Stewart 9th - 1.1, 1.6 (20Q)

Find a formula for the described function and state its domain. A rectangle has a perimeter of 20 m. Express the area of the rectangle as a function of the length of one of its sides.

Find the domain of the function. \(h(x) = \dfrac{1}{\sqrt[4]{x^2 - 5 x}}\)

Determine whether the equation defines \(y\) as a function of \(x\). \((y + 3)^3 + 1 = 2 x\)

Find a formula for the described function and state its domain. An open rectangular box with volume 2 m\({}^3\) has a square base. Express the surface area of the box as a function of the length of a side of the base.

Evaluate \(f(-3)\), \(f(0)\), and \(f(2)\) for the piecewise defined function. Then sketch the graph of the function. \( f(x) = \begin{cases} -1 & \quad \text{if } x \leq 1 \\ 7 - 2 x & \quad \text{if } x > 1 \end{cases} \)

Determine whether the table defines \(y\) as a function of \(x\).

\(x\) (Height in inches)	72	60	60	63	70
\(y\) (Shoe size)	12	8	7	9	10

If \(g(x) = \dfrac{x}{\sqrt{x + 1}}\), find \(g(0)\), \(g(3)\), \(5 g(a)\), \(\dfrac{1}{2} g(4 a)\), \(g(a^2)\), \([g(a)]^2\), \(g(a + h)\), and \(g(x - a)\).

Sketch the graph of the function. \(g(t) = |1 - 3 t|\)

Sketch the graph of the function. \(f(x) = |x + 2|\)

Evaluate the difference quotient for the given function. Simplify your answer. \(f(x) = 4 + 3 x - x^2\), \(\quad \dfrac{f(3 + h) - f(3)}{h}\)

Find the limit, if it exists. If the limit does not exist, explain why. \(\operatorname*{lim}\limits_{x \rightarrow 0^-} \left(\dfrac{1}{x} - \dfrac{1}{|x|}\right)\)

\( \operatorname*{lim}\limits_{x \rightarrow -2} \dfrac{x^2 - x - 6}{3 x^2 + 5 x - 2} \)

\( \operatorname*{lim}\limits_{x \rightarrow 4} \dfrac{x^2 + 3 x}{x^2 - x - 12} \)

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \(\operatorname*{lim}\limits_{t \rightarrow -1} \left(\dfrac{2 t^5 - t^3}{5 t^2 + 4}\right)^3\)

\( \operatorname*{lim}\limits_{x \rightarrow -5} \dfrac{2 x^2 + 9 x - 5}{x^2 - 25} \)

Show by means of an example that \(\operatorname*{lim}\limits_{x \rightarrow a} [f(x) g(x)]\) may exist even though neither \(\operatorname*{lim}\limits_{x \rightarrow a} f(x)\) nor \(\operatorname*{lim}\limits_{x \rightarrow a} g(x)\) exists.

The figure shows a fixed circle \(C_1\) with equation \((x - 1)^2 + y^2 = 1\) and a shrinking circle \(C_2\) with radius \(r\) and center the origin. \(P\) is the point \((0, r)\), \(Q\) is the upper point of intersection of the two circles, and \(R\) is the point of intersection of the line \(P Q\) and the \(x\)-axis. What happens to \(R\) as \(C_2\) shrinks, that is, as \(r \rightarrow 0^+\)?

Find the limit, if it exists. If the limit does not exist, explain why. \(\operatorname*{lim}\limits_{x \rightarrow -2} \dfrac{2 - |x|}{2 + x}\)