answer:The cross product of two vectors vec{a} = (a_1, a_2, a_3) and vec{b} = (b_1, b_2, b_3) is defined as: vec{a} times vec{b} = left( begin{array}{c} a_2b_3 - a_3b_2 a_3b_1 - a_1b_3 a_1b_2 - a_2b_1 end{array} right) So, for the given vectors vec{a} and vec{b}, we have: vec{a} times vec{b} = left( begin{array}{c} (0)(7) - (-1)(-9) (-1)(-6) - (4)(7) (4)(-9) - (0)(-6) end{array} right) vec{a} times vec{b} = left( begin{array}{c} 9 -22 -36 end{array} right) Therefore, the cross product of vec{a} and vec{b} is left( begin{array}{c} -9 -22 -36 end{array} right). The answer is vec{a} times vec{b} = left( begin{array}{c} -9 -22 -36 end{array} right)

answer:To find the fourth order expansion of f(g(x)) = tan(log(x^2)) around x = 0, first expand log(x^2) and then tan(u), where u = log(x^2). The Taylor expansion of log(x^2) about x = 0 is: log(x^2) = log(1 + 2x^2) = 2x^2 - frac{2x^4}{2!} + frac{2x^6}{3!} - dots Since we only need up to fourth order, we can ignore terms beyond x^4. Next, we substitute u = 2x^2 in the Taylor expansion of tan(u): tan(u) = u + frac{u^3}{3} + frac{2u^5}{15} + dots Substituting u = 2x^2: tan(2x^2) = 2x^2 + frac{8x^6}{3} + dots Now, consider f(g(x)) = tan(log(x^2)): f(g(x)) = 2x^2 + frac{8x^6}{3} + dots This is the fourth order expansion about x = 0. Note that the frac{x^3}{3} term from the original answer is part of the tan(u) expansion but canceled out due to u = 2x^2 and the fact that we only need terms up to x^4.

answer:Milankovitch cycles are astronomical cycles that describe the variations in the Earth's orbit around the Sun and the tilt of its axis. These cycles are named after the Serbian mathematician Milutin Milankovitch, who first proposed them in the early 20th century. The three main Milankovitch cycles are: 1. Eccentricity: This cycle describes the changes in the shape of the Earth's orbit around the Sun. The Earth's orbit is not a perfect circle but rather an ellipse. The eccentricity of the orbit is a measure of how elongated the ellipse is. The eccentricity of the Earth's orbit varies over a period of about 100,000 years. 2. Obliquity: This cycle describes the changes in the tilt of the Earth's axis of rotation. The Earth's axis is not fixed in space but rather wobbles like a spinning top. The obliquity of the Earth's axis is a measure of the angle between the Earth's axis and the axis of its orbit around the Sun. The obliquity of the Earth's axis varies over a period of about 41,000 years. 3. Precession: This cycle describes the changes in the direction of the Earth's axis of rotation. The Earth's axis of rotation is not fixed in space but rather precesses, or wobbles, like a spinning top. The precession of the Earth's axis is a measure of the angle between the Earth's axis and the axis of its orbit around the Sun. The precession of the Earth's axis varies over a period of about 26,000 years. Milankovitch cycles are thought to be the primary cause of the ice ages. When the Earth's orbit is more elliptical, the amount of solar radiation that reaches the Earth's surface varies more from season to season. This can lead to more extreme winters and summers. When the Earth's axis is more tilted, the amount of solar radiation that reaches the Earth's surface varies more from latitude to latitude. This can lead to more extreme temperatures in the tropics and more moderate temperatures in the polar regions. When the Earth's axis precesses, the amount of solar radiation that reaches the Earth's surface varies more from hemisphere to hemisphere. This can lead to more extreme temperatures in one hemisphere and more moderate temperatures in the other hemisphere. The combined effects of Milankovitch cycles can lead to significant changes in the Earth's climate. These changes can cause glaciers to advance and retreat, sea levels to rise and fall, and deserts to expand and contract.

answer:The least squares vector is: left( begin{array}{c} 0.121 -0.519 0.578 -0.023 end{array} right)