Solved: An object is put outside on a cold day at time t = 0. Its temperature, H = f(t)

The population of a city, P, in millions, is a function of t, the number of years since 1970, so P = f(t). Explain the meaning of the statement f(35) = 12 in terms of the population of this city.

The pollutant PCB (polychlorinated biphenyl) affects the thickness of pelican eggs. Thinking of the thickness, T , of the eggs, in mm, as a function of the concentration, P, of PCBs in ppm (parts per million), we have T = f(P). Explain the meaning of f(200) in terms of thickness of pelican eggs and concentration of PCBs

Describe what Figure 1.8 tells you about an assembly line whose productivity is represented as a function of the number of workers on the line

For Exercises 47, find an equation for the line that passes through the given points.

For Exercises 47, find an equation for the line that passes through the given points.

For Exercises 47, find an equation for the line that passes through the given points.

For Exercises 47, find an equation for the line that passes through the given points.

For Exercises 811, determine the slope and the y-intercept of the line whose equation is given.

For Exercises 811, determine the slope and the y-intercept of the line whose equation is given.

For Exercises 811, determine the slope and the y-intercept of the line whose equation is given.

For Exercises 811, determine the slope and the y-intercept of the line whose equation is given.

Match the graphs in Figure 1.9 with the following equations. (Note that the x and y scales may be unequal.)

Match the graphs in Figure 1.10 with the following equations. (Note that the x and y scales may be unequal.) (a) y = 2.72x (b) y = 0.01 + 0.001x (c) y = 27.9 0.1x (d) y = 0.1x 2

Estimate the slope and the equation of the line in Figure 1.11.

Find an equation for the line with slope m through the point (a, c).

Find a linear function that generates the values in Table 1.3.

Find an equation for the line through the point (2, 1) which is perpendicular to the line y = 5x 3

Find equations for the lines through the point (1, 5) that are parallel to and perpendicular to the line with equation y + 4x = 7.

Find equations for the lines through the point (a, b) that are parallel and perpendicular to the line y = mx + c, assuming m = 0.

For Exercises 2023, give the approximate domain and range of each function. Assume the entire graph is shown.

For Exercises 2023, give the approximate domain and range of each function. Assume the entire graph is shown.

For Exercises 2023, give the approximate domain and range of each function. Assume the entire graph is shown.

For Exercises 2023, give the approximate domain and range of each function. Assume the entire graph is shown.

Find domain and range in Exercises 2425.

Find domain and range in Exercises 2425.

If f(t) = t2 16, find all values of t for which f(t) is a real number. Solve f(t)=3.

The volume of a sphere is proportional to the cube of its radius, r

The average velocity, v, for a trip over a fixed distance, d, is inversely proportional to the time of travel, t.

The strength, S, of a beam is proportional to the square of its thickness, h.

The energy, E, expended by a swimming dolphin is proportional to the cube of the speed, v, of the dolphin.

The number of animal species, N, of a certain body length, l, is inversely proportional to the square of l.

In 1983 the average annual sea level in Aberdeen was 7.019 meters.

The average annual sea level in Aberdeen in 2008.

The average annual sea level in Aberdeen was the same in 1865 and 1911

The average annual sea level in Aberdeen increased by 1 millimeter from 2007 to 2008.

In December 2010, the snowfall in Minneapolis was unusually high,2 leading to the collapse of the roof of the Metrodome. Figure 1.12 gives the snowfall, S, in Minneapolis for December 615, 2010. (a) How do you know that the snowfall data represents a function of date? (b) Estimate the snowfall on December 12. (c) On which day was the snowfall more than 10 inches? (d) During which consecutive two-day interval was the increase in snowfall largest?

The value of a car, V = f(a), in thousands of dollars, is a function of the age of the car, a, in years. (a) Interpret the statement f(5) = 6 (b) Sketch a possible graph of V against a. Is f an increasing or decreasing function? Explain. (c) Explain the significance of the horizontal and vertical intercepts in terms of the value of the car.

Which graph in Figure 1.13 best matches each of the following stories?3 Write a story for the remaining graph. (a) I had just left home when I realized I had forgotten my books, and so I went back to pick them up. (b) Things went fine until I had a flat tire. (c) I started out calmly but sped up when I realized I was going to be late.

An object is put outside on a cold day at time t = 0. Its temperature, H = f(t), in C, is graphed in Figure 1.14. (a) What does the statement f(30) = 10 mean in terms of temperature? Include units for 30 and for 10 in your answer. (b) Explain what the vertical intercept, a, and the horizontal intercept, b, represent in terms of temperature of the object and time outside

A rock is dropped from a window and falls to the ground below. The height, s (in meters), of the rock above ground is a function of the time, t (in seconds), since the rock was dropped, so s = f(t). (a) Sketch a possible graph of s as a function of t. (b) Explain what the statement f(7) = 12 tells us about the rocks fall. (c) The graph drawn as the answer for part (a) should have a horizontal and vertical intercept. Interpret each intercept in terms of the rocks fall.

In a California town, the monthly charge for waste collection is $8 for 32 gallons of waste and $12.32 for 68 gallons of waste. (a) Find a linear formula for the cost, C, of waste collection as a function of the number of gallons of waste, w. (b) What is the slope of the line found in part (a)? Give units and interpret your answer in terms of the cost of waste collection. (c) What is the vertical intercept of the line found in part (a)? Give units and interpret your answer in terms of the cost of waste collection.

In a California town, the monthly charge for waste collection is $8 for 32 gallons of waste and $12.32 for 68 gallons of waste. (a) Find a linear formula for the cost, C, of waste collection as a function of the number of gallons of waste, w. (b) What is the slope of the line found in part (a)? Give units and interpret your answer in terms of the cost of waste collection. (c) What is the vertical intercept of the line found in part (a)? Give units and interpret your answer in terms of the cost of waste collection.

A company rents cars at $40 a day and 15 cents a mile. Its competitors cars are $50 a day and 10 cents a mile. (a) For each company, give a formula for the cost of renting a car for a day as a function of the distance traveled. (b) On the same axes, graph both functions. (c) How should you decide which company is cheaper?

Residents of the town of Maple Grove who are connected to the municipal water supply are billed a fixed amount monthly plus a charge for each cubic foot of water used. A household using 1000 cubic feet was billed $40, while one using 1600 cubic feet was billed $55. (a) What is the charge per cubic foot? (b) Write an equation for the total cost of a residents water as a function of cubic feet of water used. (c) How many cubic feet of water used would lead to a bill of $100?

Driving speed against time on the highway

Distance driven against time on the highway

Distance from my exit vs time on the highway

Distance between cars vs distance driven on the highway

Let f(t) be the number of US billionaires in the US in year t. (a) Express the following statements4 in terms of f. (i) In 1985 there were 13 US billionaires. (ii) In 1990 there were 99 US billionaires. (b) Find the average yearly increase in the number of US billionaires between 1985 and 1990. Express this using f. (c) Assuming the yearly increase remains constant, find a formula predicting the number of US billionaires in year t.

An alternative to petroleum-based diesel fuel, biodiesel, is derived from renewable resources such as food crops, algae, and animal oils. The table shows the recent annual percent growth in US biodiesel consumption.5 Year 2005 2006 2007 2008 2009 % growth over previous yr 237 186.6 37.2 11.7 7.3 (a) Find the largest time interval over which the percentage growth in the US consumption of biodiesel was an increasing function of time. Interpret what increasing means, practically speaking, in this case. (b) Find the largest time interval over which the actual US consumption of biodiesel was an increasing function of time. Interpret what increasing means, practically speaking, in this case

Hydroelectric power is electric power generated by the force of moving water. Figure 1.15 shows6 the annual percent growth in hydroelectric power consumption by the US industrial sector between 2004 and 2009. (a) Find the largest time interval over which the percentage growth in the US consumption of hydroelectric power was a decreasing function of time. Interpret what decreasing means, practically speaking, in this case. (b) Find the largest time interval over which the actual US consumption of hydroelectric power was a decreasing function of time. Interpret what decreasing means, practically speaking, in this case.

Solar panels are arrays of photovoltaic cells that convert solar radiation into electricity. The table shows the annual percent change in the US price per watt of a solar panel.

Table 1.4 shows the average annual sea level, S, in meters, in Aberdeen, Scotland,8 as a function of time, t, measured in years before 2008. Table 1.4 t 0 25 50 75 100 125 S 7.094 7.019 6.992 6.965 6.938 6.957 (a) What was the average sea level in Aberdeen in 2008? (b) In what year was the average sea level 7.019 meters? 6.957 meters? (c) Table 1.5 gives the average sea level, S, in Aberdeen as a function of the year, x. Complete the missing values.

A controversial 1992 Danish study9 reported that mens average sperm count has decreased from 113 million per milliliter in 1940 to 66 million per milliliter in 1990. (a) Express the average sperm count, S, as a linear function of the number of years, t, since 1940. (b) A mans fertility is affected if his sperm count drops below about 20 million per milliliter. If the linear model found in part (a) is accurate, in what year will the average male sperm count fall below this level?

The table gives the average weight, w, in pounds, of American men in their sixties for height, h, in inches.10 (a) How do you know that the data in this table could represent a linear function? (b) Find weight, w, as a linear function of height, h. What is the slope of the line? What are the units for the slope?

An airplane uses a fixed amount of fuel for takeoff, a (different) fixed amount for landing, and a third fixed amount per mile when it is in the air. How does the total quantity of fuel required depend on the length of the trip? Write a formula for the function involved. Explain the meaning of the constants in your formula.

The cost of planting seed is usually a function of the number of acres sown. The cost of the equipment is a fixed cost because it must be paid regardless of the number of acres planted. The costs of supplies and labor vary with the number of acres planted and are called variable costs. Suppose the fixed costs are $10,000 and the variable costs are $200 per acre. Let C be the total cost, measured in thousands of dollars, and let x be the number of acres planted. (a) Find a formula for C as a function of x. (b) Graph C against x. (c) Which feature of the graph represents the fixed costs? Which represents the variable costs?

You drive at a constant speed from Chicago to Detroit, a distance of 275 miles. About 120 miles from Chicago you pass through Kalamazoo, Michigan. Sketch a graph of your distance from Kalamazoo as a function of time.

(a) Consider the functions graphed in Figure 1.16(a). Find the coordinates of C.

When Galileo was formulating the laws of motion, he considered the motion of a body starting from rest and falling under gravity. He originally thought that the velocity of such a falling body was proportional to the distance it had fallen. What do the experimental data in Table 1.6 tell you about Galileos hypothesis? What alternative hypothesis is suggested by the two sets of data in Table 1.6 and Table 1.7?

Values of y on the graph of y = 0.5x 3 increase more slowly than values of y on the graph of y = 0.5 3x

The equation y = 2x + 1 indicates that y is directly proportional to x with a constant of proportionality 2.

A linear function with a positive slope and a negative xintercept

A formula representing the statement q is inversely proportional to the cube root of p and has a positive constant of proportionality

For any two points in the plane, there is a linear function whose graph passes through them

If y = f(x) is a linear function, then increasing x by 1 unit changes the corresponding y by m units, where m is the slope.

If y is a linear function of x, then the ratio y/x is constant for all points on the graph at which x = 0.

If y = f(x) is a linear function, then increasing x by 2 units adds m + 2 units to the corresponding y, where m is the slope.

Which of the following functions has its domain identical with its range?