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Q:

The Political Action Club has surveyed \( 270 \) students on your campus regarding the relationship between their political affiliation and their preference in the \( 2016 \) presidential election. The results are given in the following table. If a student is selected randomly from those surveyed, find the probability that the student is an Independent, given that the student preferred candidate B. 

Q:

Follow the steps below to find the nonnegative numbers \( x \) and \( y \) that satisfy the given requirements. Give the optimum value of the indicated expression. Complete parts (a) through (f) below. 

\( x + y = 170 \) and the product \( P = x y \) as large as possible. 

(d) Find \( \frac { d P } { d x } \) . Solve the equation \( \frac { d P } { d x } = 0 \) . 

\( \frac { d P } { d x } = 170 - 2 x \) and when \( \frac { d P } { d x } = 0 , x = 85 \) (Use a comma to separate answers as needed.) 

(e) Evaluate \( P \) at any solutions found in part (d), as well as at the endpoints of the domain found in part (c). 

To answer the first part, select the correct choice below and, if necessary, fill in the answer box(es) within your choice. 

A. There was one solution in part (d). For that solution, \( P = 7225 \) . 

B. There were two solutions in part (d). For the lesser value of \( x , P = \) . For the greater value of \( x , P = \) 

Evaluate \( P \) at the endpoints of the domain. At the lower endpoint, \( P = \square \) . At the upper endpoint, \( P = \square \) . 

Q:

A person standing close to the edge on top of a \( 112 \) -foot building throws a ball vertically upward. The quadratic function \( h = - 16 t ^ { 2 } + 96 t + 112 \) models the ball's height above the ground, \( h \) , in feet, \( t \) seconds after it was thrown. 

a) What is the maximum height of the ball? 

b) How many seconds does it take until the ball hits the ground? seconds 

Q:

Consider the function \( f ( x ) = 5 x - 8 \) and find the following: a) The average rate of change between the points \( ( - 1 , f ( - 1 ) ) \) and \( ( 4 , f ( 4 ) ) \) . 

b) The average rate of change between the points \( ( a , f ( a ) ) \) and \( ( b , f ( b ) ) \) . 

c) The average rate of change between the points \( ( x , f ( x ) ) \) and \( ( x + h , f ( x + h ) ) \) 

Q:

Using the given graph of the function \( f \) , find the following. 

(a) the intercepts, if any

 (b) its domain and range

 (c) the intervals on which it is increasing, decreasing, or constant 

(d) whether it is even, odd, or neither 

Q:

For the graph of a function \( y = f ( x ) \) shown to the right, find the absolute maximum and the absolute minimum, if they exist. Identify any local maxima or local minima. 

Q:

Let \( g ( x ) = 5 x ^ { 2 } - 9 . \) 

(a) Find the average rate of change from \( - 2 \) to \( 1 . \) 

(b) Find an equation of the secant line containing \( ( - 2 , g ( - 2 ) ) \) and \( ( 1 , g ( 1 ) ) \) . 

 

Q:

 

Write \( x ^ { 2 } + y ^ { 2 } - 2 x + 6 y - 6 = 0 \) in standard form. Then identify the center and radius. 

Q:

Find the indicated one-sided limits of \( f \) , and determine the continuity of \( f \) at the indicated point. You should also sketch a graph of \( y = f ( x ) \) , including hollow and solid circles in the appropriate places. NOTE: Type DNE if a limit does not exist. 

\( \lim _ { x \rightarrow 3 ^ { - } } f ( x ) = \) 

\( \lim _ { x \rightarrow 3 ^ { + } } f ( x ) = \) 

\( \lim _ { x \rightarrow 3 } f ( x ) = \) 

\( f ( 3 ) = \) 

Is \( f \) continuous at \( 3 \) ? (YES/NO) 

Q:

A company that manufactures a particular product has determined that the cost of producing \( x \) units of the product is described by the function \( C ( x ) = 0.0035 x ^ { 2 } + 1855 \) . Find the average rate of change of the cost as the number of units produced increases from \( 200 \) to \( 400 \) . Round to the nearest cent. 

Q:

The given graph represents the function \( f ( x ) = 2 ( 5 ) \) 

How will the appearance of the graph change if the value in the function is decreased, but remains greater than 0? 

The graph will increase at a slower rate. 

The graph will show a decreasing, rather than increasing, function. 

The graph will show an initial value that is lower on the \( y \) -axis. 

The graph will increase at a constant additive rate, rather than a multiplicative rate. 

Q:

Nick loves to read and discuss books, so he joined a book club. The first book he read as a member of the club was a novel set in ancient Rome. There is a proportional relationship between the time Nick has spent reading the novel (in hours), \( x \) , and the number of pages he has read, \( y \) . Nick read \( 120 \) pages of the novel in 

\( 2.5 \) hours. Write the equation for the relationship between \( x \) and \( y . \) 

\( y = \) 

Q:

A line with a slope of \( \frac { 7 } { 8 } \) passes through the point \( ( 0,4 ) \) . What is its equation in slope-intercept form? 

Write your answer using integers, proper fractions, and improper fractions in simplest form. 

Q:

Which equation represents an exponential function that passes through the point \( ( 2,36 ) ? \) 

\( f ( x ) = 4 ( 3 ) ^ { x } \) 

\( f ( x ) = 4 ( x ) ^ { 3 } \) 

\( f ( x ) = 6 ( 3 ) ^ { x } \) 

\( f ( x ) = 6 ( x ) ^ { 3 } \) 

Q:

Given the information about the function and it's derivatives, find the following. 

\( f ( x ) = 4 x ^ { 1 / 3 } - x ^ { 4 / 3 } f ^ { \prime } ( x ) = \frac { 4 } { 3 } x ^ { - 2 / 3 } ( 1 - x ) f ^ { \prime \prime } ( x ) = - \frac { 4 } { 9 } x ^ { - 5 / 3 } ( x + 2 ) \) 

(Do not recalculate the derivatives! Use the information above to answer.) 

Q:

\(f ( x ) = \sqrt { x } - 4 \) . Find the inverse of \( f ( x ) \) and its domain. 

A. \( f ^ { - 1 } ( x ) = x ^ { 2 } + 4 ; x \geq 0 \) 

B. \( f ^ { - 1 } ( x ) = x ^ { 2 } + 4 ; x \geq - 4 \) 

c. \( f ^ { - 1 } ( x ) = ( x + 4 ) ^ { 2 } ; x \geq - 4 \) 

D. \( f ^ { - 1 } ( x ) = ( x + 4 ) ^ { 2 } ; x \geq 0 \) 

Q:

What are the domain and range of the function represented by the set of ordered pairs? 

\( \{ ( - 6,5 ) , ( - 3,2 ) , ( - 1,0 ) , ( 5 , - 4 ) \} \) 

A. Domain: \( \{ - 4,0,2,5 \} \) 

Range: \( \{ - 6 , - 3 , - 1,5 \} \) 

B. Domain: \( - 6 \leq x \leq 5 \) 

Range: \( - 4 \leq y \leq 5 \) 

C. Domain: \( \{ - 6 , - 3 , - 1,5 \} \) 

Range: \( \{ - 4,0,2,5 \} \) 

D. Domain: \( - 4 \leq x \leq 5 \) 

Range: \( - 6 \leq y \leq 5\) 

Q:

A line passes through \(( 1,5 ) \) and \(( - 2 , - 1 ) \) . 

Find the slope-intercept form of the equation of the line. Then fill in the value of the slope, \(m\) , and the value of the \(y\) -intercept, \(b\) , below. 

 

Q:

The height \( y \) (in feet) of a ball thrown by a child is 

\( y = - \frac { 1 } { 16 } x ^ { 2 } + 6 x + 3 \) 

where \( x \) is the horizontal distance in feet from the point at which the ball is thrown. 

(a) How high is the ball when it leaves the child's hand? feet 

(b) What is the maximum height of the ball? 

(c) How far from the child does the ball strike the ground? Round your answers to the nearest \( 0.01 \) .

Q:

What is the effect on the graph of \( f ( x ) = x ^ { 2 } \) when it is transformed to \( h ( x ) = \frac { 1 } { 8 } x ^ { 2 } - 13 \) ? A. The graph of \( f ( x ) \) is horizontally stretched by a factor of \( 8 \) and shifted \( 13 \) units to the right. 

B. The graph of \( f ( x ) \) is horizontally compressed by a factor of \( 8 \) and shifted \( 13 \) units down. 

C. The graph of \( f ( x ) \) is vertically compressed by a factor of \( 8 \) and shifted \( 13 \) units to the right. 

D. The graph of \( f ( x ) \) is vertically compressed by a factor of \( 8 \) and shifted \( 13 \) units down. 

Q:

The height \( y \) (in feet) of a ball thrown by a child is 

\( y = - \frac { 1 } { 16 } x ^ { 2 } + 6 x + 5 \) 

where \( x \) is the horizontal distance in feet from the point at which the ball is thrown. (a) How high is the ball when it leaves the child's hand? (b) What is the maximum height of the ball? (c) How far from the child does the ball strike the ground? Round your answers to the nearest Ouestion Heln. 

Q:

Which of the following models could be used to find two numbers whose difference is \( 116 , \) but whose product is a minimum? 

\( \bigcirc f ( x ) = x ( 116 - x ) \) 

\( \bigcirc y = 116 - x \) 

\( \bigcirc f ( x ) = x ( x - 116 ) \) 

\( \circ f ( x ) = x y\) 

Q:

Determine the solution to the system of equations graphically. If the system is inconsistent or dependent, so state. 

\( y = - 4 x \) 

\( y = 4 x - 8\) 

Q:

At least one of the answers above is NOT correct. If \( f ( x ) = x ^ { 3 } + 3 e ^ { x } \) , find \( f ^ { \prime } ( 4 ) \) . 

\( f ^ { \prime } ( 4 ) = 211.79445009 \) 

Use this to find the equation of the tangent line to the curve \( y = x ^ { 3 } + 3 e ^ { x } \) at the point \( ( a , f ( a ) ) \) when \( a = 4 \) . The equation of this tangent line can be written in the form \( y = m x + b \) . Find \( m = \) and \( b \) . 

\( m = 211.7944501 \) 

\( b = - 491.38335\) 

Q:

The daily revenue R achieved by selling \(x\) cases of toothpaste is given by the function R \(x ( x ) = - 2 x ^ { 2 } + 32 x + 120\) . Recall that \(h = R ( h ) \) 

a. How many cases should be sold to maximize revenue? 

b. What is the maximum revenue? 

 

 

Q:

 

At least one of the answers above is NOT correct. 

Consider the function \( f ( x ) = 3 x ^ { 3 } - 4 x \) on the interval \( [ - 4,4 ] \) . 

Find the average or mean slope of the function on this interval. 

By the Mean Value Theorem, we know there exists at least one \( c \) in the open interval \( ( - 4,4 ) \) such that \( f ^ { \prime } ( c ) \) is equal to this mean slope. For this problem, there are two values of \( c \) that work: 

\( 2.309\) 

Q:

The equation \( N ( t ) = \frac { 650 } { 1 + 49 e - 0.7 t } \) models the number of people in a town who have heard a rumor after \( t \) days. As \( t \) increases without bound, what value does \( N ( t ) \) approach? Interpret your answer. How many people started the rumor? Number 

\( N ( t ) \) approaches Number 

\( N ( t ) \) is limited by the rate at which the rumor spreads. 

\(N ( t ) \) is limited by the carrying capacity of the town. 

\(N ( t ) \) is limited by the number of people who started the rumor. 

Q:

Find the derivative of \( y = ( 9 x ^ { 2 } - 2 ) ^ { \sec ( x ) } \) . Be sure to include parentheses around the arguments of any logarithmic or trigonometric functions in your answer. Provide your answer below: 

\( y ^ { \prime } = \square \) 

Q:

A T-shirt manufacturer is planning to expand its workforce. It estimates that the number of T-shirts produced by hiring \( x \) new workers is given by 

\( T ( x ) = - 0.75 x ^ { 4 } + 12 x ^ { 3 } , 0 \leq x \leq 12 \) . When is the rate of change of T-shirt production increasing and when is it decreasing? What is the point of diminishing returns and the maximum rate of change for T-shirt production? Graph \( T \) and \( T ^ { \prime } \) on the same coordinate system. The rate of change of T-shirt production is increasing on (Type your answer in interval notation.) 

Q:

Find the equation of the parabola with its focus at \( ( - 5,0 ) \) and its directrix \( y = 2 \) . 

A) \( y = 1 / 4 ( x + 5 ) ^ { 2 } + 1 \) 

B) \( y = - 1 / 4 ( x + 5 ) ^ { 2 } + 1 \) 

C) \( y = - 1 / 4 ( x + 1 ) ^ { 2 } + 5 \) 

D) \( y = - 4 ( x + 5 ) ^ { 2 } + 1\) 

Q:

The function \( f ( x ) = \frac { 1 } { 9 } ( x - 7 ) ^ { 2 } + 4 , x \geq 7 \) is one-to-one. 

(a) Find the inverse of \( f \) and check the answer. 

(b) Find the domain and the range of \( f \) and \( f ^ { - 1 } \) 

Q:

The domain of a one-to-one function \( g \) is \( ( - \infty , 0 ] \) , and its range is \( [ 4 , \infty ) \) . State the domain and the range of \( g ^ { - 1 } \) . 

Q:

An employee's monthly productivity \( M \) , in number of units produced, is found to be a function of the number \( t \) of years of service. For a certain product, a productivity function is shown below. Find the maximum productivity and the year in which it is achieved. 

\( M ( t ) = - 4 t ^ { 2 } + 200 t + 150,0 \leq t \leq 40\) 

Q:

Use transformations to graph the function \( f ( x ) = 3 \cdot ( \frac { 1 } { 4 } ) ^ { x } \) . Determine its domain, range, and horizontal asymptote. 

 

Q:

Find a so that the graph of \( f ( x ) = \log _ { a } x \) contains the point \( ( 5,2 ) \) 

Q:

Suppose that a company has just purchased a new computer for \( \$ 3500 \) . The company chooses to depreciate using the straight-line method for \( 7 \) years. 

(a) Write a linear function that expresses the book value of the computer as a function of its age. 

\( V ( x ) = 3500 - 500 x \) 

(Type your answer in slope-intercept form.) 

(b) What is the implied domain of the function found in part (a)? 

\( \square \) 

(Type your answer in interval notation.) 

Q:

The slope of the tangent line to the parabola \(y = 2 x ^ { 2 } - 7 x + 2\) at the point where \(x = - 3\) is: The equation of this tangent line can be written in the form \(y = m x + b\) where \(m\) is: and where \(b\) is: \(\square \) Message instructor Question Help: Video \(\square \) Mubmit Question 

Q:

Suppose that the functions \(f\) and \(g\) are defined as follows. 

\(f ( x ) = - 5 + 2 x ^ { 2 } \) 

\(g ( x ) = 3 - 4 x\)

 (a) Find \(( \frac { f } { g } ) ( 2 ) \) . 

(a) \(( \frac { f } { g } ) ( 2 ) = \square \) (b) Find all values that are NOT in the domain of \(\frac { f } { g } \) . If there is more than one value, separate them with commas. 

Q:

Solve the inequality \(8 x > - 7 x + 60\) . Write your answer as an inequality and in interval notation, then graph the solution set on the number line. For Graphing: If the symbol in the inequality is \(< \) or \(> \) , use an open circle. If the symbol in the inequality is \(\leq \) or \(\geq \) , use a closed circle. 

Q:

Which best describes the scale of an axis? 

The vertical axis 

The smallest number labeled on the axis 

The first number in an ordered pair 

The point where the \(x\) -axis and \(y\) -axis intersect 

The direction the axis is pointing 

The distance between tick marks on that axis 

The horizontal axis 

The second number in an ordered pair 

Q:

Use the vertex and intercepts to sketch the graph of the quadratic function. Give the equation for the parabola's axis of symmetry. Use the parabola to identify the function's domain and range. 

\(f ( x ) = 9 - ( x - 1 ) ^ { 2 } \) 

Use the graphing tool to graph the equation. Use the vertex and one of the intercepts when drawing the graph. 

Q:

A ball is thrown upward and outward from a height of \( 7 \) feet. The height of the ball, \( f ( x ) \) , in feet, can be modeled by 

\( f ( x ) = - 0.4 x ^ { 2 } + 2.7 x + 7\) 

where \( x \) is the ball's horizontal distance, in feet, from where it was thrown. Use this model to solve parts (a) through (c). 

a. What is the maximum height of the ball and how far from where it was thrown does this occur? The maximum height is \( \square \) feet, which occurs \( \square \) feet from the point of release. (Round to the nearest tenth as needed.) 

Q:

Suppose that the functions \( h \) and \( g \) are defined as follows. 

\( h ( x ) = x + 9 \) 

\( g ( x ) = ( x - 5 ) ( x + 5 ) \) 

(a) Find \( ( \frac { h } { g } ) ( 6 ) \) . 

(b) Find all values that are NOT in the domain of \( \frac { h } { g } \) 

If there is more than one value, separate them with commas. 

Q:

On the work you'll submit check whether \( - 2 \) solves the equation \( 3 x ^ { 2 } - 4 x = 2 x + \) 

24. Then select the proper choice below. Make sure you show your work.

 There is not enough information to answer the question. 

Yes, it is a solution. 

No, it is not a solution. 

Q:

Form a polynomial whose real zeros and degree are given. Zeros: \( - 1,0,2 ; \) degree: \( 3 \) Type a polynomial with integer coefficients and a leading coefficient of \( 1 . \) 

\( f ( x ) = \square \) (Simplify your answer.) 

Q:

(a) Using a graphing utility, draw a scatter diagram of the data using age, \( x \) , as the independent variable and total cholesterol, \( y \) , as the dependent variable. 

(b) Based on the scatter diagram drawn in part (a), decide on a model (linear, quadratic, cubic, exponential, logarithmic, or logistic) that you think best describes the relation between age and total cholesterol. 

A. Linear because the data follows a straight line

 B. Exponential because the data follows an exponential pattern 

C. Quadratic because the data resembles an upside-down "U shape" 

D. Logistic model because the value of cholesterol reaches some limiting value 

E. Logarithmic because the data follows a logarithmic pattern 

F. Cubic because the data resembles a cubic equation 

Q:

Solve the following. Express answers as decimals. Round to four decimal places when applicable. a) \( 2 = 3 ^ { 3 x + 4 } \) 

b) \( - 9 + \log _ { 3 } ( 4 + x ) = - 4 \) 

c) \( 5 ^ { 2 x + 1 } = 8 ^ { x - 2 } \) 

Q:

What is the domain of the following function? 

\( f ( x ) = \sqrt { x ^ { 4 } - 81 } \) 

Choose the correct domain

 A. \( \{ x | - 3 < x < 3 \} \) 

B. \( \{ x | - 3 \leq x \leq 3 \} \) 

C. \( \{ x | x < - 3 \) or \( x > 3 \} \) 

D. \( \{ x | x \leq - 3 \) or \( x \geq 3 \} \) 

E. \( ( - \infty , \infty ) \) 

F. \( \varnothing \) 

Q:

The daily production level for a product is given by \( N ( x ) = 180 - 180 e ^ { - 0.5 x } \) units, where \( x \) is the time in hours after production begins. Find the average production during the first \( 5 \) hours. Enter your answer in exact form or rounded to the nearest whole number. 

Q:

Write the following as an algebraic expression: the quotient of \( 5 \) more than a numbi 

\( y \) and \( 3 \) times less than a number \( x \) . 

\( \frac { 5 + y } { x - 3 } \) 

 

Q:

The slope of the tangent line to the parabola \( y = 2 x ^ { 2 } - 7 x + 7 \) at the point where \( x = 3 \) is:__________.

The equation of this tangent line can be written in the form \( y = m x + b \)  where \( m \) is:_______ and where \( b \) is: ________.

Q:

Let \( f ( x ) = 2 x ^ { 2 } - 3 x + 13 \) 

The slope of the tangent line to the graph of \( f ( x ) \) at the point \( ( 4,33 ) \) is The equation of the tangent line to the graph of \( f ( x ) \) at \( ( 4,33 ) \) is \( y = m x + b \) for 

\( m = \) and 

\( b = \) Hint: the slope is given by the derivative at \( x = 4 \) Question Help: \( \square \) Video \( \square \) Message instructor Submit Question 

Q:

The slope of the tangent line to the parabola \( y = 3 x ^ { 2 } - 6 x + 2 \) at the point where \( x = - 4 \) is: The equation of this tangent line can be written in the form \( y = m x + b \) where \( m \) is: Question Help: \( \square \) Video \( \square \) Message instructor Submit Question 

Q:

please include the number where b is

The slope of the tangent line to the parabola \( y = 2 x ^ { 2 } - 3 x + 3 \) at the point where \( x = 0 \) is: The equation of this tangent line can be written in the form \( y = m x + b \) where \( m \) is: and where \( b \) is: Question Help: Video Message instructor Submit Question 

Q:

The slope of the tangent line to the parabola \( y = 2 x ^ { 2 } - 3 x + 3 \) at the point where \( x = 0 \) is ?

Q:

Type the correct answer in the box. Round your answer to the nearest whole number. The population in a city is \( 650,000 \) and grows at a rate of \( 5 \) percent per year. In approximately ___years, the population of the city will reach \( 1 \) million. (Model the situation as \( P = P _ { 0 } \) 

Q:

Suppose that \( f ( x ) = 4 x - 1 \) and \( g ( x ) = - 3 x + 6 \) 

(a) Solve \( f ( x ) = 0 \) 

(c) Solve \( f ( x ) = g ( x ) \) (b) Solve \( f ( x ) > 0 \) 

(e) Graph \( y = f ( x ) \) and \( y = g ( x ) \) and find the point that represents the solution to the equation \( f ( x ) = g ( x ) \) . 

(a) For what value of \( x \) does \( f ( x ) = 0 \) ? 

\( x = \frac { 1} { 4} \)

(Type an integer or a simplified fraction.) (b) For which values of \( x \) is \( f ( x ) > 0 \) ? 

\( x > \) ___

(Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) 

Q:

Use the following equation answer the following question. No calculators allowed. 

\( y = x ^ { 2 } - 2 x - 15 \) 

What is the equation of the axis of symmetry? 

Q:

Which of the following functions is quadratic: 

\( \quad f ( x ) = - 3 x - 5\) 

b        \(b ( x ) = \frac { 5 } { x ^ { 2 } } \)

\(c \quad f ( x ) = 2 x ( x - 4 ) - 7 - 2 x ( x + 3 ) \)

\(d \quad f ( x ) = x ( x - 3 ) \)

Q:

Solve the quadratic equation \( 5 x ^ { 2 } + 30 x + 90 = 0 \) by using the Quadratic Formula. Steps 1. Write the equation in Standard Form 2. Identify the coefficients \( a , b \) and \( c \) . 3. Substitute the values into the quadratic equation 

\( x = \frac { - b \pm \sqrt { b ^ { 2 } - 4 a c } } { 2 a } \) 

4. Solve the equation for \( x _ { 1 } \) and \( x _ { 2 } \) 

\( x _ { 1 } = \frac { - b - \sqrt { b ^ { 2 } - 4 a c } } { 2 a } x _ { 2 } = \frac { - b + \sqrt { b ^ { 2 } - 4 a c } } { 2 a } \) 

5. Write your answers in Exact Form. 6. Note: If only one solution exists, \( x _ { 2 } \) will equal \( D N E\) 

Q:

\(f ( x ) = \frac { ( x + 2 ) ( x + 1 ) ^ { 3 } } { ( x + 2 ) ( x - 3 ) ^ { 2 } } \) 

Reduced \( f ( x ) = \) 

Domain: \( x \neq \) ___

 Thertical Asymptote(s):

 There is/are vertical asymptote(s) at 

There are no vertical asymptotes. 

The VA is even. 

The VA is odd. 

There is no VA. 

Hole(s): 

There is a hole at 

There is no hole in the function. 

X-intercept(s): 

There is/are \( x \) -intercept(s) at 

There are no \( x \) -intercepts. 

The \( x \) -intercept is even, meaning the graph will bounce on it. 

The \( x \) -intercept is odd, meaning the graph will cross the \( x \) -axis at the intercept. 

There is no \( x \) -intercept. 

Q:

Write an equation for a rational function with: Vertical asymptotes at \( x = - 3 \) and \( x = 4 \) 

\( x \) intercepts at \( x = 5 \) and \( x = - 2 \) 

\( y \) intercept at \( 5 \) 

\( y = \) 

Q:

Write an equation for a rational function with: Vertical asymptotes at \( x = - 3 \) and \( x = 3 \) 

\( x \) -intercepts at \( x = 4 \) and \( x = - 2 \) 

\( y \) -intercept at \( 6 \) 

\( y = \) 

Q:

(a) Use the drop-down menu to select the absolute value \( ( | x | ) \) function. The basic function \( f ( x ) = | x | \) is drawn in a dashed line with three key points labeled. Now, use the slider labeled \( k \) to slowly increase the value of \( k \) from \( 0 \) to \( 3 \) . As you do this, notice the form of the function \( g ( x ) = f ( x - h ) + k \) represented by the solid line. Repeat this for other functions available in the drop-down menu. Based on what you observe, conclude if a positive real number \( k \) is added to the outputs of a function \( y = f ( x ) \) , the graph of the new function \( g ( x ) = f ( x ) + k \) is the graph of \( f \) shifted vertically up \( k \) units. (b) Use the drop-down menu to select the square root \( ( \sqrt { x } ) \) function. The basic function \( f ( x ) = \sqrt { x } \) is drawn in a dashed line with three key points labeled. Now, use the slider labeled \( k \) to slowly decrease the value of \( k \) from \( 0 \) to \( - 3 \) . As you do this, notice the form of the function \( g ( x ) = f ( x - h ) + k \) represented by the solid line. Repeat this for other functions available in the drop-down menu. Based on what you observe, conclude if a positive real number \( k \) is subtracted from the outputs of a function 

\( y = f ( x ) \) , the graph of the new function \( g ( x ) = f ( x ) - k \) is the graph of \( f \) shifted \( k \) k units. 

Q:

A softball is thrown upward with an initial velocity of \( 32 \) feet per second from \( 5 \) feet above ground. The ball's height in feet above the ground is modeled by \( h ( t ) = - 16 t ^ { 2 } + 32 t + 5 \) , where \( t \) is the time in seconds after the ball is released. What is the maximum height of the ball? 

Q:

A ferry takes several trips between points A and B. It moves at a constant speed of \( 0.35 \) miles/minute and takes the same route on each trip. The total duration of a round trip from point A to point B and back is \( 80 \) minutes, ignoring the time spent stopped at point B. If \( d \) is the ferry's distance from point A in miles and \( t \) is the time in minutes, ] which equation models the ferry's distance from point A for the duration of the trip? 

A. \( d = - | 0.35 t - 40 | + 40 \) 

B. \( d = - | 0.35 t - 80 | + 80 \) 

C. \( d = - | 0.35 t - 28 | + 28 \) 

D. \( d = - | 0.35 t - 14 | + 14\) 

Q:

Put a dot on the vertex of the parabola below. What is the coordinates or a maximum? 

Q:

Put the equation \( y = x ^ { 2 } + 8 x + 12 \) into the form \( y = ( x - h ) ^ { 2 } + k \) : Answer: \( y = \)___

Q:

Suppose \( ( 6,6 ) \) is a point on the graph of \( y = g ( x ) \) . 

(a) What point is on the graph of \( y = g ( x + 8 ) - 4 ? \) 

(b) What point is on the graph of \( y = - 2 g ( x - 6 ) + 9 \) ? 

(c) What point is on the graph of \( y = g ( 2 x + 10 ) ?\) 

 

Q:

Select the correct answer. For the next five years, a dairy company wants to increase its daily milk processing capacity (in gallons) by the equation \( y = x ^ { 6 } - 9 x ^ { 4 } + 70 x ^ { 2 } \) . where \( x \) is the number of years. How many years will it take until the company can process \( 630 \) gallons of milk in a day? A.\(2\)years B. \( 3 \) years D. \( 4 \) years D. \( 5 \) years 

Q:

 The population of snakes at a wildlife park over the course of six months is modeled by the equation \( y = x ^ { 5 } - 5 x ^ { 4 } + 2 x ^ { 3 } - 10 x ^ { 2 } + 100 x + 5 \) , where 

\( x \) is the number of months. If the park officials count \( 505 \) snakes, how many months have passed since the initial count was taken?A. \(3\)months B. \( 4 \) months B. \( 5 \) months D. \( 6 \) months 

Q:

Graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function shown to the right. Find the domain and range of the function. 

\( g ( x ) = 4 \sqrt { x - 3 } + 5\) 

Q:

Suppose a state's income tax code states that tax liability is \( 13 \% \) on the first \( \$ 18,400 \) of taxable earning and \( 15 \% \) on the remainder. Find constants \( a \) and \( b \) for the tax function 

\( T ( x ) = \left \{ \begin{array} { l l } { a + 0.13 x } \\ { b + 0.15 ( x - 18,400 ) x \leq 18,400 } \\ { x > 18,400 } \end{array} \right . \) 

such that \( \lim _ { x \rightarrow 0 ^ { + } } T ( x ) = 0 \) and lim \( x \rightarrow 18,400\) 

Q:

Given the standard form of a quadratic function: 

\( y = 2 x ^ { 2 } - 3 x + 1 \) 

Complete the table 

Q:

Write your answer in radical form. Find a value of \( c \) satisfying the conclusion of the Mean Value Theorem. 

\( f ( x ) = x ^ { 3 } + 7 x ^ { 2 } , [ 0,3 ] \) 

\( c = \) 

Q:

Solve the equation \( 2 x ^ { 2 } - 7 x = 10 \) by factoring. 

Select one: 

a. \( x = 10 \) and \( x = 3 / 2 \) 

b. None of these 

c.\( x = 10 \) and \( x = 17 / 2 \) 

d. \( x = 0\) 

Q:

Make the indicated substitution for an unspecified function \( f ( x ) \) . 

\( u = \cos ( x ) f o r \int _ { 3 \pi / 2 } ^ { 2 \pi } 5 ( - \sin ( x ) ) f ( \cos ( x ) ) d x \) 

\( \int _ { 0 } ^ { 1 / 2 } f ( u ) d u \) 

\( \int _ { 0 } ^ { 1 } 5 f ( u ) d u \) 

\( \int _ { 0 } ^ { 1 / 2 } 5 f ( u ) d u\) 

Q:

The daily production level for a product is given by \( N ( x ) = 180 - 180 e ^ { - 0.1 x } \) units, where \( x \) is the time in hours after production begins. Find the average production during the first \( 3 \) hours. Enter your answer in exact form or rounded to the nearest whole number. Answer 

Q:

A ball is thrown vertically upward. After \( t \) seconds, its height \( h \) (in feet) is given by the function \( h ( t ) = 40 t - 16 t ^ { 2 } \) . After how long will it reach its maximum height? Do not round your answer. Time: ___ seconds 

Q:

Delbert stands at the top of a \( 300 \) -foot cliff and throws his algebra book directly upward with a velocity of \( 20 \) feet per second. The height of his book above the ground \( t \) seconds later is given by the equation where \( h \) is in feet. \( h = - 16 t ^ { 2 } + 20 t + 300 \) 

Graph the height equation on your calculator. Use your graph and a table of values to help you answer the questions. 

a. When does Delbert's book pass him on its way down?

b. Write and solve an equation to answer the question: How long will it take Debert's book to hit the ground at the bottom of the cliff? 

Q:

Determine algebraically whether the given function is even, odd, or neither. 

\( f ( x ) = \frac { 8 } { x ^ { 10 } } \) 

Is the given function even, odd, or neither? A. Even B. Odd C. Neither 

Q:

Suppose that \( f ( x ) = 4 x - 1 \) and \( g ( x ) = - 3 x + 6 \) 

(a) Solve \( f ( x ) = 0 \) (b) Solve \( f ( x ) > 0 \) 

(c) Solve \( f ( x ) = g ( x ) \) . (d) Solve \( f ( x ) \leq g ( x ) \) 

(e) Graph \( y = f ( x ) \) and \( y = g ( x ) \) and find the point that represents the solution to the equation \( f ( x ) = g ( x ) \) . 

(a) For what value of \( x \) does \( f ( x ) = 0 \) ? 

\( x = \frac { 1 } { 4 } \) 

(Type an integer or a simplified fraction.) (b) For which values of \( x \) is \( f ( x ) > 0 \) ? (Typlajour answer in interval notation. Use integers or fractions for any numbers in the expression.) 

Q:

(a) Use the drop-down menu to select the absolute value \( ( | x | ) \) function. The basic function \( f ( x ) = | x | \) is drawn in a dashed line with three key points labeled. Now, use the slider labeled \( h \) to slowly increase the value of \( h \) from \( 0 \) to \( 4 \) . As you do this, notice the form of the function \( g ( x ) = f ( x - h ) + k \) represented by the solid line. Repeat this for other functions available in the drop-down menu. Based on what you observe, conclude if the argument \( x \) of a function is replaced by \( x - h , h > 0 \) , the graph of the new function \( g ( x ) = f ( x - h ) \) is the graph of \( f \) shifted horizontally right \( k \) units. (b) Use the drop-down menu to select the absolute value \( ( | x | ) \) function. The basic function \( f ( x ) = | x | \) is drawn in a dashed line with three key points labeled. Now, use the slider labeled h to slowly decrease the value of \( h \) from \( 0 \) to \( - 4 \) . As you do this, notice the form of the function \( g ( x ) = f ( x - h ) + k \) represented by the solid line. Repeat this for other functions available in the drop-down menu. Based on what you observe, conclude if the argument \( x \) of a function is replaced by \( x + h \) , \( h > 0 \) , the graph of the new function \( g ( x ) = f ( x + h ) \) is the graph of \( f \) shifted vertically \( k \) units. 

Q:

In parts (a) and (b), use the given figure.

 (a) Solve the equation: \( f ( x ) = g ( x ) \) . 

(b) Solve the inequality \( g ( x ) \leq f ( x ) < h ( x ) \) . 

(a) For what value of \( x \) does \( f ( x ) = g ( x ) \) ? 

\( x = \square \) 

(b) For which values of \( x \) is \( g ( x ) \leq f ( x ) < h ( x ) \) ? For every \( x \) in the interval \( \square , g ( x ) \leq f ( x ) < h ( x ) \) . (Type your answer in interval notation.) 

Q:

(a) Use the drop-down menu to select the absolute value \( ( | x | ) \) function. The basic function \( f ( x ) = | x | \) is drawn in a dashed line with three key points labeled. Now, use the slider labeled \( k \) to slowly increase the value of \( k \) from \( 0 \) to \( 3 \) . As you do this, notice the form of the function \( g ( x ) = f ( x - h ) + k \) represented by the solid line. Repeat this for other functions available in the drop-down menu. Based on what you observe, conclude if a positive real number \( k \) is added to the outputs of a function \( y = f ( x ) \) , the graph of the new function \( g ( x ) = f ( x ) \) + is the graph of \( f \) shifted vertically up \( k \) units. (b) Use the drop-down menu to select the square root \( ( \sqrt { x } ) \) function. The basic function \( f ( x ) = \sqrt { x } \) is drawn in a dashed line with three key points labeled. Now, use the slider labeled \( k \) to slowly decrease the value of \( k \) from \( 0 \) to \( - 3 \) . As you do this, notice the form of the function \( g ( x ) = f ( x - h ) + k \) represented by the solid line. Repeat this for other functions available in the drop-down menu. Based on what you observe, conclude if a positive real number \( k \) is subtracted from the outputs of a function \( y = f ( x ) \) , the graph of the new function 

\( g ( x ) = f ( x ) - k \) is the graph of \( f \) shifted vertically down \( k \) units. (c) If \( y = f ( x ) \) is some function whose graph contains the point \( ( 2,4 ) \) , the graph of \( y = f ( x ) + 2 \) would contain the point \( \square \) . (Type an ordered pair.). 

Q:

Match the graph of \( h ( x ) \) with the appropriate transformation of the given function, \( g ( x ) \) 

\(h ( x ) = g ( 2 x ) + 4 \) 

\( h ( x ) = g ( \frac { 1 } { 4 } x ) + 4 \) 

\( h ( x ) = g ( \frac { 1 } { 2 } x ) + 4 \) 

\( h ( x ) = g ( 4 x ) + 4 \) 

\( h ( x ) = g ( 2 ( x - 1 ) ) + 4 \) 

\( h ( x ) = g ( \frac { 1 } { 2 } ( x - 1 ) ) + 4\) 

Q:

(a) Use the drop-down menu to select the absolute value \( ( | x | ) \) function. The basic function \( f ( x ) = | x | \) is drawn in a dashed line with three key points labeled. Set the slider labeled a to \( 1 \) . Now, use the slider labeled a to slowly increase the value of a from \( 1 \) to \( 3 \) . As you do this, notice the form of the function \( g ( x ) = \) af \( ( x ) \) and the behavior of the graph of the function \( g ( x ) = a f ( x ) \) represented by the solid line. Repeat this for other functions available in the drop-down menu. Based on what you observe, conclude when the right side of a function \( y = f ( x ) \) is multiplied by a positive number \( a > 1 \) , the graph of the new function is obtained by multiplying each 

 

\( y \) -coordinate on the graph of \( y = f ( x ) \) by \( a > 1 \) . The new graph is a vertically 

Q:

Form a polynomial whose zeros and degree are given. Zeros: \( - 1,1,8 ; \) degree: \( 3 \) Type a polynomial with integer coefficients and a leading coefficient of \( 1 \) in the box below. 

\( f ( x ) = \square \) (Simplify your answer.) 

Q:

The function \( f ( x ) = x ^ { 3 } + 9 \) is one-to-one. 

a. Find an equation for \( f ^ { - 1 } \) , the inverse function. 

b. Verify that your equation is correct by showing that \( f ( f ^ { - 1 } ( x ) ) = x \) and \( f ^ { - 1 } ( f ( x ) ) = x \) . 

a. Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answer. Use integers or fractions for any numbers in the expression.) 

A. \( f ^ { - 1 } ( x ) = \) ___, for \( x \geq \) ___

 

B. \( f ^ { - 1 } ( x ) = \)___ , for \( x \leq \) ___

C. \( f ^ { - 1 } ( x ) = \) ___,for \( x \neq \) ___

D. \( f ^ { - 1 } ( x ) = \)___ , for all \( x \) 

b. Verify that the equation is correct. 

Q:

Given the function \( f ( x ) = ( x + 2 ) ^ { 3 } \) , complete parts a through c

(a) Find an equation for \( f ^ { - 1 } ( x ) \) . 

(b) Graph \( f \) and \( f ^ { - 1 } \) in the same rectangular coordinate system. 

(c) Use interval notation to give the domain and the range of \( f \) and \( f ^ { - 1 } \) 

\(f ^ { - 1 } ( x ) = \square \) 

(Type an exact answer, using radicals as needed.) 

b) Choose the correct graph which shows \( f \) and \( f ^ { - 1 } \) graphed in the same coordinate system. 

Q:

A company will sell \( N \) units of a product after spending \( \$ x \) thousand in advertising, as given by \( N = 50 x - x ^ { 2 } \) , \( 5 \leq x \leq 25 \) . Approximately what increase in sales will result by increasing the advertising budget from \( \$ 15,000 \) to \( \$ 16,000 ? \) From \( \$ 20,000 \) to\( \$ 21,000 ? \) 

Find the differential \( dN \) . 

\( dN = ( \square dx) \) 

Q:

An aircraft factory manufactures airplane engines. The unit cost \( C \) (the cost in dollars to make each airplane engine) depends on the number of engines made. If 

\( x \) engines are made, then the unit cost is given by the function \( C ( x ) = 0.7 x ^ { 2 } - 126 x + 22,330 \) . How many engines must be made to minimize the unit cost? Do not round your answer. 

Q:

Suppose a company's revenue function is given by \( R ( q ) = - q ^ { 3 } + 370 q ^ { 2 } \) and its cost function is given by \( C ( q ) = 370 + 16 q \) , where \( q \) is hundreds of units sold/produced, while \( R ( q ) \) and \( C ( q ) \) are in total dollars of revenue and cost, respectively. 

A) Find a simplified expression for the marginal profit function. (Be sure to use the proper variable in your answer.) 

\( M P ( q ) = \) 

B) How many items (in hundreds) need to be sold to maximize profits? 

Q:

The height, \( h \) , of a falling object \( t \) seconds after it is dropped from a platform \( 300 \) feet above the ground is modeled by the function \( h ( t ) = 3001 - 16 t ^ { 2 } \) . Which expression could be used to determine the average rate at which the object falls during the first \( 3 \) seconds of its fall? 

h(3)-h(0)

h(1)-h(0)

h(3)/3

(h(3)-h(0))/3

Q:

A vertical cylinder is leaking water at a rate of \( 5 m ^ { 3 } / sec \) . If the cylinder has a height of \( 10 m \) and a radius of \( 4 m \) , at what rate is the height of the water changing when the height is \( 4 m \) ? 

\(\frac { d h } { d t } = \)

Q:

A widget manufacturer determines that if she manufactures \(x\) thousands of widgets per month and sells the widgets for \(y\) dollars each, then her monthly profit (in thousands of dollars) will be \(P = x y - \frac { 1 } { 27 } x ^ { 2 } y ^ { 3 } - x\) . If her factory is capable of producing at most \(3,000\) widgets per month, and government regulations prevent her from charging more than \(\$ 2\) per widget, how many should she manufacture, and how much should she charge for each, to maximize her monthly profit?

Q:

A plate of areal density \(\sigma ( x )\) occupying the region \(a \leq x \leq b\) \(f ( x ) \leq y \leq g ( x )\)

Q:

Find the area below \(y = e ^ { - x } \sin x\) and above \(y = 0\) from \(x = 0\) to \(x = \pi\) .

Q:

There are two distinct straight lines that pass through the point \(( 1 , - 3 )\) and are tangent to the curve \(y = x ^ { 2 }\) . Find their equations. Hint: Draw a sketch. The points of tangency are not given; let them be denoted \(( a , a ^ { 2 } )\) .

Q:

A solid is \(6 ft\) high. Its horizontal cross-section at height \(z ft\) above its base is a rectangle with length \(2 + z ft\) and width \(8 - z ft\) . Find the volume of the solid.