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mathematics
applied calculus

Questions and Answers of Applied Calculus

A quantity P is an exponential function of time t. Use the given information about the function P = P0ekt to:(a) Find values for the parameters k and P0.(b) State the initial quantity and the
The number of sales per month, S, is a function of the amount, a (in dollars), spent on advertising that month, so S = f(a). (a) Interpret the statement f(1000) = 3500. (b) Which of the
Use Figure 1.92 to estimate the function value or explain why it cannot be done. v(u(40)) 100 50 u(x) x 100 v(x) Figure 1.92 50 x
The population of the US was 281.4 million in 2000 and 316.1 million in 2013.103 Assuming exponential growth, (a) In what year is the population expected to go over 350 million? (b) What
In a Washington town, the charge for commercial waste collection is $694.55 for 5 tons and $1098.32 for 8 tons of waste. (a) Find a linear formula for the cost, C, of waste collection as a
World production, Q, of zinc in thousands of metric tons and the value, P, in dollars per metric ton are given in Table 1.31. Plot the value as a function of production. Sketch a possible supply
The number of US households with cable television was 12,168,450 in 1977 and 73,365,880 in 2003. Estimate the average rate of change in the number of US households with cable television during this
Graph the given function on the axes in Figure 1.119.y = k(sin x) − k + -2π 2π -k k -k y -2π+ + k Figure 1.119 + x 2π
Put the functions in the form P = P0ekt.P = 4(0.55)t
Financial investors know that, in general, the higher the expected rate of return on an investment, the higher the corresponding risk.(a) Graph this relationship, showing expected return as a
Use Figure 1.92 to estimate the function value or explain why it cannot be done. v(u(10)) 100 50 u(x) x 100 v(x) Figure 1.92 50 x
In 1923, koalas were introduced on Kangaroo Island off the coast of Australia. In 1996, the population was 5000. By 2005, the population had grown to 27,000, prompting a debate on how to control
Search and rescue teams work to find lost hikers. Members of the search team separate and walk parallel to one another through the area to be searched. Table 1.7 shows the percent, P, of lost
The demand curve for a product is given by q = 120,000 − 500p and the supply curve is given by q = 1000p for 0 ≤ q ≤ 120,000, where price is in dollars.(a) At a price of $100, what quantity are
Find a formula for the number of zebra mussels in a bay as a function of the number of years since 2014, given that there were 2700 at the start of 2014 and 3186 at the start of 2015.(a) Assume that
Graph the given function on the axes in Figure 1.119.y = k(cos x) + k + -2π 2π -k k -k y -2π+ + k Figure 1.119 + x 2π
Table 1.20 gives the revenues, R, of General Motors, formerly the world’s largest auto manufacturer.(a) Find the change in revenues between 2003 and 2015. (b) Find the average rate of change
Put the functions in the form P = P0ekt.P = 174(0.9)t
A tree of height y meters has, on average, B branches, where B = y − 1. Each branch has, on average, n leaves, where n = 2B2 − B. Find the average number of leaves on a tree as a function of
After an injection, the concentration of a drug in a patient’s body increases rapidly to a peak and then slowly decreases. Graph the concentration of the drug in the body as a function of the time
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 people’s bones.
A $25,000 vehicle depreciates $2000 a year as it ages. Repair costs are $1500 per year.(a) Write formulas for each of the two linear functions at time t, value, V (t), and repair costs to date, C(t).
Table 1.30 gives data for the linear demand curve for a product, where p is the price of the product and q is the quantity sold every month at that price. Find formulas for the following functions.
The company that produces Cliffs Notes (abridged versions of classic literature) was started in 1958 with $4000 and sold in 1998 for $14,000,000. Find the annual percent increase in the value of this
Graph the given function on the axes in Figure 1.119.y = −k cos x + -2π 2π -k k -k y -2π+ + k Figure 1.119 + x 2π
The total world marine catch48 of fish, in metric tons, was 17 million in 1950 and 99 million in 2001. What was the average rate of change in the marine catch during this period? Give units and
Using Table 1.38, create a table of values for f(g(x)) and for g(f(x)). Table 1.38 X -3 -2 f(x) 0 1 g(x) 3 2 لنا -1 0 12 2 3 21 20-2 3 0 -2 -2 -3
Put the functions in the form P = P0ekt.P = 10(1.7)t
The gas mileage of a car (in miles per gallon) is highest when the car is going about 45 miles per hour and is lower when the car is going faster or slower than 45 mph. Graph gas mileage as a
Pregnant women metabolize some drugs at a slower rate than the rest of the population. The half-life of caffeine is about 4 hours for most people. In pregnant women, it is 10 hours.101 (This is
Annual sales of music compact discs (CDs) have declined since 2000. Sales were 942.5 million in 2000 and 384.7 million in 2008.(a) Find a formula for annual sales, S, in millions of music CDs, as a
When the price, p, charged for a boat tour was $25, the average number of passengers per week, N, was 500. When the price was reduced to $20, the average number of passengers per week increased to
The number of passengers using a railway fell from 190,205 to 174,989 during a 5-year period. Find the annual percentage decrease over this period.
A health club has cost and revenue functions given by C = 10,000 + 35q and R = pq, where q is the number of annual club members and p is the price of a one year membership. The demand function for
Graph the given function on the axes in Figure 1.119.y = k sin x + -2π 2π -k k -k y -2π+ + k Figure 1.119 + x 2π
Table 1.19 shows the total US labor force, L. Find the average rate of change between 1940 and 2000; between 1940 and 1960; between 1980 and 2000. Give units and interpret your answers in terms of
Use the graphs in Figure 1.91. Estimate f(g(3)). -3 3 دیا -3 T 3 نرا f(x) x g(x) -3 Figure 1.91 3 نیا -3 3 X
Put the functions in the form P = P0ekt.P = 15(1.5)t
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
If the quantity of a substance decreases by 4% in 10 hours, find its half-life.
World grain production was 1241 million tons in 1975 and 2048 million tons in 2005, and has been increasing at an approximately constant rate.(a) Find a linear function for world grain production, P,
A company produces and sells shirts. The fixed costs are $7000 and the variable costs are $5 per shirt.(a) Shirts are sold for $12 each. Find cost and revenue as functions of the quantity of shirts,
If the world’s population increased exponentially from 5.937 billion in 1998 to 7.238 billion in 2014 and continued to increase at the same percentage rate in 2014 and 2015, calculate what the
A sporting goods wholesaler finds that when the price of a product is $25, the company sells 500 units per week. When the price is $30, the number sold per week decreases to 460 units.(a) Find the
In tide pools on the New England coast, snails eat algae. Describe what Figure 1.10 tells you about the effect of snails on the diversity of algae. Does the graph support the statement that diversity
Find f(5). 6 4 2 f(x) f 2 4 6 8 10
Interpret the expression in terms of Arctic Sea ice extent, the area of sea covered by ice. Let E(x) and F(t) be the Arctic Sea ice extent, both in millions of square kilometers, as a function of
Write a formula representing the function.The energy, E, expended by a swimming dolphin is proportional to the cube of the speed, v, of the dolphin.
World population69 is approximately P = 7.32(1.0109)t, with P in billions and t in years since 2015. (a) What is the yearly percent rate of growth of the world population? (b) What was the
Figure 1.24 shows the distance from home, in miles, of a person on a 5-hour trip. (a) Estimate the vertical intercept. Give units and interpret it in terms of distance from home. (b)
A 50 mg dose of quinine is given to a patient to prevent malaria. Quinine leaves the body at a rate of 6% per hour.(a) Find a formula for the amount, A (in mg), of quinine in the body t hours after
A company that makes Adirondack chairs has fixed costs of $5000 and variable costs of $30 per chair. The company sells the chairs for $50 each.(a) Find formulas for the cost and revenue functions.(b)
Interpret the expression in terms of Arctic Sea ice extent, the area of sea covered by ice. Let E(x) and F(t) be the Arctic Sea ice extent, both in millions of square kilometers, as a function of
Write a formula representing the function.The average velocity, v, for a trip over a fixed distance, d, is inversely proportional to the time of travel, t.
Solve for t using natural logarithms.7 ⋅ 3t = 5 ⋅ 2t
Find f(5). 1 f(x) 2.3 X 2 3 2.8 3.2 4 5 6 3.7 4.1 5.0 7 5.6 5.6 8 6.2
From October 2002 to October 2006 the number N(t) of Wikipedia articles was approximated by N(t) = N0et∕500, where t is the number of days after October 1, 2002. Find the doubling time for the
Use the variable u for the inside function to express each of the following as a composite function: (a) y = 23x−1 (b) P = √ 5t2 + 10 (c) w = 2 ln(3r + 4)
Delta Cephei is one of the most visible stars in the night sky. Its brightness has periods of 5.4 days, the average brightness is 4.0 and its brightness varies by ± 0.35. Find a formula that models
The surface area of a mammal, S, satisfies the equation S = KM2∕3, where M is the body mass and the constant of proportionality k depends on the body shape of the mammal. A human of body mass 70
Solve for t using natural logarithms.2et − 5 = 0
In 2014, the world’s population reached 7.17 billion and was increasing at a rate of 1.1% per year. Assume that this growth rate remains constant. (In fact, the growth rate has decreased since
The function S = f(t) gives the average annual sea level, S, in meters, in Aberdeen, Scotland,7 as a function of t, the number of years before 2012. Write a mathematical expression that represents
Find a possible formula for the graph. 90 50 10 X I t
Simplify the quantities in Problem using m(z) = z2.m(z) − m(z − ℎ)
(a) Make a table of values for y = ex using x = 0, 1, 2, 3. (b) Plot the points found in part (a). Does the graph look like an exponential growth or decay function? (c) Make a table of
Use Figure 1.25 showing how the quantity, Q, of grass (kg/hectare) in different parts of Namibia depended on the average annual rainfall, r, (mm), in two different years.Figure 1.25(a) For 1939, find
A movie theater has fixed costs of $5000 per day and variable costs averaging $6 per customer. The theater charges $11 per ticket.(a) How many customers per day does the theater need in order to make
An observer tracks the distance a plane has rolled along the runway after touching down. Figure 1.39 shows this distance, x, in thousands of feet, as a function of time, t, in seconds since
Solve for t using natural logarithms.0 = 7 − 3et
The number of species of lizards, N, found on an island off Baja California is proportional to the fourth root of the area, A, of the island. Write a formula for N as a function of A. Graph this
A cup of coffee contains 100 mg of caffeine, which leaves the body at a continuous rate of 17% per hour. (a) Write a formula for the amount, A mg, of caffeine in the body t hours after drinking
The function S = f(t) gives the average annual sea level, S, in meters, in Aberdeen, Scotland,7 as a function of t, the number of years before 2012. Write a mathematical expression that represents
Figure 1.41 shows a particle’s distance from a point as a function of time, t. What is the particle’s average velocity from t = 0 to t = 3?Figure 1.41 distance (meters) 5 3 1 2 s(t) 4 t (sec)
In the Andes mountains in Peru, the number, N, of species of bats is a function of the elevation, ℎ, in feet above sea level, so N = f(ℎ). (a) Interpret the statement f(500) = 100 in terms
Represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous.P = 15e−0.06t
Use the graphs in Figure 1.90.Estimate g(f(4)). 2 2 -f(x)- 4 6 X 4 2 Figure 1.90 2 -g(x)- 4 6 + x
Biologists estimate that the number of animal species of a certain body length is inversely proportional to the square of the body length. Write a formula for the number of animal species, N, of a
Find a possible formula for the graph. 8 y 20л + x
The table gives the number of North American houses (millions) with analog cable TV.(a) Plot the number of houses, H, in millions, with cable TV versus year, Y.(b) Could H be a linear function of Y?
A new bus worth $100,000 in 2015 depreciates linearly to $25,000 in 2030.(a) Find a formula for the value of the bus, V, as a function of time, t, in years since 2015.(b) What is the value of the bus
World milk production rose at an approximately constant rate between 2000 and 2012. See Figure 1.27. (a) Estimate the vertical intercept and interpret it in terms of milk production. (b)
Table 1.16 shows world population, P, in billions of people, world passenger automobile production, A, in millions of cars, and world cell phone subscribers, C, in millions of subscribers. (a)
The half-life of a radioactive substance is 12 days. There are 10.32 grams initially. (a) Write an equation for the amount, A, of the substance as a function of time. (b) When is the
The functions in Problems represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous.P = 3.2e0.03t
Figure 8.16 on page 348 gives a contour diagram of corn production as a function of rainfall, R, in inches and temperature, T , in ◦F. Corn production, C, is measured as a percentage of the present
Find all points where the partial derivatives of f(x, y) are both 0.f(x, y) = x2 + 2x + y2
The sales of a product, S = f(p, a), are a function of the price, p, of the product (in dollars per unit) and the amount, a, spent on advertising (in thousands of dollars).(a) Do you expect fp to be
Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these.f(x, y) = y3 − 3xy + 6x
Sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced.f(x, y) = x2 + y2
Find all points where the partial derivatives of f(x, y) are both 0.f(x, y) = x3 + 3x2 + y3 − 3y
Maximize production, Q = 5x0.3y0.8, where x and y are quantities of raw materials and the total quantity of these raw materials cannot be more than 100.
The monthly payments, P dollars, on a mortgage in which A dollars were borrowed at an annual interest rate of r% for t years is given by P = f(A, r, t). Is f an increasing or decreasing function of
Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these.f(x, y) = x3 + y2 − 3x2 + 10y + 6
Sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced.f(x, y) = xy
Figure 8.47 shows a contour diagram for the monthly payment P as a function of the interest rate, r%, and the amount, L, of a 5-year loan. Estimate ∂P∕∂r and ∂P∕∂L at the point where r =
(a) Find fx(1, 1) and fy(1, 1).(b) Use part (a) to match f(x, y) with one of the contour diagrams (I)–(III), each shown centered at (1, 1) with the same scale in the x and y directions.f(x, y) = x2
Maximize production for a manufacturing process with a $12 million budget and which uses x kilograms of one raw material and y kilograms of a second raw material to make Q = 3 ln(x + 1) + 2 ln(y + 1)
An airport can be cleared of fog by heating the air. The amount of heat required, H(T ,ω) (in calories per cubic meter of fog), depends on the temperature of the air, T (in ◦C), and the wetness of
Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these.f(x, y) = x3 + y3 − 6y2 − 3x + 9

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