Exam 5: Exponential and Logarithmic Functions

Find the derivative of the function. ​

According to estimates by Paul Kroger Associates, the percent of households that own VCRs is given by , where t is measured in years, with t = 0 corresponding to the beginning of 1985. ​ What percent of households owned VCRs at the beginning of 1985? Round your answer to the nearest tenth. ​ __________% ​ At the beginning of 1992? Round your answer to the nearest tenth. ​ __________%

The concentration of a drug in an organ at any time t (in seconds) is given by where is measured in grams/cubic centimeter . ​ How long would it take for the concentration of the drug in the organ to reach 0.15 ? ​ How long would it take for the concentration of the drug in the organ to reach 0.12 ? ​ Round your answer to the nearest hundredth. ​

The monthly demand for a certain brand of table wine is given by the demand equation where p denotes the wholesale price per case (in dollars) and x denotes the number of cases demanded. Find the rate of change of the price to the nearest hundredth of a cent per case when x = 1,000. ​

Evaluate the expression. ​ __________

Strontium 90, a radioactive isotope of strontium, is present in the fallout resulting from nuclear explosions. It is especially hazardous to animal life, including humans, because, upon ingestion of contaminated food, it is absorbed into the bone structure. Its half-life is 27 years. If the amount of strontium 90 in a certain area is found to be eight times the "safe" level, find how much time must elapse before an "acceptable level" is reached. ​

Find the second derivative of the function. ​

Eleni, who is now 50 years old, is employed by a firm that guarantees her a pension of $40,000/year at age 65. What is the present value of her first year's pension if inflation over the next 15 year(s) is at the given rate. Assume that inflation is continuously compounded. Round your answers to the nearest cent. ​ 6% ​ $__________ ​ 7% ​ $__________ ​ 18% ​ $__________

Use the laws of logarithms to simplify the expression. ​ ​

Solve the equation for . ​ ​

For children between the ages of 5 and 13 years old, the Ehrenberg equation gives the relationship between the weight W (in kilograms) and the height h (in meters) of a child. Use differentials to estimate the change in the weight of a child who grows from 1 m to 1.2 m. ​

Express the equation in logarithmic form. ​ ​

The concentration of a drug in an organ at any time t (in seconds) is given by where x(t) is measured in grams/cubic centimeter (g/cm³). ​ Select the graph of x. ​

Solve the equation for . ​

According to estimates by Paul Kroger Associates, the percent of households that own VCRs is given by where t is measured in years, with t = 0 corresponding to the beginning of 1985. What percent of households owned VCRs at the beginning of 1985? At the beginning of 1990? ​

Use logarithmic differentiation to find the derivative of the function. ​

Evaluate the expression. ​ ​

The concentration of a drug in an organ at any time (in seconds) is given by ​ ​ Where is measured in milligrams/cubic centimeter (mg/cm³). What is the concentration to four decimal places of the drug in the organ after 5 sec? ​

Sketch the graphs of the given functions on the same axes. ​ and ​

The change from religious to lay teachers at Roman Catholic schools has been partly attributed to the decline in the number of women and men entering religious orders. The percent of teachers who are lay teachers is given by where t measured in decades, with t = 0 corresponding to the beginning of 1960. ​ What percent of teachers were lay teachers at the beginning of 1990? Please round the answer to one decimal place. ​ __________ % ​ How fast was the percent of lay teachers changing at the beginning of 1990? Please round the answer to two decimal places. ​ __________ % / year ​ Find the year when the percent of lay teachers was increasing most rapidly.