Let p2 be the vector space of all polynomials of degree 2 or less, and let h be the subspace spanned by 10x2+4x−1, 3x−4x2+3, and 5x2+x−1. the dimension of the subspace h is . is {10x2+4x−1,3x−4x2+3,5x2+x−1} a basis for p2? be sure you can explain and justify your answer. a basis for the subspace h is { }. enter a polynomial or a comma separated list of polynomials.

the direct variation says that:

then the equation is of the form is given by:

            where k is the constant variation.

from the given table:

consider any value of x and m(x)=y

x = 2 and m(x)=y = 3

substitute these value in [1]; we have;

divide both sides by 2 we get;

⇒

check:

put x = -1 and m(x) = -1.5

substitute in the equation:

-1.5 = -1.5                   true

therefore, the function m(x) represents a direct variation

graph of this function as shown below:

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.