Exam 1: Functions, graphs, and Limits

After t years,the remaining mass y(in grams)of 16 grams of a radioactive element whose half-life is 34 years is given by ,for How much of the initial mass remains after 136 years? Round your answer to two decimal places. ​

​Future value.The future value that accrues when $700 is invested at 9%,compounded continuously,is ,where t is the number of years.At what rate is the money in this account growing when ?

​Find the limit (if it exists): ​

Find the x-value at which the absolute minimum of f(x)occurs on the interval [a,b]. ​ ​ ​

​Use the properties of logarithms to expand ​

​Find the absolute extrema of the function on the interval ​

​If the total cost function for a product is dollars,determine how many units x should be produced to minimize the average cost per unit? ​

​Find the equation of the tangent line to at the point ​

​You are in a boat 2 miles from the nearest point on the coast.You are to go to point Q located 3 miles down the coast and 1 mile inland (see figure).You can row at a rate of 1 miles per hour and you can walk at a rate of 2 miles per hour.Toward what point on the coast should you row in order to reach point Q in the least time? ​ ​

Assume the population P (in millions)of the United States from 1992 through 2005 can be modeled by the exponential function ,where t is the time in years,with corresponding to 1992.Use the model to estimate the population in the year 2009.Round your answer to the nearest million.

​A retail sporting goods store estimates that weekly sales and weekly advertising costs are related by the equation The current weekly advertising costs are $1,500,and these costs are increasing at a rate of $135 per week.Find the current rate of change of weekly sales. ​

​Find the derivative of the function ​

The graph shows the number of visitors V to a national park in hundreds of thousands during a one-year period,where represents January.Estimate the rate of change of V over the interval Round your answer to the nearest hundred thousand visitors per year. ​ ​

​Find the slope of the line passing through the given pair of points. ​ ​ ​

​Identify a function f(x)that has the given characteristics and then sketch the function. ​ ​; ; ​

Differentiate the given function. ​ ​

​Complete the table and use the result to estimate the limit.Round your answer to six decimal places. ​ ​ ​

​Sketch the graph of the function given below.Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. ​

​Find if

​Analyze and sketch a graph of the function ​