Find derivative of the function defined as

Problem 5E Find derivative of the function defined as follow. y =?16e2x+1

Problem 4E Find derivative of the function defined as follow. y =1.2e5x

Problem 3E Find derivative of the function defined as follow. y =?8e3x

Problem 1E Find derivative of the function defined as follow. y = e4x

Problem 2E Find derivative of the function defined as follow. y = e?2x

Problem 6E Find derivative of the function defined as follow. y =?4e?0.3x

Problem 63A The Gateway Arch The Gateway Arch in St. Louis, Missouri, is approximately 630 ft wide and 630 ft high. At first glance, the arch resembles a parabola, but its shape is actually known as a modified catenary. The height s (in feet) of the arch, measured from the ground to the middle of the arch, is approximated by the function s(x) = 693.9 ? 34.38(e0.01003x + e ?0.01003x), where x = 0 represents the center of the arch with ?299.2 ? x ? 299.2. Source: Notices of the AMS. a. What is the height of the arch when x = 0? ________________ b. What is the slope of the line tangent to the curve when x = 0? Does this make sense? ________________ c. Find the rate of change of the height of the arc when x = 150 ft.

Problem 7E Find derivative of the function defined as follow.

Problem 8E Find derivative of the function defined as follow.

Problem 9E Find derivative of the function defined as follow.

Problem 10E Find derivative of the function defined as follow.

Problem 11E Find derivative of the function defined as follow.

Problem 12E Find derivative of the function defined as follow.

Problem 13E Find derivative of the function defined as follow. y = xex

Problem 17E Find derivative of the function defined as follow.

Problem 15E Find derivative of the function defined as follow. y =(x + 3)2e4x

Problem 16E Find derivative of the function defined as follow. y =(3x3 ?4x)e?5x

Problem 14E Find derivative of the function defined as follow. y = x2e?2x

Problem 18E Find derivative of the function defined as follow.

Problem 19E Find derivative of the function defined as follow.

Problem 20E Find derivative of the function defined as follow.

Problem 21E Find derivative of the function defined as follow.

Problem 23E Find derivative of the function defined as follow.

Problem 22E Find derivative of the function defined as follow.

Problem 24E Find derivatives of the functions defined as follows.

Problem 25E Find derivative of the function defined as follow. y =73x+1

Problem 26E Find derivative of the function defined as follow. y = 4-5x+2

Problem 27E Find derivative of the function defined as follow.

Problem 28E Find derivative of the function defined as follow.

Problem 29E Find derivative of the function defined as follow.

Problem 30E Find derivative of the function defined as follow.

Problem 31E Find derivative of the function defined as follow.

Problem 32E Find derivative of the function defined as follow.

Problem 33E Find derivative of the function defined as follow.

Problem 34E Find derivative of the function defined as follow.

Problem 35E Prove that if y = y0ekt, where y0 and k are constants, then dy/dt = ky. (This says that for exponential growth and decay, the rate of change of the population is proportional to the size of the population, and the constant of proportionality is the growth or decay constant.)

Problem 38A Sales? The sales of a new personal computer (in thousands) are given by S?? ?) = 100 ? 90?? 0.3?t?, where ?t? represents time in years. Find the rate of change of sales at each time. a.? After 1 year b.? After 5 years c.? What is happening to the rate of change of sales as time goes on? d.? Does the rate of change of sales ever equal zero?

Problem 40A Product Awareness? After the introduction of a new product for tanning without sun exposure, the percent of the public that is aware of the product is approximated by A?(? = 10t? ?22?? ?, where ?t? is the time in months. Find the rate of change of the percent of the public that is aware of the product after the following numbers of months. a.? 2 b.? 4 c.? Notice that the answer to part a is positive and the answer to part b is negative. What does this tell you about how public awareness of the product has changed?

Problem 41A Product Durability Using data in a car magazine, we constructed the mathematical model y =100e?0.03045t for the percent of cars of a certain type still on the road after t years. Find the percent of cars on the road after the following numbers of years. a. 0 ________________ b. 2 ________________ c. 4 ________________ d. 6 Find the rate of change of the percent of cars still on the road after the following numbers of years. ________________ e. 0 ________________ f. 2 ________________ g. Interpret your answers to parts e and f.

Problem 37E Use graphical differentiation to verify that

Problem 39A Cost? The cost in dollars to produce ?x? DVDs can be approximated by Find the marginal cost when the following quantities are made. a.? 0 b.? 20 c.? What happens to the marginal cost as the number produced becomes larger and larger?

Problem 36E Use a graphing calculator to sketch the graph of y = [f(x + h) ? f(x)]/h using f(x) = ex and h=0.0001. Compare it with the graph of y = ex and discuss what you observe.

Problem 42A Investment The value of a particular investment changes over time according to the function where S(t) is the value after t years. Calculate the rate at which the value of the investment is changing after 8 years. (Choose one of the following.) Source: Society of Actuaries. a. 618 ________________ b. 1934 ________________ c. 2011 ________________ d. 7735 ________________ e. 10,468

Problem 43A Internet Users The growth in the number (in millions) of Internet users in the United States between 1990 and 2015 can be approximated by a logistic function with k = 0.0018, where t is the number of years since 1990. In 1990 (when t = 0), the number of users was about 2 million, and the number is expected to level out around 250 million. Source: World Bank. a. Find the growth function G(t) for the number of Internet users in the United States. Estimate the number of Internet users in the United States and the rate of growth for the following years. ________________ b. 1995 ________________ c. 2000 ________________ d. 2010 ________________ e. What happens to the rate of growth over time?

Problem 44A Population Growth In Exercise 1, the growth in world population (in millions) was approximated by the exponential function A(t) = 3100e0.0166t, where t is the number of years since 1960. Find the instantaneous rate of change in the world population at the following times. Source: United Nations. a. 2010 ________________ b. 2015 Exercise 1 Population Growth Since 1960, the growth in world population (in millions) closely fits the exponential function defined by A(t) = 3100e0.0166t, where t is the number of years since 1960. Source: United Nations. a. World population was about 3686 million in 1970. How closely does the function approximate this value? ________________ b. Use the function to approximate world population in 2000. (The actual 2000 population was about 6115 million.) ________________ c. Estimate world population in the year 2015.

Problem 46A Insect Growth The growth of a population of rare South American beetles is given by the logistic function with k =0.00001 and t in months. Assume that there are 200 beetles initially and that the maximum population size is 10,000. a. Find the growth function G(t) for these beetles. Find the population and rate of growth of the population after the following times. ________________ b. 6 months ________________ c. 3 years ________________ d. 7 years ________________ e. What happens to the rate of growth over time?

Problem 47A Clam Population The population of a bed of clams in the Great South Bay off Long Island is described by the logistic function with k = 0.0001 and t in years. Assume that there are 400 clams initially and that the maximum population size is 5200. a. Find the growth function G(t) for the clams. Find the population and rate of growth of the population after the following times. ________________ b. 1 year ________________ c. 4 years ________________ d. 10 years ________________ e. What happens to the rate of growth over time?

Problem 48A Pollution Concentration The concentration of pollutants (in grams per liter) in the east fork of the Big Weasel River is approximated by P(x) = 0.04e?4x, where x is the number of miles downstream from a paper mill that the measurement is taken. Find the following values. a. The concentration of pollutants 0.5 mile downstream ________________ b. The concentration of pollutants 1 mile downstream ________________ c. The concentration of pollutants 2 miles downstream Find the rate of change of concentration with respect to distance for the following distances. ________________ d. 0.5 mile ________________ e. 1 mile ________________ f. 2 miles

Problem 45A Minority Population In Exercise 1, we saw that the projected Hispanic population in the United States (in millions) can be approximated by the function h(t) = 37.79(1.021)t where t = 0 corresponds to 2000 and 0 ? t ? 50. Source: U.S. Census Bureau. a. Estimate the Hispanic population in the United States for the year 2015. ________________ b. What is the instantaneous rate of change of the Hispanic population in the United States when t = 15? Interpret your answer. Exercise 1 Minority Population According to the U.S. Census Bureau, the United States is becoming more diverse. Based on U.S. Census population projections for 2000 to 2050, the projected Hispanic population (in millions) can be modeled by the exponential function h(t) = 37.79(1.021)t, where t = 0 corresponds to 2000 and 0 ? t ? 50. Source: U.S. Census Bureau. a. Find the projected Hispanic population for 2005. Compare this to the actual value of 42.69 million. ________________ b. The U.S. Asian population is also growing exponentially, and the projected Asian population (in millions) can be modeled by the exponential function a(t) = 11.14(1.023)t, where t = 0 corresponds to 2000 and 0 ? t ? 50. Find the projected Asian population for 2005, and compare this to the actual value of 12.69 million. ________________ c. Determine the expected annual percentage increase for Hispanics and for Asians. Which minority population, Hispanic or Asian, is growing at a faster rate? ________________ d. The U.S. black population is growing at a linear rate, and the projected black population (in millions) can be modeled by the linear function b(t) = 0.5116t + 35.43, where t = 0 corresponds to 2000 and 0 ? t ? 50. Find the projected black population for 2005 and compare this projection to the actual value of 37.91 million. ________________ e. Graph the projected population function for Hispanics and estimate when the Hispanic population will be double its actual value for 2005. Then do the same for the Asian and black populations. Comment on the accuracy of these numbers.

Problem 50A Mortality The percentage of people of any particular age group that will die in a given year may be approximated by the formula P(t) = 0.00239e0.0957t, where t is the age of the person in years. Source: U.S. Vital Statistics. a. Find P(25), P(50), and P(75). ________________ b. Find P?(25), P?(50), and P?(75). ________________ c. Interpret your answers for parts a and b. Are there any limitations of this formula?

Problem 49A Breast Cancer It has been observed that the following formula accurately models the relationship between the size of a breast tumor and the amount of time that it has been growing. V(t) = 1100[1023e?0.02415t + 1]?4, where t is in months and V(t) is measured in cubic centimeters. Source: Cancer. a. Find the tumor volume at 240 months. ________________ b. Assuming that the shape of a tumor is spherical, find the radius of the tumor from part a. (Hint:The volume of a sphere is given by the formula V = (4/3)?r3.) c. If a tumor of size 0.5 cm3 is detected, according to the formula, how long has it been growing? What does this imply? ________________ d. Find V(t) and interpret this value. Explain whether this makes sense. ________________ e. Calculate the rate of change of tumor volume at 240 months and interpret.

Problem 53A Beef Cattle Researchers have compared two models that are used to predict the weight of beef cattle of various ages, where W1(t) and W2(t) represent the weight (in kilograms) of a t- day-old beef cow. Source: Journal of Animal Science. a. What is the maximum weight predicted by each function for the average beef cow? Is this difference significant? b. According to each function, find the age that the average beef cow reaches 90% of its maximum weight. c. Find W1?(750) and W2?(750) . Compare your results. d. Graph the two functions on [0, 2500] by [0, 525] and comment on the differences in growth patterns for each of these functions. e. Graph the derivative of these two functions on [0, 2500] by [0 1] and comment on any differences you notice between these functions.

Problem 52A Arctic Foxes The age/weight relationship of female Arctic foxes caught in Svalbard, Norway, can be estimated by the function where t is the age of the fox in days and M(t) is the weight of the fox in grams. Source: Journal of Mammalogy. a. Estimate the weight of a female fox that is 200 days old. ________________ b. Use M(t) to estimate the largest size that a female fox can attain. (Hint: Find M(t).) ________________ c. Estimate the age of a female fox when it has reached 80% of its maximum weight. ________________ d. Estimate the rate of change in weight of an Arctic fox that is 200 days old. (Hint: Recall that .) ________________ e. Use a graphing calculator to graph M(t) and then describe the growth pattern. ________________ f. Use the table function on a graphing calculator or a spreadsheet to develop a chart that shows the estimated weight and growth rate of female foxes for days 50, 100, 150, 200, 250, and 300.

Problem 55A Survival of Manuscripts Paleontologist John Cisne has demonstrated that the survival of ancient manuscripts can be modeled by the logistic equation. For example, the number of copies of the Venerable Bede’s De Temporum Ratione was found to approach a limiting value over the five centuries after its publication in the year 725. Let G(t) represent the proportion of manuscripts known to exist after t centuries out of the limiting value, so that m = 1. Cisne found that for Venerable Bede’s De Temporum Ratione, k =3.5 and G0 = 0.00369. Source: Science. a. Find the growth function G(t) for the proportion of copies of De Temporum Ratione found. Find the proportion of manuscripts and their rate of growth after the following number of centuries. ________________ b. 1 ________________ c. 2 ________________ d. 3 ________________ e. What happens to the rate of growth over time?

Problem 54A Cholesterol Researchers have found that the risk of coronary heart disease rises as blood cholesterol increases. This risk may be approximated by the function R(c) = 3.19(1.006)c, 100 ? c ? 300, where R is the risk in terms of coronary heart disease incidence per 1000 per year, and c is the cholesterol in mg/dL. Suppose a person’s cholesterol is 180 mg/dL and going up at a rate of 15 mg/dL per year. At what rate is the person’s risk of coronary heart disease going up? Source: Circulation.

Problem 51A Medical Literature It has been observed that there has been an increase in the proportion of medical research papers that use the word “novel” in the title or abstract, and that this proportion can be accurately modeled by the function p(x) = 0.001131e0.1268x, where x is the number of years since 1970. Source: Nature. a. Find p (40). ________________ b. If this phenomenon continues, estimate the year in which every medical article will contain the word “novel” in its title or abstract. ________________ c. Estimate the rate of increase in the proportion of medical papers using this word in the year 2010. ________________ d. Explain some factors that may be contributing to researchers using this word.

Problem 56A Habit Strength According to work by the psychologist C. L. Hull, the strength of a habit is a function of the number of times the habit is repeated. If N is the number of repetitions and H(N) is the strength of the habit, then H(N ) = 1000( 1 - e-kN), where k is a constant. Find H’(N) if k = 0.1 and the number of times the habit is repeated is as follows. a. 10 b. 100 c. 1000 d. Show that H’ (N) is always positive. What does this mean?

Problem 58A Radioactive Decay The amount (in grams) of a sample of lead 214 present after t years is given by A(t) = 500e?0.31t. Find the rate of change of the quantity present after each of the following years. a. 4 ________________ b. 6 ________________ c. 10 ________________ d. What is happening to the rate of change of the amount present as the number of years increases? ________________ e. Will the substance ever be gone completely?

Problem 59A Electricity In a series resistance?capacitance DC circuit, the instantaneous charge Q on the capacitor as a function of time (where t = 0 is the moment the circuit is energized by closing a switch) is given by the equation Q(t) = CV(1 ? e?t/RC), where C, V, and R are constants. Further, the instantaneous charging current IC is the rate of change of charge on the capacitor, or IC = dQ/dt. Source: Kevin Friedrich. a. Find the expression for IC as a function of time. ________________ b. If C = 10?5 farads, R = 107 ohms, and V = 10 volts, what is the charging current after 200 seconds? (Hint: When placed into the function in part a the units can be combined into amps.)

Problem 57A Online Learning The growth of the number of students taking at least one online course can be approximated by a logistic function with k = 0.0440, where t is the number of years since 2002. In 2002 (when t = 0), the number of students enrolled in at least one online course was 1.603 million. Assume that the number will level out at around 6.8 million students. Source: The Sloan Consortium. a. Find the growth function G(t) for students enrolled in at least one online course. Find the number of students enrolled in at least one online course and the rate of growth in the number in the following years. ________________ b. 2004 ________________ c. 2006 ________________ d. 2010 ________________ e. What happens to the rate of growth over time?

Problem 60A Heat Index The heat index is a measure of how hot it really feels under different combinations of temperature and humidity. The heat index, in degrees Fahrenheit, can be approximated by H(t) = T ? 0.9971e0.02086T[1 ? e0.0445(D ? 57.2)], where the temperature T and dewpoint D are both expressed in degrees Fahrenheit. Source: American Meteorological Society. a. Assume the dewpoint is D = 85° F. Find the function H(T). ________________ b. Using the function you found in part a, find the heat index when the temperature T is 80° F. ________________ c. Find the rate of change of the heat index when T = 80° F.

Problem 62A Ballooning Suppose a person is going up in a hot air balloon. The surrounding air temperature in degrees Fahrenheit decreases with height according to the formula T(h) = 80e-0.000065h, where h is the height in feet. How fast is the temperature decreasing when the person is at a height of 1000 ft and rising at a height of 800 ft/hr?

Problem 61A Track and Field In 1958, L. Lucy developed a method for predicting the world record for any given year that a human could run a distance of 1 mile. His formula is given as follows: t(n) = 218 + 31(0.933)n, where t(n) is the world record (in seconds) for the mile run in year 1950 + n. Thus, n = 5 corresponds to the year 1955. Source: Statistics in Sports. a. Find the estimate for the world record in the year 2010. ________________ b. Calculate the instantaneous rate of change for the world record at the end of year 2010 and interpret. ________________ c. Find and interpret. How does this compare with the current world record?

Problem 38A Sales The sales of a new personal computer (in thousands) are given by S(t) = 100 ? 90e?0.3t, where t represents time in years. Find the rate of change of sales at each time. a. After 1 year ________________ b. After 5 years ________________ c. What is happening to the rate of change of sales as time goes on? ________________ d. Does the rate of change of sales ever equal zero?

Problem 39A Cost The cost in dollars to produce x DVDs can be approximated by Find the marginal cost when the following quantities are made. a. 0 ________________ b. 20 ________________ c. What happens to the marginal cost as the number produced becomes larger and larger?

Problem 40A Product Awareness After the introduction of a new product for tanning without sun exposure, the percent of the public that is aware of the product is approximated by A(t) = 10t22?t, where t is the time in months. Find the rate of change of the percent of the public that is aware of the product after the following numbers of months. a. 2 ________________ b. 4 ________________ c. Notice that the answer to part a is positive and the answer to part b is negative. What does this tell you about how public awareness of the product has changed?