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mathematics
june 2018
9

Mathematics Questions and Answers June 9, 2018

Expert-verified solutions to help you succeed

Compute the first - order partial derivatives o f the given function. P = e y 8 + z 8 2 ( U s e symbolic notation and fractions where needed. ) HILUIELI MISVVEI d e l P d e l z = Incorrect Answer

( 1 2 pts ) Find the most general antiderivative for each function. ( Don ' t forget the " c 2 ) a . f ( x ) = x 4 4 3 - 7 x 4

The function y = - 2 x 2 ( x 2 ) ( x - 3 ) has derivative y ' = 2 x ( x 1 2 ) ( x 2 ) 2 ( x - 3 ) 2 Determine Determine . . . ( a ) An equation for the Tangent to the graph of y at x = - 1 ( b ) The locations and types of the Vertical Asymptotes of y 5 points ( e ) The location and type of the Horizontal Asymptote of y ( d ) The tocations ( just

The first error was made i n Step 3 . 7 l n ( x ) - 2 l n ( 7 ) - 1 3 l n ( y ) STEP 1 : = l n ( x 7 ) - l n ( 7 2 ) - l n ( y 1 3 ) STEP 2 : = l n ( x 7 ) - l n ( 4 9 ) - l n ( y 3 ) STEP 3 : = l n ( x 7 - 4 9 ) - l n ( y 3 ) STEP 4 : = l n ( x 7 - 4 9 - y 3 ) Select the error that i s made i n Step 3 o f this solution. When rewriting l n ( x 7 )

A defined on an interval, then the following holds. If A ' ( w ) > 0 for all A ' ( w ) 0 w > c A ( c ) A A ' ( w ) 0 A ' ( w ) > 0 w > c A ( c ) A A ' ( w ) A ( w ) = - w 2 3 6 w A ' ( w ) = w = w and A ' ( w ) > 0 for all w > c , then A ( c ) i s the a b s o l u t e minimum value o f A . T o apply this test w e must first find A ' ( w ) . Doing s

( 1 0 points ) Convert - 1 1 0 1 - y 2 2 x 2 + y 2 x 2 + y 2 2 x y z d z d x d y into a n integral i n cylindrical coordinates ( 4 points ) , and evaluate the integral ( 6 points ) .

Use a substitution o f the form u = a x + b t o evaluate the indefinite integral below. ( x + 2 ) 1 3 d x

4 . 5 : 1 8 s i n ( 1 x ) x 2 d x

Establish the indefinite integral: , 1 1 4 x 2 d x 4 t a n - 1 ( 4 x ) C 2 t a n - 1 ( 2 x ) C 1 4 t a n - 1 ( 4 x ) C 1 2 t a n - 1 ( 2 x ) C

Express the number a s a ratio o f integers. 0 . b a r ( 8 2 ) = 0 . 8 2 8 2 8 2 8 2 dots Step 1 W e will r e - write 0 . b a r ( 8 2 ) a s 0 . 8 2 8 2 8 2 dots = 0 . 8 2 + 0 . 0 0 8 2 + 0 . 0 0 0 0 8 2 + dots. Factoring out 0 . 8 2 from each o f these terms gives u s 0 . 8 2 ( + , + , + dots )

Question 2 Find the integral t a n 2 x s e c 6 x d x 1 7 t a n 7 x + 2 5 t a n 5 x - 1 3 t a n 3 x + C 1 7 t a n 7 x + 2 5 t a n 5 x + 1 3 t a n 3 x + C 1 5 t a n 5 x + 2 3 t a n 3 x - t a n x + C 1 5 t a n 5 x + 2 3 t a n 3 x - t a n x + c

Find the indefinite integral and che x 6 ( 1 x 7 ) 2 d x

Find simpler expressions for the following without using either e o r l n . a . e l n 9 . 1 b . e - l n ( 3 x 4 ) c . e l n ( 3 x ) - l n ( 2 y )

Let F [ x c ] = x 2 - 1 x 2 - 4 , With F ' [ x ] = - 6 x [ x 2 - 4 2 ] and F ' [ x ] = 6 [ 3 x 2 4 ] [ x 2 - 4 ] 3 Find the local extrema of f ( x ) and indicate on which intervals f ( x ) is increasing of decreasing .

compute the integral using geometry - 4 4 ( 4 - | x | ) d x

Find the derivative of the function y = 4 e t 2 t . Put the expression containing e in the second entry. Be careful, polynomials in the power must be in parentheses. y ' = 4 ( 1 ) ( )

Use a change of variables to evaluate the following indefinite integral. 2 x ( x 2 - 4 ) 9 9 d x Determine a change of variables from x to u . Choose the correct answer below. A . u = ( x 2 - 4 ) 9 9 B . u = x 2 C . u = 2 x B . y = x 2 - 4 Write the integral in terms of u . 2 x ( x 2 - 4 ) 9 0 d x = ddu Evaluate the integral. 2 x ( x 2 - 4 ) 9 9 d

Evaluate the definite integral given below. 0 3 ( - 5 s i n ( x ) - c o s ( x ) ) d x Enter an exact answer.

[ - / 3 Points ] Three integrals are given that, although they look similar, may require different techniques of integration. Evaluate the integrals. ( Demember the constant of integration. ) ( a ) 2 s i n 2 ( x ) d x ( b ) 2 s i n 3 ( x ) d x ( c ) 2 s i n ( 2 x ) d x

Question Given F ( x ) below, find F ' ( x ) . F ( x ) = 3 c o s ( x ) ( t 2 1 0 ) d t Provide your answer below: Activity Details

Find the limit: lim x 1 x 2 - x 3 x - 1 = lim x 1 x 2 - x 3 x - 1

Find all possible functions with the given derivative. f ' ( t ) = c o s 7 t + s i n ( t 2 )

Determine whether the given differential equation is exact. If it is exact, solve it . ( If it is not exact, enter NOT. ) ( 2 x - 1 ) d x ( 7 y 4 ) d y = 0

First Order [ E a s y ] ( S e p a r a b l e ) Solve the following ODE: y ' = 2 0 2 5 y

f ( x ) = 9 x 3 1 8 3 x 2 3 f ' ( x ) =

Consider the system d x d t = ( 5 - x ) ( y - x ) , d y d t = y ( 5 - 4 x - x 2 ) a ) Find all the critical points ( equilibrium solutions ) , Number of critical polnts:

QUESTION 2 0 * 1 POINT Evaluate the definite integral below. 2 3 3 2 x 1 d x Enter an exact answer. Provide your answer below: 2 3 3 9 r 1 d x =

You are trying to solve an initial value problem of the form dydt = f ( t , y ) , with a known initial condition y ( 0 ) = y 0 . What order of magnitude do you expect the global error in your solution to be if you solve using the Forward Euler method with a time - step size of t = 0 . 0 5 ?

Find 2 x 5 x + 8 d x + C

Select all statements that are correct. There may be more than one correct answer. The statements may appear in what seems to be a random order A . If the discriminant of f is positive, and f k e is negative, then we have a local maximum. B . The point ( a , b ) is a critical point for the multivariable function f ( x , y ) , if both partial

Use the properties of logarithms to evaluate the expressior e l n x 3 2 A . x 3 2 B . l n x 3 2 C . x D . e x 3 2

Find unit tangent and normal vectors T ( t ) and N ( t ) and find the curvature of r ( t ) = ( : t , t 2 , 4 : ) .

Calculate the limit . lim x n 1 1 n 1 8 ( Give an exact answer. Use symbolic notation and fractions where needed. Enter DNE if the limit does not exist. ) lim x n 1 1 n 1 8 = Use this limit to determine if the series * * = 1 n 1 1 n 1 8 converges or diverges. Does the series coaverge of diverge? Since the limit is finite, the series coaverges to

( 5 points ) Use geometry and properties o f integrals t o evaluate the following definite integrals. Show your work. 0 1 ( x + 1 - x 2 2 ) d x

Evaluate the indefinite integral given below. 8 5 x d x Provide your answer below: 8 5 x d x =

Evaluate the double integral D 1 2 xydA, where D i s the triangular region with vertices ( 0 , 0 ) , ( 1 , 2 ) , and ( 0 , 3 ) . Answer:

Find and equation for a plane that goes through ( 7 , 3 , - 1 ) and is parallel to the plane - 4 x 5 y - 1 3 z = 2 2

If f ( x ) = x + 5 , A = [ 1 , 2 ] , B = [ 0 , 3 ] , find f 1 ( A B ) .

Question 1 0 1 0 pts Find the partial fraction decomposition of: 3 x 3 - 7 x 6 x 3 - 4 x 3 3 x H M ) ( M
$Question 1 0 1 0 pts Find the partial fraction$

Find the surface area generated by rotating the lemniscate r 2 = 6 c o s ( 2 ) about the polar axis. a . 2 6 2 b . 1 2 ( 2 - 2 2 ) c . 6 2 d . 6 2 1 0 e . 6

The 2 0 1 0 census in a particular area gives us an age distribution that is approximately given ( in millions ) by the function f ( x ) = 3 9 . 4 2 . 1 3 x - 0 . 8 6 7 x 2 where x varies from 0 to 9 decades. The population of a given age group can be found by integrating this function over the interval for that age group. ( a ) Find the integral

Use continuity to evaluate the limit . ( a ) lim x 2 x 2 0 - x 2 2 ( b ) lim x s i n ( x s i n x )

A machinist is required to manufacture a circular metal disk with area 1 0 0 0 c m 2 . ( a ) What radius produces such a disk? ( Round your answer to four declmal places. ) cm places. ) c m c m

Evaluate the integral. 1 3 d x x l n ( x ) ( E x p r e s s numbers i n exact form. Use symbolic notation and fractions where needed. Enter the symbol not converge. ) 1 3 d x x l n ( x ) =

Evaluate the following indefinite integral: ( - 2 s i n ( x ) + 5 ) d x

given the parametric equations x = 2 t - 3 and y = 2 t ^ 2 + 1 , what is d ^ 2 y / dx ^ 2 when x = 4

find L [ e - 2 t ( 3 c o s 6 t - 5 s i n 6 t ) ]

Let A ( t ) = ( [ e 4 t , c o s t ] , [ 2 t , 3 t 2 - 1 ] ) Find d A d t , 0 2 A ( t ) d t , 0 t A ( s ) d s

Evaluate the given integral: 0 1 7 2 x e 7 x 2 d x

Use the trapezoid rule with n = 5 to approximate the integral 1 2 1 x dx

Consider the functions z = 9 e x l n y , x = l n ( u c o s v ) , and y = usinv. ( a ) Express d e l z d e l u and d e l z d e l v as functions of u and v both by using the Chain Rule and by expressing z directly in terms of u and v before differentiating. ( b ) Evaluate d e l z d e l u and d e l z d e l v at ( u , v ) = ( 9 , 6 ) ( a ) Find each

Evaluate the following limits . If needed, enter for and - for - . ( a ) lim x ( - 2 5 x 2 - 3 3 x 3 ) = ( b ) lim x - ( - 2 5 x 2 - 3 3 x 3 ) = I

Find F ' ( x ) . , F ( x ) = x x 8 ( 8 t 7 ) d t F ' ( x ) =

QUESTION 3 * 1 POINT Evaluate the indefinite integral given below. ( 4 x 2 7 e - 4 x ) d x Provide your answer below: ( 4 x 2 7 e - 4 x ) d x =

QUESTION 1 5 , I POINT Evaluate the indefinite integral given below. - 7 x 3 c o s ( 5 x 4 1 0 ) d x Provide your answer below: - 7 x 3 c o s ( 5 x 4 1 0 ) d x =

Find the sum using the formulas for the sums of powers of integers. 5 n ^ 2 n = 1

Find a recurrence relation for the a s awuming the solution to the differential equation x y ' ' y ' x y - 0 has a power series enteroot at x 0 - 1 of the form = = 0 a .

Use L'Hopital to determine the following limit . Use exact values. lim x 0 ( 1 s i n 8 x ) 1 x =

Observe the grid below: Using the values o f a and b obtained from the grid, compute the area A under the curve y = x 2 between x = a and x = b : A = a b x 2 d x Requirements: First determine the values o f a and b from the grid Then calculate the definite integral using these values Show all steps clearly

A projectile is thrown upward so that its distance above the ground ather t seconds is h ( t ) = - 1 6 t 2 6 0 8 t . a ) Find the maximum height the object reschos. b ) Find the time it takes the object to reach the ground. ( Pint: When an object reaches the ground, its height is 0 . )

0 8 1 8 - x 3 d x Find the Integral 1 1 . 2 x 3 - 4 x 2 - 1 5 x 5 x 2 - 2 x - 8 d x Please answer in terms of a calculus 2 student and no AI and show all work thanks!

8 . 3 6 . ( a ) Find a Laurent series for 1 ( z 2 - 4 ) ( z - 2 ) centered a t z = 2 and specify the region i n which i t converges. ( b ) Compute C [ 2 ) , 1 d z ( z 2 - 4 ) ( z - 2 ) .

Determine whether the following function is continuous at a . Use the continuity checklist to justify your answer. f ( x ) = * - 9 x - 3 if x# 3 ; a = 3 if x = 3 Question 2 . Question 3 Question 4 Select all that apply. A . The function is continuous at a = 3 . B . The function is not continuous at a = 3 because f ( 3 ) is undefined. The function

Solve the following initial - value problem ( 1 5 points ) . t y y ' = t 2 1 , y ( 1 ) = 4

Use a n appropriate substitution t o solve the differential equation d y d x - 2 x y = - 3 x y 3 . ( 8 p t s . )

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

4 ( 2 0 pts ) . ( 2 0 pts ) . Evaluate the double integral ( x ) d x d y over the region R that is bounded by y = x 2 , and y = 2 x . Choose the mont comenient order of integration.

( 1 point ) Find two parametric representations for the part of the plane z = 4 x 3 y that lies inside the cylinder x 2 y 2 = 1 6

Find the derivative of P ( t ) = ( 4 t + 2 4 0 ) / ( 0 . 3 t ^ ( 2 ) + 3 )

Find the sum using the formulas for the sums of powers of integers. 8 n n = 1

Determine the intervals on which the following function is concave up or concave down. Identify any inflection points. f ( x ) = x 4 - 2 x 3 3 Determine the intervals on which the given function is concave up or concave down. Select the correct choice below and fill in the answer box ( es ) to complete your choice. ( Simplify your answer. Type

V . For each of the following series, determine whether it converges absolutely , converges conditionally, or diverges. Clearly state which convergence test you are using and justify your reasoning. n = 1 2 n 1 n n = 1 1 n ( n 1 )

Find the function y = y ( x ) y = y ( x ) ( for x > 0 x > 0 ) which satisfies the separable differential equation dydx = 5 + 1 1 xxy 2 ; x > 0 dydx = 5 + 1 1 xxy 2 ; x > 0 with the initial condition y ( 1 ) = 4 y ( 1 ) = 4 . y =

Find the values a , b , c , d which make the quadratic function f ( x ) = ax ^ 3 + bx ^ 2 + cx + d have a local minimum value of 7 at x = 0 and a local maximum value of 1 5 at x = 2 . a = b = c = d =

Evaluate the indefinite integral. ( Remember the constant of integration. ) 7 t 7 t 8 d t

Consider the function f ( x ) = l n 4 ( e 6 x + 5 ) . Find f ' ( x ) .

Determine whether the following series converges or diverges. If it converges, find its sum. Be sure to include justification throughout your work and circle whether convergent or divergent. X n = 0 3 n + 1 + 2 2 n 5 n Conclusion: Convergent series Divergent Series

find the general solution to the linear system of the differential equation x = [ row 1 = - 2 - 3 , row 2 = 3 - 2 ]

2 . 1 Show that the expression ( 1 - c o s 2 x ) ( 1 + c o s 2 x ) c o s 2 x i s equivalent t o s i n 2 x + s e c 2 x - 1 .

( 5 points ) Given f ( x ) = 5 x 2 - 2 x 1 , find ( a ) [ f ( a ) ] 2 ( b ) f ( a h )

Determine the following limits , using \ infty or - \ infty when appropriate, or state that they do not exist. Given expressions: a . \ lim _ { x \ to 5 ^ - } \ frac { x ^ 2 - 7 x + 1 2 } { ( x - 5 ) ^ 2 } b . \ lim _ { x \ to 5 ^ + } \ frac { x ^ 2 - 7 x + 1 2 } { ( x - 5 ) ^ 2 } c . \ lim _ { x \ to 5 } \ frac { x ^ 2 - 7 x + 1 2 } { ( x - 5 ) ^

1 / 3 t ^ 3 1 / 4 t ^ 2 - t ) dt . Evaluate the integral

Find the equation of the sphere passing through Pleft parenthesis negative 4 , 3 , 8 right parenthesis and Qleft parenthesis 6 , negative 7 , 9 right parenthesis with its center at the midpoint of PQ .

A model for the surface area of some solid object is given by S = 0 . 2 7 1 w 0 . 5 2 7 h 0 . 8 3 7 , where w is the weight ( in pounds ) , h is the height ( in inches ) , and S is measured in square feet. If the errors in measurements of w and h are at most 1 % , estimate the maximum error in the calculated surface area. The estimate of the

The answer is NOT correct. Use the Laplace transform to solve the following initial value probleme y ' 6 y 2 5 y = 4 ( - 7 ) , y ( 0 ) = 0 , y ( 0 ) = 0 Use step ( t - c ) for u c ( t ) , y ( t ) =

Let f ( t ) = F ' ( t ) . Write the integral a b f ( t ) d t and evaluate it using the Fundamental Theorem of Calculus: r F ( t ) = t 3 . a = 2 , and b = 3 , then the integral is a b f ( t ) d t = a n a n where v 1 = v 2 = Then, after evaluation, a b f ( t ) d t =

Find G ' ( w ) if G ( w ) = 5 9 w 5 5 w 4 G ' ( w ) =

Let C be the curve y = x 3 3 1 4 x for 0 . 4 x 2 . 7 . Find the arc length of C shown above. First find and simplify 1 y 2 2 = Then arc length =

Suppose that lim x c f ( x ) = 1 3 and lim x c g ( x ) = - 1 1 . Find the following l i m i t : lim x c ( f ( x ) + g ( x ) )

Find the vertex, focus, and directrix of the parabola. \ table [ [ ( x - 6 ) 2 - 2 ( y 3 ) = 0

Evaluate x 2 3 d x ( 1 point ) 2 3 x - 1 3 + C x 5 2 + C x 5 3 + C 2 5 x 5 2 + C 3 5 x 5 3 + C

Find 4 e - 2 x d x + C

Evaluate the indefinite integral. ( Remember the constant of integration. ) t 5 e - t 6 d t

Use substitution to evaluate the indefinite integral. l n ( 7 x ) 3 x d x

lim x ( 2 x 3 x 5 ) x 1

Find equations for the oblique asymptotes, i f any, for the following rational function. f ( x ) = 5 x 2 - 6 x + 1 - x + 9

lim x ( 5 x 1 8 1 0 x 1 4 + 2 6 x 1 8 ) =

What is the derivative of the function f ( x ) = - 2 3 x 3 ? Select one: a . - x 2 b . - 3 x 2 c . - 2 x 2 d . - 2 x 2 9

Determine the sum of the geometric series 2 2 5 2 * 2 3 5 3 2 4 5 4 - dots, if it is convergent. 3 5 5 7 4 3 5 7 5 none of these

Let f ( x ) = x 2 - 8 x . a . Determine the slope of the secant line joining ( 1 , f ( 1 ) ) and ( 9 , f ( 9 ) ) . Slope of secant line: b . Determine the slope of the secant line joining ( 5 , f ( 5 ) ) and ( 5 h , f ( 5 h ) ) . Slope of secant line:

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