Chapter 10: Chapter 10

Each of the functions in Exercises 9-12 gives the cost to manufacture \(x\) items. Find the average cost per unit of manufacturing h more items (i.e., the average rate of change of the total cost) at a production level of \(x\), where \(x\) is as indicated and \(h=10\) and \(1 .\) Hence, estimate the instantaneous rate of change of the total cost at the given production level \(x\), specifying the units of measurement. (Use smaller values of h to check your estimates.) HINT [See Example 1.] $$ C(x)=10,000+5 x-\frac{x^{2}}{10,000} ; x=1,000 $$

Determine if the given limit leads to a determinate or indeterminate form. Evaluate the limit if it exists, or say why if not. $$ \lim _{x \rightarrow-\infty}\left(-x^{2}+5\right) $$

Compute \(f^{\prime}(a)\) algebraically for the given value of a. HINT [See Example 1.] $$ f(x)=x^{3}+2 x ; a=2 $$

Weekly sales of an old brand of TV are given by $$ S(t)=100 e^{-t / 5} $$ sets per week, where \(t\) is the number of weeks after the introduction of a competing brand. Estimate \(S(5)\) and \(\left.\frac{d S}{d t}\right|_{t=5}\) and interpret your answers.

On January 1,1996, America Online was the biggest online service provider, with \(4.5\) million subscribers, and was adding new subscribers at a rate of 60,000 per week. \(^{53}\) If \(A(t)\) is the number of America Online subscribers \(t\) weeks after January 1,1996, what do the given data tell you about values of the function \(A\) and its derivative? HINT [See Quick Example 2 on page $736 .$ ]

The percentage \(p(t)\) of children who can speak in at least single words by the age of \(t\) months can be approximated by the equation \(^{21}\) $$ p(t)=100\left(\frac{1-12,200}{t^{4.48}}\right) . \quad(t \geq 8.5) $$ Calculate \(\lim _{t \rightarrow+\infty} p(t)\) and interpret the results. HINT [See Example \(5(\mathrm{e}),(\mathrm{f}) .]\)

On January 1, 1996 Prodigy was the third-biggest online service provider, with \(1.6\) million subscribers, but was losing subscribers. \({ }^{54}\) If \(P(t)\) is the number of Prodigy subscribers \(t\) weeks after January 1,1996, what do the given data tell you about values of the function \(P\) and its derivative? HINT [See Quick Example 2 on page 736.]

Let \(p(t)\) represent the percentage of children who are able to speak at the age of \(t\) months. a. It is found that \(p(10)=60\) and $\left.\frac{d p}{d t}\right|_{t=10}=18.2 .\( What does this mean? \)^{55}\( HINT [See Quick Example 2 on page \)736 .$ ] b. As \(t\) increases, what happens to \(p\) and \(\frac{d p}{d t}\) ?

Describe the algebraic method of evaluating limits as discussed in this section and give at least one disadvantage of this method.

Let \(p(t)\) represent the number of children in your class who learned to read at the age of \(t\) years. a. Assuming that everyone in your class could read by the age of 7, what does this tell you about \(p(7)\) and \(\left.\frac{d p}{d t}\right|_{t=7}\) ? HINT [See Quick Example 2 on page 736.] b. Assuming that \(25.0 \%\) of the people in your class could read by the age of 5 , and that \(25.3 \%\) of them could read by the age of 5 years and one month, estimate \(\left.\frac{d p}{d t}\right|_{t=5}\). Remember to give its units.