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Mathematics Questions and Answers May 9, 2018

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Find ( A ) the leading term of the polynomial, ( B ) the limit as x approaches , and ( C ) the limit as x approaches - . p ( x ) = 1 1 + 5 x 4 - 8 x 5 ( A ) The leading term is ( B ) The limit of p ( x ) as x approaches is ( C ) The limit of p ( x ) as x approaches - is
Find ( A ) the leading term of the polynomial, (

find the fourth derivative of f ( x ) = 2 [ x , 3 ] + 3 sinx

3 . Let F ( x ) = ( 1 , 3 1 2 ( cos ( 1 ' ) + 2 ) dt . + A . Using the First Fundamental Theorem of Calculus, find F ' ( x ) . ( Yes , this is as easy as it seems. )

The value of a watch, w ( t ) , is modeled by the function w ( t ) = 1 2 ( 1 + 0 . 3 7 ) t , where t is time in years. Does this function represent growth or decay? What is the percent rate of change?

Find dy / dx by implicit differentiation. 8 cos ( x ) sin ( y ) = 7 dydx =

oning for Business Multiply the following rational expressions. Write the answer in lowest terms. ( c + d ) / ( 3 x ) * ( 6 3 ) / ( 2 ( c + d ) )

Find the general solution of the differential equation x 4 y ' + 8 y 2 = 0 using separation of variables. ( Use symbolic notation and fractions where needed. C is an arbitrary constant. Absorb into C as much as possible. ) y = 1 C + 8 3 x 2 Incorrect Answer
Find the general solution of the differential

Consider the following. lim x 9 + x + 8 x - 9 If x is chosen to be sufficiently close to 9 with x greater than 9 , then f ( x ) ? 0 , 0 and f ( x ) can be made - - - Select - - Determine the infinite limit . - Need Help?
Consider the following. lim x 9 + x + 8 x - 9 If

what is the derivitave of x to the 1 0

Fill in the blanks using the figure below. What is lim x 0 - f ( x ) ? Answe What is lim x 0 + f ( x ) ? Answer: What is lim x 0 f ( x ) ? Answer: What is f ( 0 ) ? Answe What is lim x 4 f ( x ) ? Answer
Fill in the blanks using the figure below. What

find all roots for ( - 6 4 ) ^ 1 / 4

Solve trigonometric integrals with products and powers of sine and cosine Question Evaluate the integral below. upper bound = ( ) lower bound = ( 3 / 2 ) 5 cos 3 ( 5 x ) d x Enter an exact answer.

1 7 The rectangle yontims 7 semi - circles and 2 quarter circles. What percentage of the rectangle is shaded?

\ int _ ( - 4 ) ^ 4 3 - | x | dx

Does 0 0 . 3 x 2 5 - x 2 = 1 7 5 s e c ( 6 )

Given the points A ( 1 , 1 , 0 ) , B ( 0 , 1 , 1 ) and C ( 1 , 0 , 1 ) , find | p r o j v e c ( A C ) vec ( A B ) | State the EXACT answer only. NO DECIMALS.
Given the points A ( 1 , 1 , 0 ) , B ( 0 , 1 , 1

What is the derivative of the function f ( x ) = ln ( x + x 2 1 ) f ( x ) = ln ( x + x 2 1 ) ? Select one: a . 1 x + x 2 1 1 x + x 2 1 b . 1 x + x 2 1 ( 1 + xx 2 1 ) 1 x + x 2 1 ( 1 + xx 2 1 ) c . 1 x + x 2 1 ( 1 x 2 x 2 1 ) 1 x + x 2 1 ( 1 x 2 x 2 1 ) d . 1 x + x 2 1 ( 1 + 2 xx 2 1 )

Find the particular solution for the differential equation d y d x = s i n x + 5 with the initial condition F ( 0 ) = 2 .
Find the particular solution for the differential

Find the 4 0 th derivative of f ( x ) = sin 2 x

integrate ( e ^ x / ( 1 + e ^ 2 x ) ) dx

give the interval ( s ) of x such that h ( x ) > 0 . Use union symbols between multiple intervals

write down an expression, in terms of x and y , for the total cost of x cakes at 9 0 cents each and y drinks at 7 5 cents each.

QUESTION 1 1 Find the limit . lim x 7 5 2 5 2 5 1 7 2 1 7
QUESTION 1 1 Find the limit . lim x 7 5 2 5 2 5 1

The revenue, , from producing and selling Awesome Hearing Aids is modeled by the function a . How many hearing aids need to be produced and sold in order to maximize the revenue? Number of hearing aids: b . What is the maximum revenue? Maximum revenue: $

1 / ( 1 + ^ 2 ) ^ 7 / 2

Question 1 2 / 3 points Find the derivative of f ( x ) using the First Principle for the following function f ( x ) = x 2 - 2 x 2 - No text entered -
Question 1 2 / 3 points Find the derivative of f

Find the area of the surface when the graph of y = \ sqrt ( 4 - x ^ ( 2 ) ) is rotated about the x - axis ( a ) for 0 = x = 1 , ( b ) for 1 = x = 2 , and ( c ) for 2 = x = 3 .
Find the area of the surface when the graph of y

( 2 points ) Find the limit: ( 2 points ) Find a parametrization of the circle of radius 4 in the x y - plane, centered at the origin, oriented clockwise. The point ( 4 , 0 ) should correspond to t = 0 . Use t as the parameter for all of your answers. x ( t ) = y ( t ) ( 2 points ) Find a vector parametric equation vec ( r ) ( t ) for the line
( 2 points ) Find the limit: ( 2 points ) Find a

4 pts ) 6 ) Convert the Cartesian point ( 3 , 5 ) into two polar points, one with a positive r and one with anegative r

A bucket that has a mass of 1 0 kg when filled with sand needs to be lifted to the top of a 1 0 meter tall building. You have a rope that has a mass of 0 . 2 kg / m that must be secured to the bucket. It takes 1 meter of rope to secure the bucket. Once the bucket reaches the top of the building it only has mass 8 kg because there was a hole in the

the second albegraic value is incorrect
the second albegraic value is incorrect

( 5 ) Sketch a graph of a function, f , that has at least all of the following properties. Draw in any vertical asymptote ( s ) , where appropriate. lim x - 8 f ( x ) = 2 lim x - 4 f ( x ) = f ( 0 ) = - 1 f ( x ) = 0 , when x = 3 f ' ( 6 ) = 0 lim x 7 - f ( x ) = 2 lim x 7 + f ( x ) = - 5 lim x f ( x ) = -
( 5 ) Sketch a graph of a function, f , that has

given expression for y ( t ) below, what is sx ( t ) with respect to y ( t ) ? y ( t ) = x ( - 4 t + 4 )

6 . What is the derivative of the function f ( x ) = 1 3 5 6 3 ? First re - write the function then use the general power rule.

Find an equation of the line through the points ( 8 , 1 ) and ( 4 , 3 ) .

Problem 1 : A tank holds 1 0 0 0 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume \ ( V \ ) of water ( in gallons ) after \ ( t \ ) minutes. a . Sketch a graph of this data. Label your axis. b . Calculate the average drainage rate ( flow rate ) over the following time intervals: \
Problem 1 : A tank holds 1 0 0 0 gallons of

[ 0 / 0 . 6 6 Points ] TAMUBUSCALC 1 2 . 2 . 0 1 0 . 1 / 6 Submissions Used Consider the following graph. Use the given graph of the function f ( x ) to find the x - values for which f ( x ) is nondifferentiable. ( Enter your answers as a comma - separated list. If an answer does not exist, enter DNE, ) x = [ 0 / 0 . 6 6 Points ] TAMUBUSCALC 1 2 .
[ 0 / 0 . 6 6 Points ] TAMUBUSCALC 1 2 . 2 . 0 1

Use a spreadsheet to approximate 2 4 x 3 d x using a right Riemann sum and 1 9 rectangles. Copy - paste your answer below. 2 4 x 3 d x ~~ Remark: note the question asks you to copy - paste, and not type. Depending on the size of your cells, your program may not display all the decimals it's computing, so copy - pasting is likely to give you a
Use a spreadsheet to approximate 2 4 x 3 d x

FREE RESPONSE OUESTION Let f be the continuous function defined on - 4 , 3 whose graph, consisting of three line segments and semicircle centered at the origin, is given at right. Let g be the function given by g ( x ) = 1 x f ( t ) d t . ( 2 points ) a ) Find the values of g ( 2 ) and g ( 3 ) . b ) For each of g ' ( - 4 ) and g ' ' ( - 3 ) , find
FREE RESPONSE OUESTION Let f be the continuous

Describe the sets of points in space whose coordinates satisfy the given inequalities. a . x 2 y 2 z 2 6 4 b . x 2 y 2 z 2 > 3 6 a . Choose the best description of x 2 y 2 z 2 6 4 . A . A ball centered on the origin with a radius of 6 4 B . An infinite cylinder centered on the z - axis with a radius of 8 C . A ball centered on the origin with a
Describe the sets of points in space whose

Evaluatelimh 0 f ( 2 + h ) f ( 2 ) h , where f ( x ) = 3 x + 4 . If the limit does not exist enter DNE. Limit =

Find dydx if y = 4 x ^ 4 + 2

Evaluate the limit , if it exists. ( If an answer does not exist, enter DNE. ) lim u 5 u + 5 u 3 + 1 2 5

A group of 6 2 campers wants to take row boats out on the lake. If each boat holds 6 campers, how many boats will the campers need? row boats

1 . ) Use partial fractions to determine what three fractions have the sum of 1 4 x 2 - 3 2 x - 3 6 x 3 - 3 x 2 - 4 x
1 . ) Use partial fractions to determine what

A particle moves along the x - axis. The velocity of the particle at time t is given by v ( t ) , and the acceleration of the particle at time t is given by a ( t ) . Which of the following gives the average velocity of the particle from time t = 0 to time t = 5 ? a ) v ( 0 ) + v ( 5 ) 5 b ) a ( 0 ) + a ( 5 ) 5 c ) 1 5 - 0 0 5 v ( t ) d t d ) 1 2
A particle moves along the x - axis. The velocity

Use the precise definition of a limit to prove lim x 5 ( 3 x - 4 ) = 1 1 . Be careful to include each step and include the "side work" along the way. Use the Intermediate Value Theorem to show that there is a solution of the equation 2 x = x - x 2 in the interval ( 2 , 3 )
Use the precise definition of a limit to prove

5 . ) Find S curlF. vec ( N ) d S , where 5 is that port of the sphere x 2 + y 2 + ( z - 2 ) 2 = 8 that lies above the x y - plane, vec ( N ) is the unit outword narmal field an S . and F = y 2 c o s ( x z ) i + x 3 e y z J - e - x y z k .
5 . ) Find S curlF. vec ( N ) d S , where 5 is

Solving a system of linear equations with Solve the following system of equations. 1 . 1 x - 0 . 4 y = 4 . 1 0 . 3 x - 1 . 2 y = 3 . 3

Sketch a graph of the region bounded by y = s i n x , y = 3 c o s ( 2 x ) , x = 0 , and x = . Then use an integral ( or integrals ) to find the area of this region. Sketch the region bounded by the lines y = x 2 , y = - 2 x 8 , and y = 2 . Then use an integral ( or integrals ) to find the area of this region. Note: You should be able to work area
Sketch a graph of the region bounded by y = s i n

Find the 1 2 th term in the sequence represented by the equation 6 + 5 ( n - 1 )

Q 4 : find the following limits a ) lim x 0 s i n - 1 x x c ) lim x l n 2 x + 3 2 - l n x + 1 2 b ) lim x 0 x 2 x 2 x - 1 d ) lim x e - x l n ( x ) Q 5 : Evaluate the following improper integrals a ) 0 e - x d x b ) - d x ( 1 + x 2 )
Q 4 : find the following limits a ) lim x 0 s i n

Use Laplace to solve the differential equation. y ' ' - 8 y ' + 1 6 y = 3 e 4 t given that y ( 0 ) = 2 and y ' ( 0 ) = 0 Please dont skip step
Use Laplace to solve the differential equation. y

2 . A particle moves on a straight line with velocity function v ( t ) = sin ( t ) cos 2 ( t ) . Findits position function s ( t ) if s ( 0 ) = 0 .

Find the limit , if it exists. ( If an answer does not exist, enter DNE. ) lim x 5 x 3 2 x + 8

Evaluate the interval, using the indicated trigonometric substitution x 3 x 2 + 4 2 dx , x = 2 t a n

The function f ( x ) = 2 ( x - 1 ) 4 increases without bound as the values of x approach 1 . Provide your answer below: lim x =
The function f ( x ) = 2 ( x - 1 ) 4 increases

If f ( x ) = ln ( 5 x + 1 ) and g ( x ) = 2 x 3 then which of the following is equal to f ( g ( x ) ) ? ln ( 2 5 x 2 + 1 0 x + 1 ) 3 ln ( 1 0 x 1 5 ) ln ( 1 0 x 1 4 ) ln ( 7 x 2 ) 3 ln ( 2 5 x 2 + 1 0 x 2 )

5 . 1 The Mean Value Theorem Calculus Name: CA # 1 \ table [ [ \ table [ [ t a
5 . 1 The Mean Value Theorem Calculus Name: CA #

Question 9 0 . 4 pts dx Oy = x 2 en - xe + Cx Oy = x 2 ea - 2 xe " + 2 e + C O y = xea + Cx O y = xe - 2 ea + 2 x - le " + Cx - 1 find the general solution to the given linear differential equation.

Evaluate the following integral. \ int _ 0 ^ 2 3 x ( x - 7 ) dx

Let f ( x ) = b x 2 e a x . It is known that f ' ( 1 ) = - 6 e - 1 . 1 3 . What is the value of a ? 1 4 . What is the value of b ?
Let f ( x ) = b x 2 e a x . It is known that f '

2 . limx 3 [ f ( x ) + g ( x ) ] limx 3 [ f ( x ) + g ( x ) ] 3 . limx 3 + [ f ( x ) + g ( x ) ] limx 3 + [ f ( x ) + g ( x ) ] 4 . limx 3 + [ f ( g ( x ) ) ] limx 3 + [ f ( g ( x ) ) ] 5 . f ( 3 ) / g ( 3 ) f ( 3 ) / g ( 3 ) 6 . limx 3 [ f ( x ) / g ( x ) ] limx 3 [ f ( x ) / g ( x ) ] 7 . limx 3 + [ f ( x ) g ( x ) ] limx 3 + [ f ( x ) g ( x ) ]
2 . limx 3 [ f ( x ) + g ( x ) ] limx 3 [ f ( x )

Find an equation of the sphere with points P such that the distance from P to A ( - 2 , 5 , 2 ) is twice the distance from P to B ( 4 , 2 , - 3 ) .
Find an equation of the sphere with points P such

Find the acute angle ( in degrees ) between the planes 3 x y 5 z = 6 and - 3 x - y - 2 z = - 9 . =
Find the acute angle ( in degrees ) between the

Suppose 4 6 6 mg . of a radioactive substance are initially present in a closed container, and additional amounts are added at the rate of 5 mg . per day. Let x represent the number of milligrams of the substance. a . Suppose the decay constant for this substance is k = - 0 . 0 1 8 6 3 day ? - 1 . Write down the IVP. x ' = , x ( 0 ) = b . Find the
Suppose 4 6 6 mg . of a radioactive substance are

consider a line that represents the equation y - 5 = ( 1 ) / ( 3 ) ( x + 6 ) what are the slope and y - intercept of the line

b ) 0 e - x 2 x 7 4 d x ; Ans: 8 2 3 solve using gamma fxn
b ) 0 e - x 2 x 7 4 d x ; Ans: 8 2 3 solve using

Solve the following inequality. ( 1 0 ) / ( x - 4 ) > = 5 Write your answer using interval notation.

Add the mixed numbers and simplify the result. 3 ( 2 ) / ( 5 ) + 5 ( 3 ) / ( 4 )

calculate limit: lim t - > Inf ( - 3 t ^ 2 + e 2 t )

0 = 2 . 1 x ^ { 3 } - 1 1 . 6 x ^ { 2 } + 1 7 . 5 x - 6

( 2 points ) Solve the initial value problem d x d t - 2 x = c o s ( 2 t ) with x ( 0 ) = 5 . x ( t ) = - c o s ( 2 t ) + s i n ( 2 t ) 8 T T 2 1 4 e 2 t
( 2 points ) Solve the initial value problem d x

integrate xcos ( 2 x ) dx

what is the asymptote of f ( x ) = 2 ^ ( x - 1 ) + 3

Find the derivative of each function. ( a ) F 1 ( x ) = 9 ( x 4 + 4 ) 5 5 F 1 ' ( x ) = 9 x 3 ( x 4 + 4 ) 4 ( b ) F 2 ( x ) = 9 5 ( x 4 + 4 ) 5 F 2 ' ( x ) = 9 x 3 ( x 4 + 4 ) 6 ( c ) F 3 ( x ) = ( 9 x 4 + 4 ) 5 5 F 3 ' ( x ) = 3 6 x 3 ( 9 x 4 + 4 ) 4 Awesome job! ( d ) F 4 ( x ) = 9 ( 5 x 4 + 4 ) 5 F 4 ' ( x ) =

complex conjugate of 2 3 6 0

Evaluate the integral. ( Use C for the constant of integration. ) cos 3 t 9 sin 2 t 9 dt

the limit as x goes to 5 from the left of ( x ^ 2 + 1 0 x + 2 5 ) / ( - x ^ 4 + 2 x ^ 2 + 3 x )

Volume by cross - sections. The shaded region R shown below is bounded by the lines x = 2 , x = 3 , y = - 1 , and the curve y = l n x . ( Copy the sketch to your own work for submission, and to annotate it . ) Suppose R is the base of a solid whose cross - sections perpendicular to the x - axis are squares ( whose side lies on the base ) . Set up
Volume by cross - sections. The shaded region R

In three dimension space, which one of the following is the function of the sphere centered at the point ( 2 , - 1 , 5 ) with radius 4 ? Question 9 options: ( x - 2 ) 2 + ( y + 1 ) 2 + ( z - 5 ) 2 = 4 ( x - 2 ) 2 + ( y + 1 ) 2 + ( z - 5 ) 2 = 1 6 x 2 + y 2 + z 2 = 1 6

Suppose that in 1 6 1 1 1 6 1 1 , a man bought a diamond worth the equivalent of $ 2 7 2 7 . Suppose that the man had instead been able to put the equivalent of $ 2 7 2 7 in a bank at 3 3 % interest compounded continuously. What would that $ 2 7 2 7 have been worth in 2 0 0 0 2 0 0 0 ? Question content area bottom Part 1 In 2 0 0 0 2 0 0 0 , the $

b ) Consider the piecewise - defined function: f ( x , y ) = { e - 1 x 2 + y 2 x 2 + y 2 , ( x , y ) ( 0 , 0 ) a , ( x , y ) = ( 0 , 0 ) where a is a constant. Task: Determine the value of a that makes the function f ( x , y ) continuous at ( 0 , 0 ) .
b ) Consider the piecewise - defined function: f

Simplify the expression, expressing your answer with no negative exponents. ( x ^ ( 3 ) yz ^ ( 3 ) ) / ( ( xyz ^ ( - 1 ) ) ^ ( - 1 ) )

Given F ( x ) = x c o s t d t , F ' ( x ) = a ) s i n x b ) c o s x c ) c o s x + 1 d ) c o s x - 1
Given F ( x ) = x c o s t d t , F ' ( x ) = a ) s

Using the graph of g ( x ) below, answer the following questions. a ) Where is g ( x ) > 0 ? ( Enter an interval ) b ) Where is g ( x ) 0 ? ( Enter an interval ) c ) Where is g ( x ) = 0 ? ( Enter a list of values, separated by commas ) d ) Where is g ' ( x ) > 0 ? ( Enter an interval ) e ) Where is g ' ( x ) 0 ? ( Enter an interval ) f ) Where is
Using the graph of g ( x ) below, answer the

Evaluate the indefinite integral. 1 4 6 4 - 3 6 x - 9 x 2 2 d x = Remark: here, and in other assessments, do not leave any expression of the form trig ( arctrig ) in your answer.
Evaluate the indefinite integral. 1 4 6 4 - 3 6 x

Note: The curve between \ ( t = 0 \ ) and \ ( t = 1 \ ) is the first quarter of a sinusoidal function of time.
Note: The curve between \ ( t = 0 \ ) and \ ( t =

Suppose that the piecewise defined function f is defined by f ( x ) = { 1 - 2 x , x 2 - 3 x 2 + 1 0 x - 1 1 , x > 2 Determine which of the following statements are true. Select the correct answer below: f is neither differentiable nor continuous at x = 2 . f is continuous but not differentiable at x = 2 . f is both differentiable and continuous at

Solve the equation 6 , x , squared, plus, 6 , x , minus, 1 3 , equals, 0 6 x 2 + 6 x - 1 3 = 0 to the nearest tenth.

Determine the order of the following differential equation. y ' ' 5 y = 4 y 7 Order = Next Question
Determine the order of the following differential

Now integrate and simplify V = 2 1 ( 1 6 x 4 + 3 2 x 2 ) dx . V = 1 6 3 x 3 + 3 2 2 1 = 1 6 3 2 3 + 3 2 2 1 6 3 1 3 + 3 2 1 = 4 6 + 1 6 3 + 3 2 = 4 6 + 1 6 3 = 6 = 6

\ int _ ( - 1 ) ^ 5 ( x - 1 ) e ^ ( x ^ ( 2 ) - 2 x - 8 ) d

equation. Slope = 5 , passing through ( - 5 , 8 ) Type the point - slope form of the equation of th

Compute the derivative of the function f ( x ) = x 2 + 1 x 2 1 f ( x ) = x 2 + 1 x 2 1 . Select one: a . ( x 2 + 1 x 2 1 ) 1 / 2 ( x 2 + 1 x 2 1 ) 1 / 2 b . ( x 2 + 1 x 2 1 ) 1 / 2 2 x 3 ( x 2 1 ) 2 ( x 2 + 1 x 2 1 ) 1 / 2 2 x 3 ( x 2 1 ) 2 c . ( x 2 + 1 x 2 1 ) 1 / 2 2 x ( x 2 1 ) 2 ( x 2 + 1 x 2 1 ) 1 / 2 2 x ( x 2 1 ) 2 d . ( x 2 + 1 x 2 1 ) 1

integral of the absolute value of sinx from the lower bound pi / 3 to upper bound 7 pi / 6

A sprinkler waters a oircular seetion of lawn. Write an equation to represent the boundary of the sprinkler area if the endpoints of a diameter are at ( - 1 2 , 1 6 ) and ( 1 2 , - 1 6 ) . x ^ ( 2 ) + y ^ ( 2 ) = 2 0 x ^ ( 2 ) - y ^ ( 2 ) = 4 0 0 x ^ ( 2 ) + y ^ ( 2 ) = 4 0 0 ( x - 1 2 ) ^ ( 2 ) + ( y - 1 6 ) ^ ( 2 ) = 4 0 0

( b . ) 8 days after a scandal in City Hall was reported, a poll showed that 1 0 0 0 0 0 people have heard the news. Using this information and the differential equation, solve for the number of people who have heard the news after t days.

Based on our definitions show that the statements below hold. That is find a to a given > 0 that verifies the statement according to one of our ( equivalent ) definitions of functional limit . ( a ) [ 1 0 pts ] limx 3 ( 3 x + 2 ) = 1 1 . ( b ) [ 1 5 pts ] limx 2 x 3 = 8 .

Evaluate the following integral or state that it diverges. Integral from 3 to infinity StartStartFraction 2 cosine left parenthesis StartFraction pi Over x EndFraction right parenthesis OverOver x squared EndEndFraction 3 2 cosxx 2 dx

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