The force, F, between two atoms depends on the distance r separating them. See Figure

A line with slope 2 and x-intercept 5.

A parabola opening downward with its vertex at (2, 5)

A parabola with x-intercepts 1 and y-intercept 3

The bottom half of a circle centered at the origin and with radius 2.

The top half of a circle with center (1, 2) and radius 3.

A cubic polynomial having x-intercepts at 1, 5, 7

A rational function of the form y = ax/(x + b) with a vertical asymptote at x = 2 and a horizontal asymptote of y = 5

A cosine curve with a maximum at (0, 5), a minimum at (,5), and no maxima or minima in between

When a patient with a rapid heart rate takes a drug, the heart rate plunges dramatically and then slowly rises again as the drug wears off. Sketch the heart rate against time from the moment the drug is administered.

If g(x) = (4 x2)/(x2 + x), find the domain of g(x). Solve g(x)=0.

The entire graph of f(x) is shown in Figure 1.98. (a) What is the domain of f(x)? (b) What is the range of f(x)? (c) List all zeros of f(x). (d) List all intervals on which f(x) is decreasing

For f(n)=3n22 and g(n) = n+1, find and simplify: (a) f(n) + g(n) (b) f(n)g(n) (c) The domain of f(n)/g(n) (d) f(g(n)) (e) g(f(n))

Let m = f(A) be the minimum annual gross income, in thousands of dollars, needed to obtain a 30-year home mortgage loan of A thousand dollars at an interest rate of 6%. What do the following quantities represent in terms of the income needed for a loan? (a) f(100) (b) f1(75)

For Exercises 1417, solve for t using logs

For Exercises 1417, solve for t using logs

For Exercises 1417, solve for t using logs

For Exercises 1417, solve for t using logs

In Exercises 1819, put the functions in the form P = P0ekt.

In Exercises 1819, put the functions in the form P = P0ekt.

For Exercises 2021, find functions f and g such that h(x) = f(g(x)). [Note: Do not choose f(x) = x or g(x) = x.]

For Exercises 2021, find functions f and g such that h(x) = f(g(x)). [Note: Do not choose f(x) = x or g(x) = x.]

Find the amplitudes and periods in Exercises 2223.

Find the amplitudes and periods in Exercises 2223.

Consider the function y = 5 + cos(3x). (a) What is its amplitude? (b) What is its period? (c) Sketch its graph

Determine the end behavior of each function as x + and as x . (a) f(x) = x7 (b) f(x)=3x+7x312x4 (c) f(x) = x4 (d) f(x) = 6x3 5x2 + 2 x3 8

In Exercises 2627, which function dominates as x ?

In Exercises 2627, which function dominates as x ?

Find possible formulas for the graphs in Exercises 2841.

Find possible formulas for the graphs in Exercises 2841.

Find possible formulas for the graphs in Exercises 2841.

Find possible formulas for the graphs in Exercises 2841.

Find possible formulas for the graphs in Exercises 2841.

Find possible formulas for the graphs in Exercises 2841.

Find possible formulas for the graphs in Exercises 2841.

Find possible formulas for the graphs in Exercises 2841.

Find possible formulas for the graphs in Exercises 2841.

Find possible formulas for the graphs in Exercises 2841.

Find possible formulas for the graphs in Exercises 2841.

Find possible formulas for the graphs in Exercises 2841.

Find possible formulas for the graphs in Exercises 2841.

Find possible formulas for the graphs in Exercises 2841.

Are the functions in Exercises 4243 continuous on [1, 1]?

Are the functions in Exercises 4243 continuous on [1, 1]?

Use Figure 1.99 to estimate the limits if they exist: (a) lim x0 f(x) (b) lim x1 f(x) (c) lim x2 f(x) (d) lim x3 f(x

For the functions in Exercises 4546, use algebra to evaluate the limits lim xa+ f(x), lim xa f(x), and lim xa

For the functions in Exercises 4546, use algebra to evaluate the limits lim xa+ f(x), lim xa f(x), and lim xa

The yield, Y , of an apple orchard (in bushels) as a function of the amount, a, of fertilizer (in pounds) used on the orchard is shown in Figure 1.100. (a) Describe the effect of the amount of fertilizer on the yield of the orchard. (b) What is the vertical intercept? Explain what it means in terms of apples and fertilizer. (c) What is the horizontal intercept? Explain what it means in terms of apples and fertilizer. (d) What is the range of this function for 0 a 80? (e) Is the function increasing or decreasing at a = 60? (f) Is the graph concave up or down near a = 40?

The graph of Fahrenheit temperature, F, as a function of Celsius temperature, C, is a line. You know that 212F and 100C both represent the temperature at which water boils. Similarly, 32F and 0C both represent waters freezing point. (a) What is the slope of the graph? (b) What is the equation of the line? (c) Use the equation to find what Fahrenheit temperature corresponds to 20C. (d) What temperature is the same number of degrees in both Celsius and Fahrenheit?

The demand function for a certain product, q = D(p), is linear, where p is the price per item in dollars and q is the quantity demanded. If p increases by $5, market research shows that q drops by two items. In addition, 100 items are purchased if the price is $550. (a) Find a formula for (i) q as a linear function of p (ii) p as a linear function of q (b) Draw a graph with q on the horizontal axis.

A flight from Dulles Airport in Washington, DC, to LaGuardia Airport in New York City has to circle LaGuardia several times before being allowed to land. Plot a graph of the distance of the plane from Washington, DC, against time, from the moment of takeoff until landing.

The force, F, between two atoms depends on the distance r separating them. See Figure 1.101. A positive F represents a repulsive force; a negative F represents an attractive force. (a) What happens to the force if the atoms start with r = a and are (i) Pulled slightly further apart? (ii) Pushed slightly closer together? (b) The atoms are said to be in stable equilibrium if the force between them is zero and the atoms tend to return to the equilibrium after a minor disturbance. Does r = a represent a stable equilibrium? Explain.

When the Olympic Games were held outside Mexico City in 1968, there was much discussion about the effect the high altitude (7340 feet) would have on the athletes. Assuming air pressure decays exponentially by 0.4% every 100 feet, by what percentage is air pressure reduced by moving from sea level to Mexico City

The population of the Ukraine fell from 45.7 million in 2009 to 45.42 million in 2010.40 Assuming exponential decline, in what year is the population predicted to be 45 million?

During April 2006, Zimbabwes inflation rate averaged 0.67% a day. This means that, on average, prices went up by 0.67% from one day to the next. (a) By what percentage did prices in Zimbabwe increase in April of 2006? (b) Assuming the same rate all year, what was Zimbabwes annual inflation rate during 2006?

Hydroelectric power is electric power generated by the force of moving water. The table shows the annual percent change in hydroelectric power consumption by the US industrial sector.41 (a) According to the table, during what single year(s), if any, did the US consumption of hydroelectric power energy increase by at least 10%? Decrease by 10% or more? (b) True or False: The hydroelectric power consumption nearly doubled from 2008 to 2009. (c) True or False: The hydroelectric power consumption decreased by about 36% from 2006 to 2009

Hydroelectric power is electric power generated by the force of moving water. The table shows the annual percent change in hydroelectric power consumption by the US industrial sector.41 (a) According to the table, during what single year(s), if any, did the US consumption of hydroelectric power energy increase by at least 10%? Decrease by 10% or more? (b) True or False: The hydroelectric power consumption nearly doubled from 2008 to 2009. (c) True or False: The hydroelectric power consumption decreased by about 36% from 2006 to 2009

The graph of f(x) is a parabola that opens upward and the graph of g(x) is a line with negative slope. Describe the graph of g(f(x)) in words

Each of the functions in the table is increasing over its domain, but each increases in a different way. Match the functions f, g, h to the graphs in Figure 1.102

A culture of 100 bacteria doubles after 2 hours. How long will it take for the number of bacteria to reach 3,200?

If f(x) = a ln(x + 2), how does increasing a affect (a) The y-intercept? (b) The x-intercept?

What is the doubling time of prices which are increasing by 5% a year?

Find the half-life of a radioactive substance that is reduced by 30% in 20 hours.

The air in a factory is being filtered so that the quantity of a pollutant, P (in mg/liter), is decreasing according to the function P = P0ekt, where t is time in hours. If 10% of the pollution is removed in the first five hours: (a) What percentage of the pollution is left after 10 hours? (b) How long is it before the pollution is reduced by 50%? (c) Plot a graph of pollution against time. Show the results of your calculations on the graph. (d) Explain why the quantity of pollutant might decrease in this way.

The half-life of radioactive strontium-90 is 29 years. In 1960, radioactive strontium-90 was released into the atmosphere during testing of nuclear weapons, and was absorbed into peoples bones. How many years does it take until only 10% of the original amount absorbed remains?

What is the period of the motion of the minute hand of a clock?

In an electrical outlet, the voltage, V , in volts, is given as a function of time, t, in seconds, by the formula V = V0 sin(120t). (a) What does V0 represent in terms of voltage? (b) What is the period of this function? (c) How many oscillations are completed in 1 second?

In a US household, the voltage in volts in an electric outlet is given by V = 156 sin(120t), where t is in seconds. However, in a European house, the voltage is given (in the same units) by V = 339 sin(100t). Compare the voltages in the two regions, considering the maximum voltage and number of cycles (oscillations) per second.

(a) How does the parameter A affect the graph of y = A sin(Bx)? (Plot for A = 1, 2, 3 with B = 1.) (b) How does the parameter B affect the graph of y = A sin(Bx)? (Plot for B = 1, 2, 3 with A = 1.)

Water is flowing down a cylindrical pipe of radius r. (a) Write a formula for the volume, V , of water that emerges from the end of the pipe in one second if the water is flowing at a rate of (i) 3 cm/sec (ii) k cm/sec (b) Graph your answer to part (a)(ii) as a function of (i) r, assuming k is constant (ii) k, assuming r is constant

Values of three functions are given in Table 1.22, rounded to two decimal places. Two are power functions and one is an exponential. One of the power functions is a quadratic and one a cubic. Which one is exponential? Which one is quadratic? Which one is cubic?

Figure 1.103 shows the hat function hN (x) = 0 if x

The point P moves around the circle of radius 5 shown in Figure 1.104. The angle , in radians, is given as a function of time, t, by the graph in Figure 1.105. (a) Estimate the coordinates of P when t = 1.5. (b) Describe in words the motion of the point P on the circle.

Match the following functions with the graphs in Figure 1.106. Assume 0

Use a computer or calculator to sketch the functions y(x) = sin x and zk(x) = kex for k = 1, 2, 4, 6, 8, 10. In each case find the smallest positive solution of the equation y(x) = zk(x). Now define a new function f by f(k) = {Smallest positive solution of y(x) = zk(x)}. Explain why the function f(k) is not continuous on the interval 0 k 10.

For each value of  in Problems 7576, find a positive value of such that the graph of the function leaves the window a

For each value of  in Problems 7576, find a positive value of such that the graph of the function leaves the window a

If possible, choose k so that the following function is continuous on any interval: f(x) = 5x3 10x2 x 2 x = 2 k x = 2

Find k so that the following function is continuous on any interval: j(x) = k cos x x 0 ex k x> 0

(a) Factor f(x) = x4 + bx3 cx3 a2x2 bcx2 a2bx + a2cx + a2bc using a computer algebra system. (b) Assuming a, b, c are constants with 0

(a) Using a computer algebra system, factor f(x) = x5 + 11x4 46x3 + 90x2 81x + 27. (b) Use your answer to part (a) to make a hand sketch of the graph of f. Explain how you know its shape.

Let f(x) = e6x+e5x2e4x10e3x8e2x+16ex+16. (a) What happens to the value of f(x) as x ? As x ? Explain your answer. (b) Using a computer algebra system, factor f(x) and predict the number of zeros of the function f(x). (c) What are the exact values of the zeros? What is the relationship between successive zeros?

Let f(x) = x2 x. (a) Find the polynomials f(f(x)) and f(f(f(x))) in expanded form. (b) What do you expect to be the degree of the polynomial f(f(f(f(f(f(x))))))? Explain.

(a) Use a computer algebra system to rewrite the rational function f(x) = x3 30 x 3 in the form f(x) = p(x) + r(x) q(x) , where p(x), q(x), r(x) are polynomials and the degree of r(x) is less than the degree of q(x). (b) What is the vertical asymptote of f? Use your answer to part (a) to write the formula for a function whose graph looks like the graph of f for x near the vertical asymptote. (c) Use your answer to part (a) to write the formula for a function whose graph looks like the graph of f for x and x . (d) Using graphs, confirm the asymptote you found in part (b) and the formula you found in part (c).

Use the trigonometric capabilities of your computer algebra system to express sin(5x) as a polynomial in sin x.

Use the trigonometric capabilities of your computer algebra system to express cos(4x) as a polynomial in (a) sin x (b) cos x.