f(x) = 4x2 − 3 and g(x) = 4x − 5. Find the value, if possible. (f − g)(−7) = ___________ (f − g)(−7) = ___________

The equation of the circle that satisfies the given conditions, with endpoints of a diameter at \( P(-2,2) \) and \( Q(6,8) \), is **\((x - 2)^2 + (y - 4)^2 = 36\)**.

To find the equation of a circle given the endpoints of a diameter, we can use the midpoint formula to find the center of the circle. The midpoint of the diameter is the center of the circle. Let's find the midpoint using the coordinates of \( P(-2,2) \) and \( Q(6,8) \):

Midpoint \( M \) = \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)

Midpoint \( M \) = \(\left(\frac{-2 + 6}{2}, \frac{2 + 8}{2}\right)\)

Midpoint \( M \) = \(\left(\frac{4}{2}, \frac{10}{2}\right)\)

Midpoint \( M \) = \((2, 5)\)

The coordinates of the midpoint \( M \) give us the center of the circle, which is \( (2, 5) \).

Next, we need to find the radius of the circle. We can use the distance formula to find the distance between \( P(-2,2) \) and \( Q(6,8) \), which is equal to twice the radius. Let's calculate the distance:

\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

\(d = \sqrt{(6 - (-2))^2 + (8 - 2)^2}\)

\(d = \sqrt{8^2 + 6^2}\)

\(d = \sqrt{64 + 36}\)

\(d = \sqrt{100}\)

\(d = 10\)

Since the distance between the endpoints is equal to twice the radius, the radius of the circle is \( \frac{10}{2} = 5 \).

Now that we have the center and radius, we can write the equation of the circle using the standard form:

\((x - h)^2 + (y - k)^2 = r^2\), where \( (h, k) \) is the center and \( r \) is the radius.

Plugging in the values, we get:

\((x - 2)^2 + (y - 5)^2 = 5^2\)

\((x - 2)^2 + (y - 4)^2 = 25\)

Therefore, the equation of the circle that satisfies the given conditions, with endpoints of a diameter at \( P(-2,2) \) and \( Q(6,8) \), is \((x - 2)^2 + (y - 4)^2 = 36\).

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Given differential equation is:(9x^3 8y/x)dx (y^2 8lnx)dy=0.

If a differential equation is of the form M(x,y)dx + N(x,y)dy = 0, then it is called an exact differential equation

if:∂M/∂y = ∂N/∂x

Here, M = 9x³ + 8y/x and N = y² + 8lnx.

Therefore, ∂M/∂y = 8 and ∂N/∂x = 8/x.

Thus, the given differential equation is an exact differential equation.

Now, to find the solution of an exact differential equation, we integrate either M or N with respect to x or y, respectively.

Let's integrate M w.r.t x. So, we get:

∫Mdx = ∫(9x³ + 8y/x)dx= 9/4 x⁴ + 8y ln x + h(y) (put h(y) = 0,

since ∂(∂M/∂y)/∂y = ∂(∂N/∂x)/∂x )

Differentiating the above w.r.t y, we get:(d/dy) ∫Mdx = 8x + h'(y)

Comparing the above with N = y² + 8lnx

We get, h'(y) = y²∴ h(y) = y³/3 + c Here, c is a constant of integration.

The general solution of  is 9/4 x⁴ + 8y ln x + y³/3 = c.

Yes the differential equation is exact

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For the polynomial x⁶-64, we can apply the Difference of Squares but not the Difference of Cubes.

The Difference of Squares is a factoring pattern that can be used when we have the difference of two perfect squares, which means two terms that are both perfect squares and are being subtracted. In this case, x⁶-64 can be written as (x³)² - 8². This can be factored further as (x³ - 8)(x³ + 8).

However, the Difference of Cubes is a factoring pattern that can be used when we have the difference of two perfect cubes, which means two terms that are both perfect cubes and are being subtracted. Since x⁶-64 does not fit this pattern, we cannot apply the Difference of Cubes.

In summary, for the polynomial x⁶-64, we can apply the Difference of Squares by factoring it as (x³ - 8)(x³ + 8), but we cannot apply the Difference of Cubes.

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The graph of the function z = f(x, y) = x² + y² is a surface in three-dimensional space. It represents a paraboloid centered at the origin with its axis aligned with the z-axis.

The shape of the graph is similar to an upward-opening bowl or a circular cone. As you move away from the origin along the x and y axes, the function increases quadratically, resulting in a smooth and symmetric surface.

The contour lines of the graph are concentric circles centered at the origin, with each circle representing a specific value of z. The closer the contour lines are to the origin, the smaller the corresponding values of z. As you move away from the origin, the values of z increase.

The surface has rotational symmetry around the z-axis. This means that if you rotate the graph about the z-axis by any angle, the resulting shape remains the same.

In summary, the graph of the function z = f(x, y) = x² + y² is a smooth, upward-opening paraboloid centered at the origin, with concentric circles as its contour lines. It exhibits symmetry around the z-axis and represents a quadratic relationship between x, y, and z.

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The next equation in the suspected number pattern is 4 = 4 × 13. This statement is true because the left side of the equation simplifies to 4, which is equal to the right side of the equation when evaluated.

By observing the given equations, we can identify a pattern. In the first equation, 3 is obtained by multiplying 1 with 33 and adding 11. In the second equation, 73 is obtained by multiplying 2 with 33 and adding 11. In the third equation, 11 + 19 results from multiplying 3 with 33 and adding 11.

Therefore, it appears that the common factor in these equations is the multiplication of a variable, which seems to correspond to the number of the equation itself, with 33, followed by the addition of 11. Applying this pattern to the next equation, we can predict that it will be 4 = 4 × 13.

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