91Ó°ÊÓ

Find the slope of the tangent line to the exponential function at the point \((0,1) .\)

Population Growth The population \(P\) (in thousands) of Houston, Texas from 1980 through 2005 can be modeled by \(P=1576 e^{0.01 t},\) where \(t=0\) corresponds to 1980 . (a) According to this model, what was the population of Houston in \(2005 ?\) (b) According to this model, in what year will Houston have a population of \(2,500,000 ?\)

Carbon Dating In Exercises \(83-86,\) you are given the ratio of carbon atoms in a fossil. Use the information to estimate the age of the fossil. In living organic material, the ratio of radioactive carbon isotopes to the total number of carbon \(4.1 .\) When organic materia. (See Example 2 in Section 4.15 years. So, the ratio with a half-life carbout 5715 years. So, the ratio \(R=10^{-12}\left(\frac{1}{2}\right)^{t / 5715}\), where \(t\) is the time (in years) and \(t=0\) represents the time when the organic material died. $$ R=0.27 \times 10^{-12} $$

Minimum Average cost The cost of producing \(x\) units of a product is modeled by \(C=100+25 x-120 \ln x, \quad x \geq 1\) (a) Find the average cost function \(\bar{C}\). (b) Analytically find the minimum average cost. Use a graphing utility to confirm your result.