Exam 5: Exponential and Logarithmic Functions

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 is measured in decades, with corresponding to the beginning of 1960. Find the year when the percent of lay teachers was increasing most rapidly.

Sketch the graphs of the functions on the same axes.

Online retail sales stood at $22.7 billion for 2000.For the next 2 years, they grew by 32.4% and 27.4% per year, respectively.For the next 6 years, online retail sales are projected to grow at 30.5%, 19.6%, 24.2%, 13.4%, 17.4%, and 10.1% per year, respectively.What are the projected online sales for 2008? Round your answer to the nearest tenth. $__________ billion

Find the present value of $50,000 due in 3 year(s) at the given rate of interest. 5% / year compounded monthly __________ 9% / year compounded daily __________

Find the derivative of the function.

Find the inflection points of the function .

The growth rate of the bacterium, a common bacterium found in the human intestine, is proportional to its size.Under ideal laboratory conditions, when this bacterium is grown in a nutrient broth medium, the number of cells in a culture doubles approximately every 20 min.If the initial cell population is 100, determine the function Q(t) that expresses the exponential growth of the number of cells of this bacterium as a function of time t (in minutes).How long will it take for a colony of 100 cells to increase to a population of 1 million? If the initial cell population were 1,000, how would this alter the model?

Use logarithmic differentiation to find the derivative of the function.

How long will it take an investment of $4,000 to triple if the investment earns interest at the rate of 6% /year compounded daily?

The percent of a certain brand of computer chips that will fail after t years of use is estimated to be . What percent of this brand of computer chips are expected to be usable after 5 years? Round your answer to one decimal place. __________%

Find the derivative of the function.

Sketch the graph of the equation.

Use the definition of a logarithm to prove .

Find the interest rate needed for an investment of $5,000 to grow to an amount of $7,500 in 3 year(s) if interest is compounded quarterly.Round your answer to two decimal places.

A certain piece of machinery was purchased 4 year(s) ago by Garland Mills for $400,000.Its present resale value is $256,000.Assuming that the machine's resale value decreases exponentially, what will it be 2 year(s) from now?

Given that a quantity Q(t) exhibiting exponential decay is described by the function where t is measured in years.What is the decay constant? What quantity is present initially? Complete the table of values.Round your answer to the nearest integer.

Use the curve-sketching guideline, to select the graph of the function.

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 four times the "safe" level, find how much time must elapse before an "acceptable level" is reached. __________ years

Find the derivative of the function.

Find the interest rate needed for an investment of $5,000 to grow to an amount of $6,000 in 3 year(s) if interest is compounded continuously.Round your answer to the nearest hundredth. Interest rate is __________% per year.