Question 1

Using calculus, we can show that $$( 1 + x ) ^ { n } = 1 + n x + \frac { n ( n - 1 ) } { 2 ! } x ^ { 2 } + \frac { n ( n - 1 ) ( n - 2 ) } { 3 ! } x ^ { 3 } + \ldots , \text { for } | x | < 1$$ This
formula can be used to evaluate binomial expressions raised to noninteger exponents. Use the first four
terms of this infinite series to approximate the given expression. Round to 3 decimal places if necessary.
-$$( 1.3 ) ^ { 4 / 3 }$$
A) 1.423
B) 1.416
C) 1.419
D) 1.422

Accepted Answer

The given expression can be approximated using the binomial series expansion formula with $n = \frac{4}{3}$ and $x = 0.3$. Substituting these values into the first four terms of the series gives: $$1 + \frac{4}{3}(0.3) + \frac{\frac{4}{3}(\frac{4}{3}-1)}{2!}(0.3)^2 + \frac{\frac{4}{3}(\frac{4}{3}-1)(\frac{4}{3}-2)}{3!}(0.3)^3$$ Calculating this expression yields approximately 1.419.

Question 2

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The Fundamental ________of ________indicates that if one event can occur in m different ways, and a second event can occur in n different ways, then the two events can occur in sequence in________different ways.

Accepted Answer

The answer of Write the word or phrase that best...

Question 3

Find the indicated term of the binomial expansion.
-$\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 14 }$; Find the term containing $x ^ { 33 }$.
A) $- 2,912 x ^ { 33 } y ^ { 6 }$
B) $- 2,002 x ^ { 30 } y ^ { 8 }$
C) $- 728 x ^ { 33 } y ^ { 6 }$
D) $- 16,016 x ^ { 30 } y ^ { 8 }$

Accepted Answer

The term containing $x^{33}$ in the expansion of $\left(x^3 - 2y^2\right)^{14}$ can be found using the binomial theorem. The general term in the expansion is given by $\binom{n}{k} (x^3)^{n-k} (-2y^2)^k$, where $n=14$ and $k$ is the term number. To find the term containing $x^{33}$, we set $3(n-k) = 33$, solving for $k$ gives $k = 3$. Substituting $n=14$ and $k=3$ into the general term formula and simplifying gives the coefficient as $-2,912$, making the term $-2,912 x^{33} y^6$.

Question 4

Stirling's formula (named after Scottish mathematician, James Stirling: 1692-1770) is used to
approximate large values of n!. Stirling's formula is $n ! \approx \sqrt { 2 \pi n } \left( \frac { n } { e } \right) ^ { n }$ a. Use Stirling's formula to approximate the given expression. Round to the nearest whole unit.
b. Compute the actual value of the expression.
-$8 !$
A) a. 39,902
b. 40,312
B) a. 40,111
b. 40,320
C) a. 40,111
b. 40,312
D) a. 39,902
b. 40,320

Accepted Answer

Stirling's formula approximation for $8!$ is $\sqrt{2\pi \times 8} \left(\frac{8}{e}\right)^8 \approx 39,902$ when rounded to the nearest whole number. The actual value of $8!$ is $8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320$.

Question 5

Expand the binomial by using the binomial theorem.
-$$( x + y - 9 ) ^ { 3 }$$
A) $x ^ { 3 } + 3 x ^ { 2 } y - 27 x ^ { 2 } + 3 x y ^ { 2 } - 54 x y + 243 x + y ^ { 3 } - 27 y ^ { 2 } + 243 y - 729$
B) $x ^ { 2 } + 2 x y - 18 x + y ^ { 2 } - 18 y + 81$
C) $x ^ { 3 } + y ^ { 3 } - 729$
D) $x ^ { 3 } + 3 x ^ { 2 } y + 27 x ^ { 2 } + 3 x y ^ { 2 } + 54 x y - 243 x + y ^ { 3 } + 27 y ^ { 2 } - 243 y + 729$

Accepted Answer

The correct expansion using the binomial theorem for $( x + y - 9 ) ^ { 3 }$ involves calculating each term by combining the coefficients from the binomial theorem with the powers of each part of the binomial. The correct expansion results in $x ^ { 3 } + 3 x ^ { 2 } y - 27 x ^ { 2 } + 3 x y ^ { 2 } - 54 x y + 243 x + y ^ { 3 } - 27 y ^ { 2 } + 243 y - 729$, which matches option A.

Question 6

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-If n items are arranged in order, then each arrangement is called a________ of n items.

Accepted Answer

The answer of Write the word or phrase that best...

Question 7

Expand the binomial by using the binomial theorem.
-$$\left( \frac { 1 } { 3 } v - 4 w \right) ^ { 4 }$$
A) $\frac { 1 } { 81 } v ^ { 4 } - \frac { 16 } { 27 } v ^ { 3 } w + \frac { 32 } { 3 } v ^ { 2 } w ^ { 2 } - \frac { 256 } { 3 } v w ^ { 3 } + 256 w ^ { 2 }$
B) $\frac { 1 } { 81 } v ^ { 4 } + 256 w ^ { 2 }$
C) $\frac { 1 } { 3 } v ^ { 4 } - 4 \mathrm { w } ^ { 4 }$
D) $\frac { 1 } { 81 } v ^ { 4 } - \frac { 256 } { 3 } v ^ { 3 } w + \frac { 32 } { 3 } v ^ { 2 } w ^ { 2 } - \frac { 16 } { 27 } v w ^ { 3 } + 256 w ^ { 2 }$

Accepted Answer

The binomial theorem states that $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k$. Applying this to $\left( \frac { 1 } { 3 } v - 4 w \right) ^ { 4 }$, we get the coefficients and powers as in option A, correctly applying the binomial coefficients and the powers of each term.

Question 8

Find the indicated term of the binomial expansion.
-$\left( \sqrt { 3 } x ^ { 3 } + y ^ { 5 } \right) ^ { 9 } ;$ fourth term
A) $13,608 x ^ { 12 } y ^ { 15 }$
B) $2,268 x ^ { 13 } y ^ { 14 }$
C) $2,268 x ^ { 12 } y ^ { 15 }$
D) $13,608 x ^ { 13 } y ^ { 14 }$

Accepted Answer

The fourth term of the binomial expansion is given by the formula:$T_4 = \binom{9}{3} (\sqrt { 3 } x ^ { 3 } ) ^ { 6 } ( y ^ { 5 } ) ^ { 3 }$$= \frac{9!}{3!6!} (\sqrt { 3 } ) ^ { 6 } x ^ { 18 } y ^ { 15 }$$= \frac{9 \times 8 \times 7}{3 \times 2 \times 1} \times 3 ^ { 3 } x ^ { 18 } y ^ { 15 }$$= 84 \times 27 x ^ { 18 } y ^ { 15 }$$= 2,268 x ^ { 12 } y ^ { 15 }$

Question 9

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The number of ways that n distinguishable items can be arranged in various order is ________!

Accepted Answer

The answer of Write the word or phrase that best...

Question 10

Expand the binomial by using the binomial theorem.
-$$\left( e ^ { x } - e ^ { - x } \right) ^ { 4 }$$
A) $e ^ { 4 x } + 4 e ^ { 3 x } + 6 e ^ { 2 x } + 4 e ^ { x } + 1$
B) $e ^ { 4 x } + 4 e ^ { 2 x } + 6 + \frac { 4 } { e ^ { 2 x } } + \frac { 1 } { e ^ { 4 x } }$
C) $e ^ { 4 x } - 4 e ^ { 3 x } + 6 e ^ { 2 x } - 4 e ^ { x } + 1$
D) $e ^ { 4 x } - 4 e ^ { 2 x } + 6 - \frac { 4 } { e ^ { 2 x } } + \frac { 1 } { e ^ { 4 x } }$

Accepted Answer

The binomial theorem states that $(a - b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}(-b)^k$. Applying this to $(e^x - e^{-x})^4$, we get terms with alternating signs and powers of $e^x$ and $e^{-x}$, leading to the correct expansion in option D.

Question 11

Use the Binomial Theorem to find the value of the number raised to the given power.
-$( 1.1 ) ^ { 3 }$ Hint: Write the expression as $( 1 + 0.1 ) ^ { 3 }$.
A) $1.331$
B) $1.21$
C) $1.001$
D) $3.3$

Accepted Answer

Using the Binomial Theorem, $(1 + 0.1)^3 = 1^3 + 3(1^2)(0.1) + 3(1)(0.1)^2 + (0.1)^3 = 1 + 0.3 + 0.03 + 0.001 = 1.331$.

Question 12

Simplify the difference quotient: $$\frac { f ( x + h ) - f ( x ) } { h }$$
-$$x ^ { 4 } - 3 x ^ { 2 } - 1$$
A) $4 x ^ { 3 } h + 6 x ^ { 2 } h ^ { 2 } + 4 x h ^ { 3 } + h ^ { 4 } - 6 x ^ { 2 } h + h ^ { 2 }$
B) $x ^ { 4 } + 4 x ^ { 3 } + 6 x ^ { 2 } h + 4 x h ^ { 2 } + h ^ { 3 } - 6 x ^ { 2 } + h$
C) $4 x ^ { 3 } + 6 x ^ { 2 } h + 4 x h ^ { 2 } + h ^ { 3 } - 6 x - 3 h$
D) $4 x ^ { 3 } + 6 x ^ { 2 } h + 4 x h ^ { 2 } + h ^ { 3 } - 6 x ^ { 2 } + h - 1$

Accepted Answer

The difference quotient simplifies to the derivative of the function as $ h $ approaches zero. The derivative of $ x^4 - 3x^2 - 1 $ is $ 4x^3 - 6x $. The terms involving $ h $ in the difference quotient are $ 6x^2h + 4xh^2 + h^3 - 3h $, which matches option C.

Question 13

Find the indicated term of the binomial expansion.
-$\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 13 }$; eighth term
A) $- 329,472 x ^ { 15 } y ^ { 16 }$
B) $- 3,432 x ^ { 18 } y ^ { 14 }$
C) $- 2,574 x ^ { 15 } y ^ { 16 }$
D) $- 219,648 x ^ { 18 } y ^ { 14 }$

Accepted Answer

The eighth term of the binomial expansion $(a - b)^n$ can be found using the formula for the $k$th term: $T_k = \binom{n}{k-1} a^{n-k+1} b^{k-1}$. For the eighth term ($k = 8$) of $(x^3 - 2y^2)^{13}$, we have $n = 13$, $a = x^3$, and $b = -2y^2$. Thus, $T_8 = \binom{13}{7} (x^3)^{13-7} (-2y^2)^7 = \binom{13}{7} x^{18} (-2)^7 y^{14}$. Calculating $\binom{13}{7} = 1716$ and $(-2)^7 = -128$, we get $T_8 = 1716 \times -128 x^{18} y^{14} = -219,648 x^{18} y^{14}$.

Question 14

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-A ________diagram can be used to illustrate the possible outcomes in a sequence of events.

Accepted Answer

The answer of Write the word or phrase that best...

Question 15

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-Consider a set of n elements of which one element is repeated $$r _ { 1 }$$ times. Then the number of
permutations of the elements of the set is given by $\frac { \square } { \square } \text {. }$

Accepted Answer

The answer of Write the word or phrase that best...

Question 16

Simplify the difference quotient: $$\frac { f ( x + h ) - f ( x ) } { h }$$
-$$6 x ^ { 3 } + 6$$
A) $18 h x ^ { 2 } + 18 x h ^ { 2 } + 6 h ^ { 3 }$
B) $x ^ { 3 } + 18 x ^ { 2 } + 18 x h + 6 h ^ { 2 }$
C) $18 x ^ { 2 } + 18 x h + 6 h ^ { 2 } + 6$
D) $18 x ^ { 2 } + 18 x h + 6 h ^ { 2 }$

Accepted Answer

To simplify the given difference quotient with $f(x) = 6x^3 + 6$, we substitute $f(x+h)$ and $f(x)$ into the formula. Simplifying the expression leads to the elimination of the constant term and results in $18x^2 + 18xh + 6h^2$, which matches option D.

Question 17

Find the indicated term of the binomial expansion.
-$( x - y ) ^ { 11 } ;$ sixth term
A) $- 462 x ^ { 6 } y ^ { 5 }$
B) $- 462 x ^ { 5 } y ^ { 6 }$
C) $462 x ^ { 5 } y ^ { 6 }$
D) $462 x ^ { 6 } y ^ { 5 }$

Accepted Answer

The answer of Find the indicated term of the binomial...

Question 18

Use the Binomial Theorem to find the value of the complex number raised to the given power. Recall that $i = \sqrt { - 1 }$
-$$( - 5 + 3 i ) ^ { 4 }$$
A) $- 644 - 960 i$
B) $644 + 960 i$
C) $644 - 960 i$
D) $- 644 + 960 i$

Accepted Answer

Using the Binomial Theorem, $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$, we expand $(-5 + 3i)^4$ and calculate each term. The real part sums to -644 and the imaginary part sums to -960.

Question 19

Find the indicated term of the binomial expansion.
-$\left( 2 r + s ^ { 2 } \right) ^ { 10 }$; eighth term
A) $180 r ^ { 2 } s ^ { 16 }$
B) $90 r ^ { 2 } s ^ { 8 }$
C) $960 r ^ { 3 } s ^ { 14 }$
D) $240 r ^ { 3 } s ^ { 7 }$

Accepted Answer

The eighth term of the binomial expansion $(a + b)^n$ can be found using the formula for the $k$th term: $T_k = \binom{n}{k-1} a^{n-(k-1)} b^{k-1}$. For the eighth term ($k = 8$) of $(2r + s^2)^{10}$, we use $a = 2r$, $b = s^2$, and $n = 10$. Thus, $T_8 = \binom{10}{7} (2r)^{10-7} (s^2)^7 = \binom{10}{7} (2r)^3 (s^2)^7 = 120 \cdot 8r^3 s^{14} = 960r^3s^{14}$.

Question 20

Expand the binomial by using the binomial theorem.
-$\left( \frac { a } { 3 } - b \right) ^ { 5 }$
A) $\frac { 1 } { 3 } a ^ { 5 } - \frac { 5 } { 3 } a ^ { 4 } b + \frac { 10 } { 3 } a ^ { 3 } b ^ { 2 } - \frac { 20 } { 3 } a ^ { 3 } b ^ { 3 } + \frac { 10 } { 3 } a ^ { 2 } b ^ { 3 } - \frac { 5 } { 3 } a b ^ { 4 } + b ^ { 5 }$
B) $\frac { 1 } { 3 } a ^ { 5 } - \frac { 5 } { 3 } a ^ { 4 } b + \frac { 10 } { 3 } a ^ { 3 } b ^ { 2 } - \frac { 10 } { 3 } a ^ { 2 } b ^ { 3 } - \frac { 5 } { 3 } a b ^ { 4 } - b ^ { 5 }$
C) $\frac { 1 } { 243 } a ^ { 5 } - \frac { 5 } { 81 } a ^ { 4 } b + \frac { 10 } { 27 } a ^ { 3 } b ^ { 2 } - \frac { 10 } { 9 } a ^ { 2 } b ^ { 3 } + \frac { 5 } { 3 } a b ^ { 4 } - b ^ { 5 }$
D) $\frac { 1 } { 243 } a ^ { 5 } - \frac { 5 } { 81 } a ^ { 4 } b + \frac { 10 } { 27 } a ^ { 3 } b ^ { 2 } - \frac { 20 } { 27 } a ^ { 3 } b ^ { 3 } + \frac { 10 } { 9 } a ^ { 2 } b ^ { 3 } - \frac { 5 } { 3 } a b ^ { 4 } + b ^ { 5 }$

Accepted Answer

The correct expansion using the binomial theorem involves calculating each term by combining the coefficients from the binomial coefficient $\binom{n}{k}$, where $n=5$ in this case, with the powers of $\frac{a}{3}$ and $b$. The coefficients correctly account for the powers of $\frac{1}{3}$ raised to each power of $a$, resulting in the denominator being a power of 3 for each term, which matches option C.

Quiz 8: Sequences, Series, Induction, and Probability

Find the indicated term of the binomial expansion. - $\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 14 }$ ; Find the term containing $x ^ { 33 }$ .

Expand the binomial by using the binomial theorem. - $( x + y - 9 ) ^ { 3 }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If n items are arranged in order, then each arrangement is called a________ of n items.

Expand the binomial by using the binomial theorem. - $\left( \frac { 1 } { 3 } v - 4 w \right) ^ { 4 }$

Find the indicated term of the binomial expansion. - $\left( \sqrt { 3 } x ^ { 3 } + y ^ { 5 } \right) ^ { 9 } ;$ fourth term

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The number of ways that n distinguishable items can be arranged in various order is ________!

Expand the binomial by using the binomial theorem. - $\left( e ^ { x } - e ^ { - x } \right) ^ { 4 }$

Use the Binomial Theorem to find the value of the number raised to the given power. - $( 1.1 ) ^ { 3 }$ Hint: Write the expression as $( 1 + 0.1 ) ^ { 3 }$ .

Simplify the difference quotient: $\frac { f ( x + h ) - f ( x ) } { h }$ - $x ^ { 4 } - 3 x ^ { 2 } - 1$

Find the indicated term of the binomial expansion. - $\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 13 }$ ; eighth term

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -A ________diagram can be used to illustrate the possible outcomes in a sequence of events.

Simplify the difference quotient: $\frac { f ( x + h ) - f ( x ) } { h }$ - $6 x ^ { 3 } + 6$

Find the indicated term of the binomial expansion. - $( x - y ) ^ { 11 } ;$ sixth term

Use the Binomial Theorem to find the value of the complex number raised to the given power. Recall that $i = \sqrt { - 1 }$ - $( - 5 + 3 i ) ^ { 4 }$

Find the indicated term of the binomial expansion. - $\left( 2 r + s ^ { 2 } \right) ^ { 10 }$ ; eighth term

Expand the binomial by using the binomial theorem. - $\left( \frac { a } { 3 } - b \right) ^ { 5 }$

Quiz 8: Sequences, Series, Induction, and Probability

Find the indicated term of the binomial expansion. - (x3−2y2) 14\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 14 }(x3−2y2) 14 ; Find the term containing x33x ^ { 33 }x33 .

Expand the binomial by using the binomial theorem. - (x+y−9) 3( x + y - 9 ) ^ { 3 }(x+y−9) 3

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If n items are arranged in order, then each arrangement is called a________ of n items.

Expand the binomial by using the binomial theorem. - (13v−4w) 4\left( \frac { 1 } { 3 } v - 4 w \right) ^ { 4 }(31​v−4w) 4

Find the indicated term of the binomial expansion. - (3x3+y5) 9;\left( \sqrt { 3 } x ^ { 3 } + y ^ { 5 } \right) ^ { 9 } ;(3​x3+y5) 9; fourth term

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The number of ways that n distinguishable items can be arranged in various order is ________!

Expand the binomial by using the binomial theorem. - (ex−e−x) 4\left( e ^ { x } - e ^ { - x } \right) ^ { 4 }(ex−e−x) 4

Use the Binomial Theorem to find the value of the number raised to the given power. - (1.1) 3( 1.1 ) ^ { 3 }(1.1) 3 Hint: Write the expression as (1+0.1) 3( 1 + 0.1 ) ^ { 3 }(1+0.1) 3 .

Simplify the difference quotient: f(x+h) −f(x) h\frac { f ( x + h ) - f ( x ) } { h }hf(x+h) −f(x) ​ - x4−3x2−1x ^ { 4 } - 3 x ^ { 2 } - 1x4−3x2−1

Find the indicated term of the binomial expansion. - (x3−2y2) 13\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 13 }(x3−2y2) 13 ; eighth term

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -A ________diagram can be used to illustrate the possible outcomes in a sequence of events.

Simplify the difference quotient: f(x+h) −f(x) h\frac { f ( x + h ) - f ( x ) } { h }hf(x+h) −f(x) ​ - 6x3+66 x ^ { 3 } + 66x3+6

Find the indicated term of the binomial expansion. - (x−y) 11;( x - y ) ^ { 11 } ;(x−y) 11; sixth term

Use the Binomial Theorem to find the value of the complex number raised to the given power. Recall that i=−1i = \sqrt { - 1 }i=−1​ - (−5+3i) 4( - 5 + 3 i ) ^ { 4 }(−5+3i) 4

Find the indicated term of the binomial expansion. - (2r+s2) 10\left( 2 r + s ^ { 2 } \right) ^ { 10 }(2r+s2) 10 ; eighth term

Expand the binomial by using the binomial theorem. - (a3−b) 5\left( \frac { a } { 3 } - b \right) ^ { 5 }(3a​−b) 5

Find the indicated term of the binomial expansion. - $\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 14 }$ ; Find the term containing $x ^ { 33 }$ .

Expand the binomial by using the binomial theorem. - $( x + y - 9 ) ^ { 3 }$

Expand the binomial by using the binomial theorem. - $\left( \frac { 1 } { 3 } v - 4 w \right) ^ { 4 }$

Find the indicated term of the binomial expansion. - $\left( \sqrt { 3 } x ^ { 3 } + y ^ { 5 } \right) ^ { 9 } ;$ fourth term

Expand the binomial by using the binomial theorem. - $\left( e ^ { x } - e ^ { - x } \right) ^ { 4 }$

Use the Binomial Theorem to find the value of the number raised to the given power. - $( 1.1 ) ^ { 3 }$ Hint: Write the expression as $( 1 + 0.1 ) ^ { 3 }$ .

Simplify the difference quotient: $\frac { f ( x + h ) - f ( x ) } { h }$ - $x ^ { 4 } - 3 x ^ { 2 } - 1$

Find the indicated term of the binomial expansion. - $\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 13 }$ ; eighth term

Simplify the difference quotient: $\frac { f ( x + h ) - f ( x ) } { h }$ - $6 x ^ { 3 } + 6$

Find the indicated term of the binomial expansion. - $( x - y ) ^ { 11 } ;$ sixth term

Use the Binomial Theorem to find the value of the complex number raised to the given power. Recall that $i = \sqrt { - 1 }$ - $( - 5 + 3 i ) ^ { 4 }$

Find the indicated term of the binomial expansion. - $\left( 2 r + s ^ { 2 } \right) ^ { 10 }$ ; eighth term

Expand the binomial by using the binomial theorem. - $\left( \frac { a } { 3 } - b \right) ^ { 5 }$