- Solve the system of equations $3x-y+2z=4, 4x+z=7$. Start by picking one of the variables to equal $t$, and then solve for all the others in terms of $t$.
- Find the distance between the two points $(3,5,-2)$ and $(-1,6,4)$.
- Give an equation of the sphere that passes through the point $(3,5,-2)$ and has the center $(-1,6,4)$.
- For this one, have everyone in the group use chalk at the same time. In 3D, plot the points $(1,2,0)$, $(1,0,3)$, $(0,2,3)$, $(1,2,3)$, $(-2,4,-3)$.
- Given the function $f(x) = \cos(x)$, find a fourth degree polynomial $P(x) = a_0+a_1x^1+a_2x^2+a_3x^3+a_4x^4$ so that $f(0)=P(0)$, $f^{\prime}(0)=P^{\prime}(0) $, $f^{\prime\prime}(0)=P^{\prime\prime}(0) $, $f^{\prime\prime\prime}(0)=P^{\prime\prime\prime}(0) $, and $f^{ (4) }(0)=P^{ (4) }(0) $. In other words, you want the function and its first four derivatives to be equal at $x=0$. Notice that you have 5 equations to find the 5 unknowns $a_0, a_1, a_2, a_3, a_4.$
- Repeat the previous problem, but this time find a 20th degree polynomial.
- Change the function to $f(x)=\sqrt{x+1}$ and find a fourth degree polynomial.
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