x ) x

Proceed as in this example to find a solution of the initial - value problem with the given piecewise - defined forcing function. y ^ ( ' ' ) + y = f ( x ) , y ( 0 ) = 1 , y ^ ( ' ) ( 0 ) = - 1 where f ( x ) = { ( 0 , x 0 ) : } 1 0 , 0 = x = 3 \ pi 0 , x > 3 \ pi y ( x ) = { ( , x 0 ) : } , 0 = x = 3 \ pi , x > 3 \ pi

Length of Infants at Birth Medical records of infants delivered at a certain hospital show that the percentage of infants whose length at birth is between 1 8 and 2 0 in . is given by P = 1 0 0 1 8 2 0 1 2 . 4 2 2 e - ( 1 2 ) [ x - 1 9 2 . 6 ] 2 d x Use Simpson's Rule with n = 1 0 to estimate P . ( Round your answer to two decimal places. ) P = , %

( 8 points ) Window 2 : Q 1 4 and Q 1 6 ( p . 4 5 6 ) : Find the volume o f the solid obtained b y rotating the region bounded b y the given curves about the specified line. Sketch the region, the solid, and a typical disk o r washer. 2 x = y 2 , x = 0 , y = 4 about the y axis Section 6 . 2 ( 8 points ) Window 2 : Q 1 4 and Q 1 6 ( p . 4 5 6 ) :

Eliminating a Parameter In Exercises 3 9 - 4 2 , eliminate the parameter and obtain the standard form of the rectangular equation. 3 9 . Line through ( x 1 , y 1 ) and ( x 2 , y 2 ) : x = x 1 r ( x 2 - x 1 ) , y = y 1 r ( y 2 - y 1 ) Circle: x = h r c o s , y = k r s i n Ellipse: x = h acos , y = k bsin Hyperbola: x = h asec , y = k btan

Find d y d x by implicit differentiation. x y x = 2 - 1 y x 1 x y - 1 x y 1 y x

Describe the end behavior o f the function f ( x ) = 6 x 4 + 3 x 5 + 1 3 b y finding lim x f ( x ) and lim x - f ( x ) .

Consider the area function. A ( x ) = 1 x ( 4 0 t 7 9 0 t 4 2 ) d t Find the derivative of A with respect to x . d d x 1 x ( 4 0 t 7 9 0 t 4 2 ) d t =

Rearranging Series Use the fact that 1 - 1 2 1 3 - 1 4 1 5 - - = l n 2 to rearrange the terms in the alwenating harmonic series so be sam of the rearranged series is .

Consider the following function. w = 1 0 x + y 8 z 6 y Find all the first partial derivatives.

Find the limit of the sequence with the given n th term. a n = 3 n 5 3 n

Find the Maclaurin Series for f ( x ) = s i n ( x 3 ) , given s i n x = x - x 3 3 ! + x 5 5 ! - x 7 7 ! + dotsdots Write answer using summation notation.

Evaluate x s i n x d x , using integration by parts. - x c o s x - s i n x C - x c o s x - 1 2 s i n 2 x C - x c o s x 1 2 s i n 2 x 0 C - x c o s x s i n x C

exact form. Use symbolic notation and fractions where needed. ) 2 l n ( x ) 8 x 2 d x = Identify the true statement ( s ) about the series and the integral. The Integral Test does not apply since the function is only increasing when x 1 . The Integral Test does not apply since f ( x ) is not continous on ( 1 , ) . f ' ( x ) 0 for all x 2 , so f is
$exact form. Use symbolic notation and fractions$

Find the position and nature o f the stationary poin t s o n the curve z = f ( x , y ) , where f ( x , y ) = x 2 - 4 x y + y 3 + 4 y

Find the equation of the tangent line to the curve of the intersection of the surface z = x ^ 2 y ^ 2 with the plane x = 8 and the point ( 8 , 1 , 6 3 )

Find the indicated midpoint rule approximation to the following integral. 0 1 5 s i n ( x ) d x using n = 6 subintervals 0 1 5 s i n ( x ) d x ~~ ( Round to four decimal places as needed. )

Given F ( x ) below, find F ' ( x ) . F ( x ) = 3 s i n ( x ) ( t 2 8 ) d t Provide your answer below:

SCalcET 7 1 6 . 8 . AE . 0 0 2 . EXAMPLE 2 Use Stokes' Theorem to compute the integral curl F d S , where F ( x , y , z ) = z y z j x y k and S is the sphere x 2 y 2 z 2 = 4 9 that lies inside the cylinder x 2 y 2 = 3 6 and above the x y - plane. ( See the figure. ) SOLUTION To find the boundary curve C we solve the equations x 2 y 2 z 2 = 4 9 and

How long does i t take for $ 1 5 7 5 t o double i f i t i s invested a t 5 . 5 % compounded continuously? Round your answer t o two decimal places, i f necessary.

Find the derivative of f ( x ) = - 4 x 5 6 x 5 - 5 Be sure to simplify and collect like terms. f ' ( x ) =

Calculate the limit below \ lim _ ( x - > 1 ) ( - 4 * sin ( ( 1 1 * \ pi * x ) / ( 2 ) ) - 5 * x ^ ( 2 ) )
$Calculate the limit below \ lim _ ( x - style=$

Find the general solution of the differential equation. ( Use C for the constant of integration. ) d y d x = 1 2 x 2 7 x 3 2 y = .

Example Indicate which test should be used to determine whether the series below converges or diverges. n = 1 n 3 5 2 7 n 3 6 n 2 2 Solution Since a n is an algebraic function of n , we compare the given series with a p - series. The comparison series for the is I b n , where b n = n 3 2 7 n 3 = n 3 2 7 n 3 = Need Help?

Strontium - 9 0 has a half - ife of 2 8 days. ( a ) A sample has initial mass of 5 0 mg . Find a formula for the mass remaining after t days, y ( t ) . y ( t ) = Impressive mork. ( b ) Find the mass ( in mg ) remaining after 2 0 days. ( Round your answer to one decimal place. ) mg

H 4 0 : Differentiating Exponential Functions Find f ' ( x ) . f ( x ) = 2 x e - 3 e x f ' ( x ) = ( Type an exact answer in terms of e . Find d y d x for the function y = 2 9 x . d y d x = )

DUE TONIGHT PLEASE HELP!!! lesults for this submission 1 of the answers is NOT correct. Solve the following differential equation by variation of parameters. Fully evaluate all integrals. y ' ' 1 6 y = s e c ( 4 x ) a . Find the most general solution to the associated homogeneous differential equation. Use c 1 and c 2 in your answer to denote

Question content area top Part 1 The population of a certain state from 2 0 0 0 to 2 0 1 0 was recorded. During that decade, the state grew from 2 0 . 6 million in 2 0 0 0 to 2 6 . 6 million in 2 0 1 0 . Use an exponential growth model to predict the population of the state in 2 0 3 5 .

Evaluate the iterated integral. 0 6 0 4 0 1 ( x y z ) d x d z d y

5 x + 1 ( 2 x + 1 ) ( x - 1 ) d x

Find the tangent plane to the equation z = - 4 x 2 4 y 2 5 y at the point ( - 3 , 5 , 8 9 )

Question 1 2 Rewrite a s the sum, difference, o r multiple o f several logarithms: l n ( x + 7 3 ( x + 3 ) 3 ) 1 3 l n ( x + 7 ) - 3 l n ( x + 3 ) l n ( 3 x + 7 ) - l n ( 3 x + 9 ) 1 3 l n ( x ) + l n ( 7 ) - 3 l n ( x + 3 ) l n ( x ) + l n ( 7 ) - l n ( 3 x ) + l n ( 9 )

Differentiate the following function. r ( t ) = ( : 5 t e - t , 8 t l n t , t s i n t : )

Q 1 ( 2 points ) Does the series n = 1 n 2 - n 3 2 l n ( n ) 2 n n 2 converge or diverge? Show and justify your work.

2 5 2 1 5 = A . 4 2 0 B . 4 1 5 C . 8 1 5 D . 4 5

Compute T 2 ( x ) a t x = 1 for y = e x and use a calculator t o compute the error | e x - T 2 ( x ) | a t x = 1 . 5 . T 2 ( x ) = | e x - T 2 ( x ) | = Please provide a n accurate answer.

solve this differential equation using power series x ^ 2 y + ( ( 9 / 5 ) x + x ^ 2 ) y ( 1 / 5 ) y = 0 .

if y = 5 . 2 - 0 . 6 7 x what is the slope and y - intercept of the line

Consider the following differential equation and initial value. y = 2 x 3 y + 1 , y ( 1 ) = 4 ; y ( 1 . 2 ) Use Euler's method to obtain a four - decimal approximation of the indicated value. Carry out the recursion of the equation yn + 1 = yn + hf ( xn , yn ) . First, use increment h = 0 . 1 . y ( 1 ) = 4 y y ( 1 . 2 ) Then, use increment h = 0 .

Find the interval of convergence for the following power series. Be sure to check the endpoints if necessary. n = 0 ( 3 x ) n ( 2 n ) !

Consider the function f ( x ) = { ( 6 * x + 5 , x = - 2 ) , ( 6 * ( x + 1 ) ^ ( 3 ) , - 2 = 1 ) : } Tick all of the following statements that are correct. lim _ ( x - > 1 ) f ( x ) exists

Determine whether each function i s a polynomial function o r another type o f function. Justify your decision. a

Question 8 ( 1 0 points ) Use the method of Lagrange multipliers to find the extreme values of the function f ( x , y ) = x 2 2 y 2 on the circle x 2 y 2 = 1 Question 9 ( 1 0 points ) Assessment Details Calculator

6 . 7 . Identify the mental blocks you encountered in Exercises 6 . 3 through 6 . 6 .

Choose a correct expression for s i n ( 2 a r c t a n ( x 3 ) ) . - 6 x x 2 9 6 x x 2 9 9 - x 2 9 x 2 x 2 - 9 x 2 9

Sample test 4 Show all your work. Find d y d x if y = 9 x 2 l o g x d d x ( 9 x 2 l o g x ) Find d y d z if y = s i n ( e - 7 z 7 ) Find d y d x if y = e 2 3 e 2 - 2 Find lim x 1 - ( x - 1 ) t a n ( 2 x ) Find y ' if t a n y = e - x 3 l o g 3 x 2 Find f ' ( x ) if f ( x ) = x 3 l n ( c o s x 4 ) s i n 3 x Find f ' ( x ) if f ( x ) = 5 - 3 x c o t

Let F ( x ) = f ( x 4 ) and G ( x ) = ( f ( x ) ) 4 . You also know that a 3 = 1 1 , f ( a ) = 2 , f ' ( a ) = 6 , f ' ( a 4 ) = 1 5 Then F ' ( a ) = and G ' ( a ) =

Find the domain o f the following function. f ( x , y ) = 1 6 - 4 x 2 - 4 y 2 2 Select the correct choice and, i f necessary, fill i n any answer box ( e s ) within your choice. A . { ( x , y ) : x 2 + y 2 B . { ( x , y ) : x 2 + y 2 C . and y , ( U s e a comma t o separate answers a s needed. ) D . { ( x , y ) : x 2 + y 2 E . R 2

If f ( x ) = 6 - x 2 8 x 2 , find: f ' ( x ) =

Find the vertical, horizontal, and oblique asymptotes, i f any, for f ( x ) = 3 x 3 + 2 0 x 2 + 1 4 x - 6 5 x 2 + 8 x + 1 5 . vertical: x = 5 , x = 3 slant: y = 3 x + 4 vertical: x = - 5 , x = - 3 slant: y = 3 x - 4 vertical: x = - 5 , x = 3 horizontal: y = 0 horizontal: y = 0 slant: y = 3 x - 4

The average rate o f change o f the function y = x 2 - x - 1 from x = - 1 t o x = 4 i s a

f ( x ) = 2 x 2 1 2 x - 4 ( a ) The function's vertex is at the point ( Type an ordered pair, using integers or fractions ) ( b ) Determine whether the parabola has a maximum value or a minimum value and find the value. Select the correct choice below and fill in the answer box to complete your choice. ( Type an integer or a fraction. ) A . The

For the function f ( x ) = x 2 7 x , find f ' ' ( x ) . Then find f ' ' ( 0 ) and f ' ' ( 4 ) . f ' ' ( x ) = Select the correct choice below and fill in any answer boxes in your choice. A . f ' ( 0 ) = ( Simplify your answer. Type an exact answer. ) B . P ' ' ( 0 ) is undefined. Select the correct choice below and fill in any answer boxes in your

Eliminating the parameter from the parametric equations x = h + asect and y = k + btant gives the equation ( x - h ) 2 a 2 - ( y - k ) 2 b 2 = 1 , which i s a hyperbola. Find a set o f parametric equations for a hyperbola with vertices a t ( 0 , 2 4 ) and ( 0 , - 2 4 ) and foci a t ( 0 , 2 5 ) and ( 0 , - 2 5 ) . x = , y = ( S i m p l i f y your

Evaluate the integral. ( Remember to use absolute values where appropriate. Use C for th d u u 2 - u 2 2

Now w e need t o find o f the constant c for which the left and right l i m i t s are equal. Comparing the two l i m i t s and solving for c , w e have 3 6 c + 4 2 = 2 1 6 - 6 c 4 2 c = c =

The regression equation is = 2 9 . 2 9 0 . 8 4 X , the sample size is 8 , and the standard error of the slope is 0 . 2 2 . What is the test statistic to test the significance of the slope? Multiple Choice z = 3 . 8 1 8 z = + 3 . 8 1 8 t = 3 . 8 1 8 t = 0 . 8 4 0

Find the general indefinite integral. ( Remember the cd s e c ( t ) ( 9 s e c ( t ) 2 t a n ( t ) ) d t

Evaluate the given integral. - 2 2 ( 6 c o s 5 t ) ( t ) d t - 2 2 ( 6 c o s 5 t ) ( t ) d t =

Evaluate the integer 0 4 ( 1 t a n t ) 3 s e c 2 t d t

Find the derivative of f ( x ) = 1 9 * 6 x . f ' ( x ) =

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