Explanation

1. Commutative Law: In an Abelian group, for all elements $a$ and $b$ in the group, the operation satisfies $a\cdot b=b\cdot a$.

2. Named after Niels Henrik Abel: The term "Abelian" is derived from the name of the mathematician Niels Henrik Abel.

Explanation

1. Intersection of $D_f$ and $D_g$: The function $\frac{f}{g}$ is defined only where both $f$ and $g$ are defined.

2. Excluding zeros of $g$: The function $\frac{f}{g}$ is undefined where $g(x)=0$ because division by zero is not allowed.

Explanation

To integrate √(2x + 1), we can use the power rule of integration and the chain rule.

Let u = 2x + 1, then du/dx = 2, and du = 2dx.

∫√(2x + 1) dx = (1/2) ∫u^(1/2) du

= (1/2) (2/3) u^(3/2) + C

= (1/3) (2x + 1)^(3/2) + C

Explanation

To find the slope of the tangent to the parabola y = 2x^2 - 13x + 5, we need to find the derivative of y with respect to x.

dy/dx = 4x - 13

We want to find if there is a point where the slope of the tangent is -1.

4x - 13 = -1

4x = 12

x = 3

Now, we need to find the corresponding y-coordinate.

y = 2(3)^2 - 13(3) + 5

y = 18 - 39 + 5

y = -16

So, the point of tangency is (3, -16).

The equation of the tangent line is:

y - (-16) = -1(x - 3)

y + 16 = -x + 3

y = -x - 13

Explanation

Calculation:

• Current total age = 80 years

• Three years ago, each person was 3 years younger.

• Since there are three people, the total reduction in age is $3\times 3=9$ years.

So, the total of their ages three years ago is:

$$\mathrm{<b><font\; color="\#0000ff">80</font></b>}<b><font\; color="\#0000ff">-</font></b>\mathrm{<b><font\; color="\#0000ff">9</font></b>}<b><font\; color="\#0000ff">=</font></b>\mathrm{<b><font\; color="\#0000ff">71</font></b>}\text{<b><font color="\#0000ff"> years</font></b>}$$

Explanation

Explanation

Harmonic Mean (H.M): The reciprocal of the arithmetic mean of the reciprocals of the numbers.

Geometric Mean (G.M): The nth root of the product of the numbers.

Arithmetic Mean (A.M): The sum of the numbers divided by the count of numbers.

Explanation

Explanation

Explanation

If the correlation coefficient is 0, the two regression lines are perpendicular.

This is because a correlation coefficient of 0 indicates no linear relationship between the variables.

This causing the regression lines to be at right angles to each other.