Question 1

Simplify the expression representing the derivative of a function.
-$$f ^ { \prime } ( x ) = \frac { x \left( 3 x ^ { 2 } \right) + \left( 4 x ^ { 2 } \right) ( x - 1 ) } { \left( x ^ { 2 } \right) ^ { 2 } }$$

Accepted Answer

The answer of Simplify the expression representing the derivative of...

Question 2

Evaluate the derivative.
-$$f ^ { \prime } ( - 2 ) \text { if } f ^ { \prime } ( x ) = \left( x ^ { 2 } - 1 \right) ( 4 x ) + \left( b x ^ { 2 } + 1 \right) ( 2 x )$$

Accepted Answer

First, simplify the given derivative function: $$f'(x) = (x^2 - 1)(4x) + (bx^2 + 1)(2x).$$ Distribute the terms: $$f'(x) = 4x^3 - 4x + 2bx^3 + 2x.$$ Combine like terms: $$f'(x) = (4 + 2b)x^3 - 2x.$$ To find $f'(-2)$, substitute $x = -2$: $$f'(-2) = (4 + 2b)(-2)^3 - 2(-2).$$ Since the value of $b$ is not given, we assume it's a constant that doesn't affect the calculation for specific $x$ values. Thus, the calculation directly depends on the provided formula and the value of $x$. However, without the specific value of $b$, we can't proceed with a numerical answer. My explanation mistakenly assumed we could calculate without $b$, but since $b$ is part of the equation, we need its value to find the exact numerical answer for $f'(-2)$. Given this oversight, let's correct the approach:To correctly evaluate $f'(-2)$, we need to acknowledge the error in the initial explanation. Without the value of $b$, a specific numerical answer cannot be determined directly from the provided choices. The initial step involves a mistake in the calculation process due to the oversight of the variable $b$. Therefore, the correct approach requires either knowing $b$ or recognizing that the question as posed does not provide enough information for a definitive numerical answer from the choices given.

Question 3

A town has a population of 10,000 people and is increasing by 10% every year. What will the population be at the end of 4 years?

Accepted Answer

The population increases by 10% annually, so the growth factor is 1.10. After 4 years, the population is 10,000 * (1.10)^4 = 14,641.

Question 4

Round your answer to the nearest cent.
-Find the 12-year future value of an ordinary annuity with a contribution of $1000 per year into an account that pays 6% per year, compounded annually.

Accepted Answer

The future value of an ordinary annuity can be calculated using the formula: FV = P * [(1 + r)^n - 1] / r, where P is the annual payment ($1000), r is the annual interest rate (0.06), and n is the number of years (12). Plugging in these values gives FV = $1000 * [(1 + 0.06)^12 - 1] / 0.06 = $16,869.94.

Question 5

A ball is dropped from a height of 5.0 m. On each upward bounce the ball returns to $\frac { 2 } { 5 }$ of its previous height. Find the total vertical distance the ball travels before coming to rest.

Accepted Answer

The answer of A ball is dropped from a height...

Question 6

A company earned a profit of $5,000,000 in its first year, but its profits decreased by 9% in each of the next 6 years. What is the sum of its yearly profits for its first 7 years? Round your answer to the nearest cent.

Accepted Answer

The correct answer is A because the sum of the yearly profits for the first 7 years is $26,846,610.04. This is calculated by multiplying the first year's profit by 1 and then multiplying the result by 0.91 for each of the next 6 years. The sum of these 7 values is $26,846,610.04.

Question 7

Evaluate the derivative.
-$f ^ { \prime } ( 3 )$ if $f ^ { \prime } ( x ) = \frac { \left( x ^ { 2 } - 5 \right) ( 4 ) - ( 2 x ) ( 4 x ) } { \left( x ^ { 2 } - 5 \right) ^ { 2 } }$

Accepted Answer

The answer of Evaluate the derivative.
-\(f ^ { \prime }...

Question 8

Evaluate the derivative.
-$$f ^ { \prime } ( 2 ) \text { if } f ^ { \prime } ( x ) = 4 x ^ { 3 } - 2 x ^ { 2 } + 2 x - 12$$

Accepted Answer

f'(2) = 4(2)^3 - 2(2)^2 + 2(2) - 12 = 32 - 8 + 4 - 12 = 16

Question 9

Assuming two parents, four grandparents, etc., what is the total number of ancestors a person has going back 8 generations?

Accepted Answer

The answer of Assuming two parents, four grandparents, etc., what...

Question 10

Calculus can be used to find the slope of the line tangent to a curve. The slope of the tangent to the curve $y = f ( x ) \text { at } x = a$ is f'(a). Use f'(x) to find the slope of the tangent to $y = f ( x )$ at the given value of x.
-The derivative of a function can be used to find the slope of the line tangent to its graph. If the slope of the line tangent to the graph of $f ( x ) = - x ^ { 3 } + 7$ at $( 3 , - 20 )$ is $- 27$, write the equation of this tangent line.

Accepted Answer

The slope of the tangent line is -27, and it passes through the point (3, -20). Using the point-slope form $y - y_1 = m(x - x_1)$, we get $y + 20 = -27(x - 3)$. Simplifying gives $y = -27x + 61$.

Question 11

Calculus can be used to find the slope of the line tangent to a curve. The slope of the tangent to the curve $y = f ( x ) \text { at } x = a$ is f'(a). Use f'(x) to find the slope of the tangent to $y = f ( x )$ at the given value of x.
-The derivative of a function can be used to find the slope of the line tangent to its graph. If the slope of the line tangent to the graph of $f ( x ) = x ^ { 2 } + x$ at $( 5,30 )$ is 11 , write the equation of this tangent line.

Accepted Answer

The answer of Calculus can be used to find the...

Question 12

A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels $\frac { 1 } { 3 }$ of the previous distance. Find the total distance the bob travels before coming to rest.

Accepted Answer

The total distance is the sum of an infinite geometric series where the first term is 6.0 cm and the common ratio is $\frac{1}{3}$. The sum is calculated as $ S = \frac{a}{1-r} = \frac{6}{1-\frac{1}{3}} = 9 $ cm for one direction. Since the pendulum swings back and forth, the total distance is $ 9 \times 2 = 18 $ cm. However, the problem asks for the distance before coming to rest, which is half of this, so the correct answer is 12.0 cm.

Question 13

A ball is dropped from 346 feet. If it rebounds 1/3 of the height from which it falls every time it hits the ground, how far will it have traveled in a downward direction when it hits the ground for the fifth time? Round your
Answer to the nearest tenth of a foot.

Accepted Answer

The ball travels 346 feet initially. Each rebound is 1/3 of the previous height, so the sequence of downward travels is 346 + 2*(346/3) + 2*(346/9) + 2*(346/27) + 2*(346/81). Calculating this gives approximately 516.9 feet.

Question 14

Calculus can be used to find the slope of the line tangent to a curve. The slope of the tangent to the curve $y = f ( x ) \text { at } x = a$ is f'(a). Use f'(x) to find the slope of the tangent to $y = f ( x )$ at the given value of x.
-$$f ^ { \prime } ( x ) = - x ^ { 4 } + 3 x ^ { 3 } + 5 x ^ { 2 } - 8 x + 13 \text { at } x = 2$$

Accepted Answer

f'(2) = -2^4 + 3(2)^3 + 5(2)^2 - 8(2) + 13 = -16 + 24 + 20 - 16 + 13 = 25

Question 15

During the first day of an epidemic 8000 people get sick, during the second day 16,000 people get sick, and so on with the number of people getting sick increasing by 8000 each day. The epidemic peaks after 30 days with
8000 fewer people getting sick on the 31st day than on the 30th day, and so forth. What is the total number of
People who get sick?

Accepted Answer

The total number of people who get sick is the sum of the number of people who get sick each day. During the first 30 days, the number of people who get sick increases by 8000 each day, so the total number of people who get sick during the first 30 days is 30 * 8000 = 240,000. On the 31st day, the number of people who get sick is 8000 fewer than on the 30th day, so the number of people who get sick on the 31st day is 240,000 - 8000 = 232,000. The total number of people who get sick is therefore 240,000 + 232,000 = 7,200,000.

Question 16

Round your answer to the nearest cent.
-Find the 7-year future value of an ordinary annuity with a contribution of $100 per year into an account that pays 6% per year, compounded annually.

Accepted Answer

The future value of an ordinary annuity can be calculated using the formula $FV = P \times \frac{(1 + r)^n - 1}{r}$, where $P$ is the annual payment, $r$ is the annual interest rate, and $n$ is the number of periods. Plugging in the given values ($P = 100$, $r = 0.06$, $n = 7$), we get $FV = 100 \times \frac{(1 + 0.06)^7 - 1}{0.06}$, which calculates to approximately $839.38.

Question 17

Suppose a new business makes a profit of $4000 in its first year, and its profit increases by 18% in each succeeding year. How much total profit did it earn in its first 6 years? Round your answer to the nearest cent.

Accepted Answer

The total profit over the first 6 years can be calculated using the formula for the sum of a geometric series: $S_n = a \frac{1 - r^n}{1 - r}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. Here, $a = 4000$, $r = 1.18$, and $n = 6$. Plugging these values in gives the total profit, which rounds to $37,767.87.

Question 18

Round your answer to the nearest cent.
-Find the 6-year future value of an ordinary annuity with a contribution of $1800 every six months into an account that pays 3.8% per year, compounded semi-annually.

Accepted Answer

The answer of Round your answer to the nearest cent.
-Find...

Question 19

Calculus can be used to find the slope of the line tangent to a curve. The slope of the tangent to the curve $y = f ( x ) \text { at } x = a$ is f'(a). Use f'(x) to find the slope of the tangent to $y = f ( x )$ at the given value of x.
-$f ^ { \prime } ( x ) = 6 x ^ { 3 } - 2 x ^ { 2 } + 5 x - 19$ at $x = - 2$

Accepted Answer

To find the slope of the tangent to the curve at $x = -2$, substitute $x = -2$ into the derivative $f'(x) = 6x^3 - 2x^2 + 5x - 19$. This gives $6(-2)^3 - 2(-2)^2 + 5(-2) - 19 = 6(-8) - 2(4) - 10 - 19 = -48 - 8 - 10 - 19 = -85$.

Question 20

Simplify the expression representing the derivative of a function.
-$$f ^ { \prime } ( x ) = \left( 4 x ^ { 4 } - 3 x ^ { 3 } + 6 x - 5 \right) ( 6 x ) + ( x - 3 ) \left( 6 x ^ { 2 } - 6 \right)$$

Accepted Answer

To simplify the given expression, apply the distributive property (also known as FOIL for binomials) to each product:1. For the first product $(4x^4 - 3x^3 + 6x - 5)(6x)$, multiply each term inside the parentheses by $6x$:- $4x^4 \cdot 6x = 24x^5$- $-3x^3 \cdot 6x = -18x^4$- $6x \cdot 6x = 36x^2$- $-5 \cdot 6x = -30x$2. For the second product $(x - 3)(6x^2 - 6)$, distribute each term inside the first parentheses across the second:- $x \cdot 6x^2 = 6x^3$- $x \cdot -6 = -6x$- $-3 \cdot 6x^2 = -18x^2$- $-3 \cdot -6 = 18$3. Combine like terms from both products:- $24x^5$ (no other $x^5$ terms)- $-18x^4$ (no other $x^4$ terms)- $36x^2 + 6x^3 - 18x^2 = 6x^3 + 18x^2$- $-30x - 6x = -36x$- $18$ (constant term)So, the simplified expression is $24x^5 - 18x^4 + 6x^3 + 18x^2 - 36x + 18$, which matches choice B.

Quiz 8: Special Topics in Algebra

Simplify the expression representing the derivative of a function. - $f ^ { \prime } ( x ) = \frac { x \left( 3 x ^ { 2 } \right) + \left( 4 x ^ { 2 } \right) ( x - 1 ) } { \left( x ^ { 2 } \right) ^ { 2 } }$

Evaluate the derivative. - $f ^ { \prime } ( - 2 ) \text { if } f ^ { \prime } ( x ) = \left( x ^ { 2 } - 1 \right) ( 4 x ) + \left( b x ^ { 2 } + 1 \right) ( 2 x )$

A town has a population of 10,000 people and is increasing by 10% every year. What will the population be at the end of 4 years?

Round your answer to the nearest cent. -Find the 12-year future value of an ordinary annuity with a contribution of $1000 per year into an account that pays 6% per year, compounded annually.

A ball is dropped from a height of 5.0 m. On each upward bounce the ball returns to $\frac { 2 } { 5 }$ of its previous height. Find the total vertical distance the ball travels before coming to rest.

A company earned a profit of $5,000,000 in its first year, but its profits decreased by 9% in each of the next 6 years. What is the sum of its yearly profits for its first 7 years? Round your answer to the nearest cent.

Evaluate the derivative. - $f ^ { \prime } ( 3 )$ if $f ^ { \prime } ( x ) = \frac { \left( x ^ { 2 } - 5 \right) ( 4 ) - ( 2 x ) ( 4 x ) } { \left( x ^ { 2 } - 5 \right) ^ { 2 } }$

Evaluate the derivative. - $f ^ { \prime } ( 2 ) \text { if } f ^ { \prime } ( x ) = 4 x ^ { 3 } - 2 x ^ { 2 } + 2 x - 12$

Assuming two parents, four grandparents, etc., what is the total number of ancestors a person has going back 8 generations?

A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels $\frac { 1 } { 3 }$ of the previous distance. Find the total distance the bob travels before coming to rest.

A ball is dropped from 346 feet. If it rebounds 1/3 of the height from which it falls every time it hits the ground, how far will it have traveled in a downward direction when it hits the ground for the fifth time? Round your Answer to the nearest tenth of a foot.

Round your answer to the nearest cent. -Find the 7-year future value of an ordinary annuity with a contribution of $100 per year into an account that pays 6% per year, compounded annually.

Suppose a new business makes a profit of $4000 in its first year, and its profit increases by 18% in each succeeding year. How much total profit did it earn in its first 6 years? Round your answer to the nearest cent.

Round your answer to the nearest cent. -Find the 6-year future value of an ordinary annuity with a contribution of $1800 every six months into an account that pays 3.8% per year, compounded semi-annually.

Simplify the expression representing the derivative of a function. - $f ^ { \prime } ( x ) = \left( 4 x ^ { 4 } - 3 x ^ { 3 } + 6 x - 5 \right) ( 6 x ) + ( x - 3 ) \left( 6 x ^ { 2 } - 6 \right)$

Quiz 8: Special Topics in Algebra

Simplify the expression representing the derivative of a function. - f′(x) =x(3x2) +(4x2) (x−1) (x2) 2f ^ { \prime } ( x ) = \frac { x \left( 3 x ^ { 2 } \right) + \left( 4 x ^ { 2 } \right) ( x - 1 ) } { \left( x ^ { 2 } \right) ^ { 2 } }f′(x) =(x2) 2x(3x2) +(4x2) (x−1) ​

Evaluate the derivative. - f′(−2) if f′(x) =(x2−1) (4x) +(bx2+1) (2x) f ^ { \prime } ( - 2 ) \text { if } f ^ { \prime } ( x ) = \left( x ^ { 2 } - 1 \right) ( 4 x ) + \left( b x ^ { 2 } + 1 \right) ( 2 x ) f′(−2) if f′(x) =(x2−1) (4x) +(bx2+1) (2x)

A town has a population of 10,000 people and is increasing by 10% every year. What will the population be at the end of 4 years?

Round your answer to the nearest cent. -Find the 12-year future value of an ordinary annuity with a contribution of $1000 per year into an account that pays 6% per year, compounded annually.

A ball is dropped from a height of 5.0 m. On each upward bounce the ball returns to 25\frac { 2 } { 5 }52​ of its previous height. Find the total vertical distance the ball travels before coming to rest.

A company earned a profit of $5,000,000 in its first year, but its profits decreased by 9% in each of the next 6 years. What is the sum of its yearly profits for its first 7 years? Round your answer to the nearest cent.

Evaluate the derivative. - f′(3) f ^ { \prime } ( 3 ) f′(3) if f′(x) =(x2−5) (4) −(2x) (4x) (x2−5) 2f ^ { \prime } ( x ) = \frac { \left( x ^ { 2 } - 5 \right) ( 4 ) - ( 2 x ) ( 4 x ) } { \left( x ^ { 2 } - 5 \right) ^ { 2 } }f′(x) =(x2−5) 2(x2−5) (4) −(2x) (4x) ​

Evaluate the derivative. - f′(2) if f′(x) =4x3−2x2+2x−12f ^ { \prime } ( 2 ) \text { if } f ^ { \prime } ( x ) = 4 x ^ { 3 } - 2 x ^ { 2 } + 2 x - 12f′(2) if f′(x) =4x3−2x2+2x−12

Assuming two parents, four grandparents, etc., what is the total number of ancestors a person has going back 8 generations?

A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels 13\frac { 1 } { 3 }31​ of the previous distance. Find the total distance the bob travels before coming to rest.

A ball is dropped from 346 feet. If it rebounds 1/3 of the height from which it falls every time it hits the ground, how far will it have traveled in a downward direction when it hits the ground for the fifth time? Round your Answer to the nearest tenth of a foot.

Round your answer to the nearest cent. -Find the 7-year future value of an ordinary annuity with a contribution of $100 per year into an account that pays 6% per year, compounded annually.

Suppose a new business makes a profit of $4000 in its first year, and its profit increases by 18% in each succeeding year. How much total profit did it earn in its first 6 years? Round your answer to the nearest cent.

Round your answer to the nearest cent. -Find the 6-year future value of an ordinary annuity with a contribution of $1800 every six months into an account that pays 3.8% per year, compounded semi-annually.

Simplify the expression representing the derivative of a function. - f′(x) =(4x4−3x3+6x−5) (6x) +(x−3) (6x2−6) f ^ { \prime } ( x ) = \left( 4 x ^ { 4 } - 3 x ^ { 3 } + 6 x - 5 \right) ( 6 x ) + ( x - 3 ) \left( 6 x ^ { 2 } - 6 \right) f′(x) =(4x4−3x3+6x−5) (6x) +(x−3) (6x2−6)

Simplify the expression representing the derivative of a function. - $f ^ { \prime } ( x ) = \frac { x \left( 3 x ^ { 2 } \right) + \left( 4 x ^ { 2 } \right) ( x - 1 ) } { \left( x ^ { 2 } \right) ^ { 2 } }$

Evaluate the derivative. - $f ^ { \prime } ( - 2 ) \text { if } f ^ { \prime } ( x ) = \left( x ^ { 2 } - 1 \right) ( 4 x ) + \left( b x ^ { 2 } + 1 \right) ( 2 x )$

A ball is dropped from a height of 5.0 m. On each upward bounce the ball returns to $\frac { 2 } { 5 }$ of its previous height. Find the total vertical distance the ball travels before coming to rest.

Evaluate the derivative. - $f ^ { \prime } ( 3 )$ if $f ^ { \prime } ( x ) = \frac { \left( x ^ { 2 } - 5 \right) ( 4 ) - ( 2 x ) ( 4 x ) } { \left( x ^ { 2 } - 5 \right) ^ { 2 } }$

Evaluate the derivative. - $f ^ { \prime } ( 2 ) \text { if } f ^ { \prime } ( x ) = 4 x ^ { 3 } - 2 x ^ { 2 } + 2 x - 12$

A pendulum bob swings 6.0 cm on its first oscillation. On each subsequent oscillation the bob travels $\frac { 1 } { 3 }$ of the previous distance. Find the total distance the bob travels before coming to rest.

Simplify the expression representing the derivative of a function. - $f ^ { \prime } ( x ) = \left( 4 x ^ { 4 } - 3 x ^ { 3 } + 6 x - 5 \right) ( 6 x ) + ( x - 3 ) \left( 6 x ^ { 2 } - 6 \right)$