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If y(ξ) = 0 then there exists a neighborhood (ξ − , ξ + ), the point ξ such that y(x) = 0 for x ∈ (ξ − , ξ + ). > 0 of Intermediate Value Theorem. Let u(x) be continuous on [a, b]. If u(a) ≤ µ ≤ u(b) then there exists ξ ∈ [a, b] such that u(ξ) = µ. This is known as the intermediate value theorem. A corollary of this is that if u(a) and u(b) are of opposite sign then u(x) has at least one zero on the interval (a, b). Maxima and Minima. If u(x) is continuous on [a, b] then u(x) has a maximum and a minimum on [a, b].

A function which is continuous on a closed interval is bounded in that closed interval. Nonzero in a Neighborhood. If y(ξ) = 0 then there exists a neighborhood (ξ − , ξ + ), the point ξ such that y(x) = 0 for x ∈ (ξ − , ξ + ). > 0 of Intermediate Value Theorem. Let u(x) be continuous on [a, b]. If u(a) ≤ µ ≤ u(b) then there exists ξ ∈ [a, b] such that u(ξ) = µ. This is known as the intermediate value theorem. A corollary of this is that if u(a) and u(b) are of opposite sign then u(x) has at least one zero on the interval (a, b).

4 Find the slope and x-intercept of the line. 5 The inverse of the function is the reflection of the function across the line y = x. 6 The formula for geometric growth/decay is x(t) = x0 rt , where r is the rate. 7 Note that p(x) and q(x) appear as a ratio, they are determined only up to a multiplicative constant. We may take the leading coefficient of q(x) to be unity. f (x) = ax2 + bx + c p(x) = 2 q(x) x + βx + χ Use the properties of the function to solve for the unknown parameters. 8 Write the polynomial in factored form.