Find one and two$-$sided limits from a graph. Remember that the limit is independent of the value of the function at a given value.

a$.\underset{x-3^{-}}{lim}f(x)=$

f$.\underset{x{2}^{-}}{lim}f(x)=$

b$.\underset{x-3^{+}}{lim}f(x)=$

c$.\underset{x-3}{lim}f(x)=$

g$.\underset{x{2}^{+}}{lim}f(x)=$

h$.\underset{x2}{lim}f(x)=$

d$.f(-3)=$

$f(2)=$

e$.\underset{x0}{lim}f(x)=$

$\underset{x-2}{lim}f(x)=$

Use limits and limit notation to find asymptotes.

a$.$ Vertical asymptotes:

i$.$ Find zeros of denominator.

ii$.$ If numerator is not zero, value is a vertical asymptote.

iii. If numerator and denominator are both zero, find limit If the limit exists, it's a hole.

$f(x)=\frac{x+3}{x^2-x-12}$

b$.$ Horizontal asymptotes:

Use limit notation to find the limit as $x$

ii$.$ Find the horizontal asymptotes using the following guidelines:

If the limit is $0,$ asymptote is $y=0$

If the limit is $L,$ asymptote is $y=L$

If the limit is $$ or $-,$ there is no asymptote.

$f(x)=\frac{3x^3-2x^2+4}{-2x^3-7x+9}$

$f(x)=\frac{3x^2-7}{-4x^3+6x-5}$

$f(x)=\frac{-4x^3+6x-5}{3x^2-7}$

Use a sign chart to solve an inequality; give answers in interval notation.

$\frac{x+3}{x^2-x}>0$

Let $f(x)=8x^4+3x^5+17.$ Describe the end behavior of the function by finding the following:

$\underset{x-}{lim}f(x)$

$\underset{x}{lim}f(x)$

Let $f(x)=\frac{9x^4-18x^2}{12x^5+6}.$ Find the following limit$.$

$\underset{x}{lim}f(x)$Use correct notation to outline the four$-$step process and use the process to find the derivative of a quadratic function.

$f(x)=x^2-5x+3$

Use differentiation rules, including rules for negative and fractional exponents. Be clear about notation; use $f(x)$ and $f^{\text{'}}(x)$ accurately.

a$.f(x)=4x^3$

b$.f(x)=\frac{-2}{x^3}$

c$.f(x)=\sqrt[3]{x^2}$

Apply the derivative to tangent lines; find the slope of a tangent line, the equation for a tangent line, and the values where the tangent line is horizontal.

Let $f(x)=x^4-50x^2+7$

a$.$ Find $f^{\text{'}}(x).$

b$.$ Find the slope of the graph of $y=f(x)$ at $x=1.$

c$.$ Find the equation of the tangent line at $x=1.$

d$.$ Find the values of $x$ where the tangent line is horizontal.

Interpret values of a function and its derivative involving rate of change. Units on the derivative are always rates: $q,$ per

The total sales in millions of dollars $t$ months from now will be given by

$S(t)=0\mathrm{.}04t^3+0\mathrm{.}3t^2+7t+5$

Find $S(6)$ and $S^{\text{'}}(6)$ and interpret the results.

$10.$ Use price$-$demand functions to find revenue and marginal revenue. Interpret results.

The price p $($in dollars$)$ and the demand x for a particular steam iron are related by the equation

$x=1,000-20p$

a$)$ Express the price $p$ in terms of the demand $x.$

b$)$ Find the revenue $R(x)$ from the sale of $x.$

c$)$ Find the marginal revenue at a production level of $400$ steam irons and interpret the results.The total cost $($in dollars$)$ of producing $x$ food processors is $C(x)=2,000+50x-0\mathrm{.}5x^2$

a$)$ Find the exact cost of producing the $21$st food processor.

b$)$ Use marginal cost to approximate the cost of producing the $21$ st food processor.

Find average and marginal profit functions. Interpret results at a specific value.

The profit from sales of $x$ lawn mowers is given by $P(x)=-0\mathrm{.}04x^2+40x-350$

a$.$ Find $\frac{?}{\text{b}\text{a}\text{r}}(P)(x).$

b$.$ Find $\frac{?}{\text{b}\text{a}\text{r}}(P{)}^{\text{'}}(x).$

c$.$ Find $P(30)$ and interpret the results.

d$.$ Find $\frac{?}{\text{b}\text{a}\text{r}}(P)(30)$ and interpret the results.

e$.$ Find $\frac{?}{\text{b}\text{a}\text{r}}(P{)}^{\text{'}}(30)$ and interpret the results.