IBM-Peru uses second derivatives to assess the relative success of various advertising

For the function g(x) graphed in Figure 2.39, are the following nonzero quantities positive or negative? (a) g(0) (b) g(0) 4 3 2 1 0 1 2 3 4 1 2 4 x Figure 2.39

At one of the labeled points on the graph in Figure 2.40 both dy/dx and d2y/dx2 are negative. Which is it? A B C D E y x Figure 2.40

Graph the functions described in parts (a)(d). (a) First and second derivatives everywhere positive. (b) Second derivative everywhere negative; first derivative everywhere positive. (c) Second derivative everywhere positive; first derivative everywhere negative. (d) First and second derivatives everywhere negative.

For Problems 49, give the signs of the first and second derivatives for the following functions. Each derivative is either positive everywhere, zero everywhere, or negative everywhere.

For Problems 49, give the signs of the first and second derivatives for the following functions. Each derivative is either positive everywhere, zero everywhere, or negative everywhere.

For Problems 49, give the signs of the first and second derivatives for the following functions. Each derivative is either positive everywhere, zero everywhere, or negative everywhere.

For Problems 49, give the signs of the first and second derivatives for the following functions. Each derivative is either positive everywhere, zero everywhere, or negative everywhere.

For Problems 49, give the signs of the first and second derivatives for the following functions. Each derivative is either positive everywhere, zero everywhere, or negative everywhere.

For Problems 49, give the signs of the first and second derivatives for the following functions. Each derivative is either positive everywhere, zero everywhere, or negative everywhere.

In Problems 1011, use the graph given for each function. (a) Estimate the intervals on which the derivative is positive and the intervals on which the derivative is negative. (b) Estimate the intervals on which the second derivative is positive and the intervals on which the second derivative is negative.

In Problems 1011, use the graph given for each function. (a) Estimate the intervals on which the derivative is positive and the intervals on which the derivative is negative. (b) Estimate the intervals on which the second derivative is positive and the intervals on which the second derivative is negative.

In Problems 1213, use the values given for each function. (a) Does the derivative of the function appear to be positive or negative over the given interval? Explain. (b) Does the second derivative of the function appear to be positive or negative over the given interval? Explain.

In Problems 1213, use the values given for each function. (a) Does the derivative of the function appear to be positive or negative over the given interval? Explain. (b) Does the second derivative of the function appear to be positive or negative over the given interval? Explain.

Sketch the graph of a function whose first derivative is everywhere negative and whose second derivative is positive for some x-values and negative for other x-values. 1

IBM-Peru uses second derivatives to assess the relative success of various advertising campaigns. They assume that all campaigns produce some increase in sales. If a graph of sales against time shows a positive second derivative during a new advertising campaign, what does this suggest to IBM management? Why? What does a negative second derivative suggest? 1

Values of f(t) are given in the following table. (a) Does this function appear to have a positive or negative first derivative? Second derivative? Explain. (b) Estimate f(2) and f(8). t 0 2 4 6 8 10 f(t) 150 145 137 122 98 56 1

The table gives the number of passenger cars, C = f(t), in millions,24 in the US in the year t. (a) Do f(t) and f(t) appear to be positive or negative during the period 19751990? (b) Do f(t) and f(t) appear to be positive or negative during the period 1990-2000? (c) Estimate f(2005). Using units, interpret your answer in terms of passenger cars. t 1975 1980 1985 1990 1995 2000 2005 C 106.7 121.6 127.9 133.7 128.4 133.6 136.6 1

Sketch a graph of a continuous function f with the following properties: f(x) > 0 for all x f(x) < 0 for x < 2 and f(x) > 0 for x > 2. 1

Sketch the graph of a function f such that f(2) = 5, f(2) = 1/2, and f(2) > 0. 2

At exactly two of the labeled points in Figure 2.41, the derivative f is 0; the second derivative f is not zero at any of the labeled points. On a copy of the table, give the signs of f, f , f at each marked point. B C A D Figure 2.41 Point f f f A B C D 2

For three minutes the temperature of a feverish person has had positive first derivative and negative second derivative. Which of the following is correct? (a) The temperature rose in the last minute more than it rose in the minute before. (b) The temperature rose in the last minute, but less than it rose in the minute before. (c) The temperature fell in the last minute but less than it fell in the minute before. (d) The temperature rose for two minutes but fell in the last minute. 2

Yesterdays temperature at t hours past midnight was f(t) C. At noon the temperature was 20C. The first derivative, f(t), decreased all morning, reaching a low of 2C/hour at noon, then increased for the rest of the day. Which one of the following must be correct? (a) The temperature fell in the morning and rose in the afternoon. (b) At 1 pm the temperature was 18C. (c) At 1 pm the temperature was 22C. (d) The temperature was lower at noon than at any other time. (e) The temperature rose all day. 2

A function f has f(5) = 20, f(5) = 2, and f(x) < 0, for x 5. Which of the following are possible values for f(7) and which are impossible? (a) 26 (b) 24 (c) 22 2

An industry is being charged by the Environmental Protection Agency (EPA) with dumping unacceptable levels of toxic pollutants in a lake. Over a period of several months, an engineering firm makes daily measurements of the rate at which pollutants are being discharged into the lake. The engineers produce a graph similar to either Figure 2.42(a) or Figure 2.42(b). For each case, give an idea of what argument the EPA might make in court against the industry and of the industrys defense. a year ago now time (a) rate of discharge a year ago now time (b) rate of discharge Figure 2.42 2

Winning the war on poverty has been described cynically as slowing the rate at which people are slipping below the poverty line. Assuming that this is happening: (a) Graph the total number of people in poverty against time. (b) If N is the number of people below the poverty line at time t, what are the signs of dN/dt and d2N/dt2? Explain. 2

In economics, total utility refers to the total satisfaction from consuming some commodity. According to the economist Samuelson:25 As you consume more of the same good, the total (psychological) utility increases. However, . ..with successive new units of the good, your total utility will grow at a slower and slower rate because of a fundamental tendency for your psychological ability to appreciate more of the good to become less keen. (a) Sketch the total utility as a function of the number of units consumed. (b) In terms of derivatives, what is Samuelson saying? 2

Let P(t) represent the price of a share of stock of a corporation at time t. What does each of the following statements tell us about the signs of the first and second derivatives of P(t)? (a) The price of the stock is rising faster and faster. (b) The price of the stock is close to bottoming out. 2

Each of the graphs in Figure 2.43 shows the position of a particle moving along the x-axis as a function of time, 0 t 5. The vertical scales of the graphs are the same. During this time interval, which particle has (a) Constant velocity? (b) The greatest initial velocity? (c) The greatest average velocity? (d) Zero average velocity? (e) Zero acceleration? (f) Positive acceleration throughout? 5 t (I) x 5 t (II) x 5 t (III) x 5 t (IV) x Figure 2.43 2

In 2009, a study was done on the impact of sea-level rise in the mid-Atlantic states.26 Let a(t) be the depth of the sea in millimeters (mm) at a typical point on the Atlantic Coast, and let m(t) be the depth of the sea in mm at a typical point on the Gulf of Mexico, with time t in years since data collection started. (a) The study reports Sea level is rising and there is evidence that the rate is accelerating. What does this statement tell us about a(t) and m(t)? (b) The study also reports The Atlantic Coast and the Gulf of Mexico experience higher rates of sea-level rise (2 to 4 mm per year and 2 to 10 mm per year, respectively) than the current global average (1.7 mm per year). What does this tell us about a(t) and m(t)? (c) Assume the rates at which the sea level rises on the Atlantic Coast and the Gulf of Mexico are constant for a century and within the ranges given in the report. (i) What is the largest amount the sea could rise on the Atlantic Coast during a century? Your answer should be a range of values. (ii) What is the shortest amount of time in which the sea level in the Gulf of Mexico could rise 1 meter? 3

Table 2.11 shows the number of Facebook subscribers, N in millions, worldwide at 3-month intervals.27 (a) Calculate the average rate of change of N per month for the time intervals shown between March 2011 and March 2012. (b) What can you say about the sign of d2N/dt2 during the period March 2011March 2012? Table 2.11 Month, t Mar 2011 Jun 2011 Sep 2011 Dec 2011 Mar 2012 N 664.0 710.7 756.9 799.1 835.5 3

In 1913, Carlson28 conducted the classic experiment in which he grew yeast, Saccharomyces cerevisiae, in laboratory cultures and collected data every hour for 18 hours. Table 2.12 shows the yeast population, P, at representative times t in hours. (a) Calculate the average rate of change of P per hour for the time intervals shown between 0 and 18 hours. (b) What can you say about the sign of d2P/dt2 during the period 018 hours?