Let

$$\Delta f = f ( 1 + h ) - f ( 1 ),$$

where

$$f ( x ) = x ^ { - 1 }.$$

Show directly that

$$E = \left| \Delta f - f ^ { \prime } ( 1 ) h \right|$$

is equal to

$$h ^ { 2 } / ( 1 + h ).$$

Then prove that

$$E \leq 2 h ^ { 2 }$$

if

$$- \frac { 1 } { 2 } \leq h \leq \frac { 1 } { 2 }.$$

Hint: In this case,

$$\frac { 1 } { 2 } \leq 1 + h \leq \frac { 3 } { 2 }.$$

In $\triangle G H J, \angle G$ is a right angle, $\mathrm{m} \angle H=60^{\circ}$, and $G J=3$. How are $G H$ and $H J$ related? Determine $G H$ and $H J$. Then, give exact values for $\sin 60^{\circ}$, $\cos 60^{\circ}$, and $\tan 60^{\circ}$.

In this exercise, write an equation of the parabola in vertex form.

(2,-2), (4,6)

Find and simptify the difference quotient

$$\frac { f ( x + h ) - f ( x ) } { h } , h \neq 0$$

for the given fherction.

$$f ( x ) = - x ^ { 2 } + 2 x + 4$$

Sketch a graph of the parabola.

$$4\text{x}^\text{2}= -2\text{y}$$

Qui est Airano Dicp?

Find each sum, difference, or product. Give your answer in the simplest radical form. $\\$$(-6-$$\sqrt{6})-(-1-3$$\sqrt{24})$

Find and simplify the difference quotient

$$\frac { f ( x + h ) - f ( x ) } { h } , h \neq 0$$

for the given function.

$$f ( x ) = x ^ { 2 } - 5 x + 8$$

Find each sum, difference, or product. Give your answer in the simplest radical form. $\\$$(4-2$$\sqrt{27})(1+$$\sqrt{75})$

Balance the following redox equations. All occur in basic solution.\

$$\mathrm{Fe}(\mathrm{OH})_2(\mathrm{~s})+\mathrm{CrO}_4^{2-}(\mathrm{aq}) \rightarrow$$

$$\mathrm{Fe}(\mathrm{OH})_{\mathrm{s}}(\mathrm{s})+\left[\mathrm{Cr}(\mathrm{OH})_4\right]^{-}(\mathrm{aq})$$

Write an equation of a parabola with a vertex at the origin and the given directrix.

directrix $y=5$

Simplify each expression.

$$\frac{\cos (x+h)+\cos x}{h}$$

What redox reaction, if any, will occur at $25^{\circ} \mathrm{C}$ when $\mathrm{Zn}$ metal is placed in a solution that is $1.0 \mathrm{M}$ in $\mathrm{Sr}^{2+}$ ?

(a) $\mathrm{Zn}^{2+}+\mathrm{Sr} \longrightarrow \mathrm{Zn}+\mathrm{Sr}^{2+}$

(b) $\mathrm{Sr}^{2+}+\mathrm{Zn} \longrightarrow \mathrm{Sr}+\mathrm{Zn}^{2+}$

(c) $\mathrm{Zn}^{2+}+\mathrm{Sr}^{2+} \longrightarrow \mathrm{Zn}+\mathrm{Sr}$

(d) $\mathrm{Zn}+\mathrm{Sr} \longrightarrow \mathrm{Zn}^{2+}+\mathrm{Sr}^{2+}$

(e) No redox reaction will occur.

Use fraction strips to find the sum. Write your answer in simplest form.

$$\frac { 3 } { 5 } + \frac { 1 } { 2 }$$