91Ó°ÊÓ

Calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. HINT [See Example 1.] $$ \begin{array}{|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 1 & 2 & 3 \\ \hline \boldsymbol{f ( x )} & 3 & 5 & 2 & -1 \\ \hline \end{array} $$ Interval: \([1,3]\)

Calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. HINT [See Example 1.] $$ \text { Cisco Systems Stock Price (\$) } $$$$ \text { Interval: }[1,5] $$

Compute \(f^{\prime}(a)\) algebraically for the given value of a. HINT [See Example 1.] $$ f(x)=m x+b ; a=43 $$

Calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. HINT [See Example 1.] $$ f(x)=3 x^{2}-\frac{x}{2} ;[3,4] $$

Calculate the average rate of change of the given function fover the intervals [a, \(a+h]\) where \(h=1,0.1\), 0.01,0.001, and 0.0001. (Technology is recommended for the cases \(h=0.01,0.001\), and \(0.0001)\) HINT [See Example 4.] \(f(x)=2 x^{2} ; a=0\)

Compute the indicated derivative. $$ U(t)=-1.3 t^{2}-4.5 t ; U^{\prime}(1) $$

Crude Oil Prices The price per barrel of crude oil in constant 2008 dollars can be approximated by $$ P(t)=0.45 t^{2}-12 t+105 \text { dollars } \quad(0 \leq t \leq 28) $$ where \(t\) is time in years since the start of \(1980 .^{40}\) a. What, in constant 2008 dollars, was the average rate of change of the price of oil from the start of 1981 ( \(t=1\) ) to the start of \(2006(t=26)\) ? HINT [See Example 3.] b. Your answer to part (a) is quite small. Can you conclude that the price of oil hardly changed at all over the 25 -year period 1981 to 2006 ? Explain.

Calculate the limits in Exercises 21-72 algebraically. If a limit does not exist, say why. $$ \lim _{x \rightarrow-2} \frac{x^{2}+8}{x^{2}+3 x+2} $$

What is wrong with the following statement? "If \(f(a)\) is defined, then \(\lim _{x \rightarrow a} f(x)\) exists and equals \(f(a)\) "

If \(f\) is a linear function of \(x\) with slope \(m\), what is its average rate of change over any interval \([a, b] ?\)