1ce690…: “Subsequent to a heavy sea”

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We encouter quadratic function more than any other in Calculus. We need to know them backwards and forwards.

Analyze Q(w)

\[ Q(w) = 3 w^2 -5 w + 1 \, \text { with its natural domain } \]

Algebraic Language

$\blacktriangleright $ Domain: We are given that the domain is the natural domain and the natural domain of all polynomials is $(-\infty , \infty )$.

$\blacktriangleright $ Continuity: All polynomials are continuous on their domain. So, $Q$ has no discontinuities. Since, the domain is all real numbers, there can be no singularities.

$\blacktriangleright $ Zeros: $Q$ doesn’t seem to factor easily, so the quadratic formula will give the zeros.

\[ \frac {-(-5) \pm \sqrt {(-5)^2 - 4 (3) (1)}}{2(3)} = \frac {5 \pm \sqrt {13}}{6} \]

There are two real zeros $\frac {5 - \sqrt {13}}{6}$ and $\frac {5 + \sqrt {13}}{6}$

We can now factor $Q$.

\[ Q(w) = \left (w - \left ( \frac {5 - \sqrt {13}}{6} \right ) \right ) \left (w - \left ( \frac {5 + \sqrt {13}}{6} \right ) \right ) \]

Both roots have an odd multipicity, which means the sign of $Q$ will change over them.

$\blacktriangleright $ End-Behavior: Polynomials with even degree (like quadratics) have the same end-behavior on either side. Since $Q$ has a positive leading coefficient, $Q$ is unbounded positively as $w$ tends to $\infty $ or $-\infty $.

\[ \lim \limits _{w \to -\infty } Q(w) = \infty \]

\[ \lim \limits _{w \to \infty } Q(w) = \infty \]

$\blacktriangleright $ Behavior:

Since the leading coefficient of $Q$ is positive, we know $Q$ has a global maximum at the critical number $\frac {-b}{2 a}= \frac {5}{6}$. We also know that $Q$ decreases and then increases.

$Q$ decreases on $\left ( -\infty , \frac {5}{6} \right )$

$Q$ increases on $\left ( \frac {5}{6}, \infty \right )$

$\blacktriangleright $ Extrema:

Since $Q$ decreases and then increase, $Q$ has a global (and one local) minimum of $Q\left ( \frac {5}{6} \right ) = - \frac {13}{12}$ at $\frac {5}{6}$.

Since $Q$ is unbounded positively, it has not global maximum. Since there is only one critical number, $Q$ has no local maximums.

$\blacktriangleright $ Range:

That range of $Q$ is $\left [ -\frac {13}{12}, \infty \right )$.

Graphical Language

$\blacktriangleright $ The graph of $y = Q(w)$ is a parabola and has no holes or breaks.

$\blacktriangleright $ The graph has two intercepts:

\[ \left ( \frac {5 - \sqrt {13}}{6}, 0 \right ) \, \text { and } \, \left ( \frac {5 + \sqrt {13}}{6}, 0 \right ) \]

$\blacktriangleright $ The graph slopes down to the right until it hits its lowest point, $\left ( \frac {5}{6}, -\frac {12}{13} \right )$, then it slopes up to the right. The graph has no highest point and tends to infinity as $w$ tends to $-\infty $ or $\infty $.

2025-01-09 20:20:29