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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verify that the given differential equation is exact; then solve it. (9x^3 8y/x)dx (y^2 8lnx)dy=0
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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Equations are given below illustrating a suspected number pattern. Determine what the next equation would be, and verify that it is indeed a true statement. 3=1×33+11=2×73+11+19=3×11 Select the correct answer below and fill in any answer boxes within your choice. (Type the terms of your expression in the same order as they appear in the original expression. Do not perform the calculation. Use the multiplication symbol in the math palette as needed. ) A. The next equation is It is a false statement because the left side of the equation simplifies to and the right side of the equation simplifies to B. The next equation is It is a true
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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