Exam 1: Functions, graphs, and Limits

Suppose the number y of medical degrees conferred in the United States can be modeled by ,for ,where t is the time in years,with corresponding to 1972.Use the test for increasing and decreasing functions to estimate the years during which the number of medical degrees is increasing and the years during which it is decreasing. ​

​The profit P (in thousands of dollars)for a company spending an amount s (in thousands of dollars)on advertising is The point of diminishing returns is the point at which the rate of growth of the profit function begins to decline.Find the point of diminishing returns.Round your answer to the nearest thousand dollars. ​

​A firm has total revenue given by for x units of a product.Find the maximum revenue from sales of that product. ​

What is the annual percentage yield (or effective annual rate)for a nominal rate of 8.7% compounded quarterly? Round your answer to two decimal places.

​If ,is y a function of x? ​

Identify the open intervals where the function is increasing or decreasing. ​

​Find the point(s),if any,at which the graph of f has a horizontal tangent line. ​ ​

A point is moving along the graph of the function such that centimeters per second. Find when ​

​Find the derivative of ​

​Find the limit. ​ ​ ​

​Find an equation of the tangent line to the graph of f at the given point. ​ ,at ​

​Production.Suppose that the total number of units produced by a worker in t hours of an 8-hour shift can be modeled by the production function Find the number of hours before the rate of production is maximized.That is,find the point of diminishing returns. ​

​Find the derivative of the following function. ​ ​

​Assume that x and y are differentiable functions of t.Given ,find when and ​

​This problem contains a function and its graph,where Use the graph to determine,as well as you can,the vertical asymptote.Check your conclusion by using the function to determine the vertical asymptote analytically. ​ ​ ​ ​ ​

Determine the range of the function

​Find the derivative of the following function. ​ ​

​Find a linear equation that expresses the relationship between the temperature in degrees Celsius and degrees Fahrenheit.Use the fact that water freezes at 0° C (32° F)and boils at 100° C (212° F ).Use the equation to convert 82° F to Celsius.Round to the nearest degree. ​

​Locate the absolute extrema of the function on the closed interval ​

​Sketch the graph of the function ​