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|[x_(1)+1,x_(1)^(2)+1,cdots,x_(1)^(n)+1],[x_(2)+1,x_(2)^(2)+1,cdots,x_(2)^(n)+1],[vdots,vdots,,vdots],[x_(n)+1,x_(n)^(2)+1,cdots,x_(n)^(n)+1]|=

$\left|\begin{array}{cccc}x_{1}+1 & x_{1}^{2}+1 & \cdots & x_{1}{ }^{n}+1 \\ x_{2}+1 & x_{2}^{2}+1 & \cdots & x_{2}{ }^{n}+1 \\ \vdots & \vdots & & \vdots \\ x_{n}+1 & x_{n}^{2}+1 & \cdots & x_{n}^{n}+1\end{array}\right|=$

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Transformations of functions

Full solution

Q.

\left|\begin{array}{cccc}x_{1}+1 & x_{1}^{2}+1 & \cdots & x_{1}{ }^{n}+1 \\ x_{2}+1 & x_{2}^{2}+1 & \cdots & x_{2}{ }^{n}+1 \\ \vdots & \vdots & & \vdots \\ x_{n}+1 & x_{n}^{2}+1 & \cdots & x_{n}^{n}+1\end{array}\right|=

Write Matrix Determinant: Write down the matrix to find its determinant. $\newline$ Matrix: $\newline$ $\begin{bmatrix} x_1+1 & x_1^2+1 & \cdots & x_1^n+1 \\ x_2+1 & x_2^2+1 & \cdots & x_2^n+1 \\ \vdots & \vdots & \ddots & \vdots \\ x_n+1 & x_n^2+1 & \cdots & x_n^n+1 \\ \end{bmatrix}$
Notice Element Form: Notice that each element in the matrix is of the form $x_i^j + 1$ , where $i$ is the row index and $j$ is the column index.
Subtract First Column: Subtract the first column from all other columns to simplify the determinant calculation. $\newline$ New Matrix: $\newline$ $\begin{bmatrix} x_1+1 & x_1^2 & \cdots & x_1^n \\ x_2+1 & x_2^2 & \cdots & x_2^n \\ \vdots & \vdots & \ddots & \vdots \\ x_n+1 & x_n^2 & \cdots & x_n^n \\ \end{bmatrix}$
Factor Out x_i: Factor out $x_i$ from the $i$ th row for columns $2$ to n. $\newline$ New Matrix: $\newline$ $\begin{bmatrix} x_1+1 & x_1 & \cdots & x_1 \\ x_2+1 & x_2 & \cdots & x_2 \\ \vdots & \vdots & \ddots & \vdots \\ x_n+1 & x_n & \cdots & x_n \\ \end{bmatrix}$
Factor Out Common Factor: Notice that all rows except the first one have a common factor in columns $2$ to n. Factor out $x_i$ from the $i$ th row for columns $2$ to n. $\newline$ New Matrix: $\newline$ $\begin{bmatrix} x_1+1 & 1 & \cdots & 1 \\ x_2+1 & x_2 & \cdots & x_2 \\ \vdots & \vdots & \ddots & \vdots \\ x_n+1 & x_n & \cdots & x_n \\ \end{bmatrix}$

More problems from Transformations of functions

Question

What does the transformation

f(x) \mapsto f(x) - 6

do to the graph of

f(x)

\newline

\newline

\text{translates it } 6 \text{ units down}

\newline

\text{translates it } 6 \text{ units right}

\newline

\text{translates it } 6 \text{ units left}

\newline

\text{translates it } 6 \text{ units up}

Posted 2 months ago

Question

g(x)

g(x)

is the reflection across the

y

f(x) = -4x + 1

\newline

\newline

\text{[g(x) = 4x - 1]}

\newline

\text{[g(x) = -4x + 1]}

\newline

\text{[g(x) = 4x + 1]}

\newline

\text{[g(x) = -4x - 1]}

Posted 2 months ago

Question

What kind of transformation converts the graph of

f(x) = -8(x - 4)^2 + 6

into the graph of

g(x) = -8(x + 6)^2 + 6

\newline

\newline

\text{[A]translation 10 units left}

\newline

\text{[B]translation 10 units right}

\newline

\text{[C]translation 10 units up}

\newline

\text{[D]translation 10 units down}

Posted 2 months ago

Question

g(x)

g(x)

is the reflection across the x-axis of

f(x) = -6(x - 7)^2 + 2

\newline

\newline

g(x) = 6(x - 7)^2 - 2

\newline

g(x) = -6(x + 7)^2 + 2

\newline

g(x) = -6(x - 7)^2 - 2

\newline

g(x) = 6(x + 7)^2 - 2

Posted 2 months ago

Question

\lim _{x \rightarrow \frac{\pi}{4}} \cos (x)=?

\newline

1

\newline

\frac{1}{2}

\newline

1

\newline

\frac{\sqrt{2}}{2}

\newline

(D) The limit doesn't exist.

Posted 3 months ago

Question

Tess tried to solve the differential equation

\frac{d y}{d x}=2 x y-6 x

. This is her work:

\newline

\frac{d y}{d x}=2 x y-6 x

\newline

1

\quad \frac{d y}{d x}=(y-3) 2 x

\newline

2

\quad \int \frac{1}{y-3} d y=\int 2 x d x

\newline

3

\quad \ln |y-3|=x^{2}+C_{1}

\newline

4

\quad e^{\ln |y-3|}=e^{x^{2}}+C_{1}

\newline

5

\quad|y-3|=e^{x^{2}}+C_{1}

\newline

6

\quad y-3= \pm e^{x^{2}}+C_{2}

\newline

7

\quad y= \pm e^{x^{2}}+C

\newline

Is Tess's work correct? If not, what is her mistake?

\newline

1

\newline

(A) Tess's work is correct.

\newline

1

is incorrect. We're not allowed to factor an expression when solving differential equations.

\newline

4

is incorrect. The right-hand side of the equation should be

e^{x^{2}+C_{1}}

\newline

7

is incorrect. Tess didn't account for adding

3

to the right side.

Posted 1 month ago

Question

Aditya tried to solve the differential equation

\frac{d y}{d x}=6 x-30

. This is his work:

\newline

\frac{d y}{d x}=6 x-30

\newline

1

\quad \int 30 d y=\int 6 x d x

\newline

2

\quad 30 y=3 x^{2}+C

\newline

3

\quad y=\frac{x^{2}}{10}+C

\newline

Is Aditya's work correct? If not, what is his mistake?

\newline

1

\newline

(A) Aditya's work is correct.

\newline

1

is incorrect. The separation of variables wasn't done correctly.

\newline

2

is incorrect. Aditya didn't integrate

30

\newline

3

is incorrect. The right-hand side of the equation should be

\frac{30}{3 x^{2}+C}

Posted 2 months ago

Question

Jonael dropped a sandbag from a hot air balloon at a target that is

250

meters below the balloon. At this moment, the sandbag is

75

meters below the balloon.

\newline

2

of the following expressions represents the distance between the sandbag and the target?

\newline

2

\newline

|-250+75|

\newline

|250+75|

\newline

|-75+250|

Posted 2 months ago

Question

f(x) = -(x+1)(x+7)

\newline

The function represents a parabola in the

xy

-plane. Which of the following is an equivalent form of

f

y

-intercept of the graph of

f

appears as a constant or coefficient?

\newline

1

\newline

f(x) = -(x+4)^{2}+9

\newline

f(x) = -(x+4)^{2}-9

\newline

f(x) = -x^{2}+8x+7

\newline

f(x) = -x^{2}-8x-7

Posted 2 months ago

Question

Determine whether the function

f(x)

is continuous at

x=-3

\newline

f(x)=\left\{\begin{array}{ll} 18-x^{2}, & x \leq-3 \\ 15+3 x, & x>-3 \end{array}\right.

\newline

f(x)

is discontinuous at

x=-3

\newline

f(x)

is continuous at

x=-3

Posted 2 months ago