Complete the given sentence. The closed-form function \(f(x)=\frac{1}{x-1}\) is continuous for all \(x\) except HINT [See Quick Example on page 707.]

The median home price in the U.S. over the period 2004-2009 can be approximated by $$ P(t)=-5 t^{2}+75 t-30 \text { thousand dollars } \quad(4 \leq t \leq 9) $$ where \(t\) is time in years since the start of \(2000 .^{52}\) a. Compute the average rate of change of \(P(t)\) over the interval \([5,9]\), and interpret your answer. HINT [See Section \(10.4\) Example 3.] b. Estimate the instantaneous rate of change of \(P(t)\) at \(t=5\), and interpret your answer. HINT [See Example 2(a).] c. The answer to part (b) has larger absolute value than the answer to part (a). What does this indicate about the median home price?

Annual U.S. sales of bottled water rose through the period 2000–2008 as shown in the following chart The function \(R(t)=12 t^{2}+500 t+4,700\) million gallons \(\quad(0 \leq t \leq 8)\) gives a good approximation, where \(t\) is time in years since 2000 . Find the derivative function \(R^{\prime}(t)\). According to the model, how fast were annual sales of bottled water increasing in \(2005 ?\)

The oxygen consumption of a bird embryo increases from the time the egg is laid through the time the chick hatches. In a typical galliform bird, the oxygen consumption (in milliliters per hour) can be approximated by \(c(t)=-0.0027 t^{3}+0.14 t^{2}-0.89 t+0.15 \quad(8 \leq t \leq 30)\) where \(t\) is the time (in days) since the egg was laid. \({ }^{57}\) (An egg will typically hatch at around \(t=28 .\) ) Use technology to graph \(c^{\prime}(t)\) and use your graph to answer the following questions. HINT [See Example 5.] 1\. Over the interval \([8,30]\) the derivative \(c^{\prime}\) is (A) increasing, then decreasing (B) decreasing, then increasing (C) decreasing (D) increasing \- When, to the nearest day, is the oxygen consumption increasing the fastest? When, to the nearest day, is the oxygen consumption increasing at the slowest rate?

In each of Exercises 55-58, estimate the given quantity. \(f(x)=e^{x} ;\) estimate \(f^{\prime}(0)\)

Compute the derivative function \(f^{\prime}(x)\) algebraically. (Notice that the functions are the same as those in Exercises \(1-14 .)\) HINT [See Examples 2 and \(3 .]\) $$ f(x)=2 x^{2}+x $$