- (a) Figure 1.81 shows exponential growth. Starting at t = 0, estimate the time for the population to double.(b) Repeat part (a), but this time start at t = 3.(c) Pick any other value of t for the
- Write the functions in Problems in the form P = P0at. Which represent exponential growth and which represent exponential decay?P = 2e−0.5t
- For each table in Problem 16 that could represent a linear function, find a formula for that function.Problem 16 Which of the following tables could represent linear functions? (a) (b) (c) x 0123 y
- Which of the following tables of values could correspond to an exponential function, a linear function, or neither? For those which could correspond to an exponential or linear function, find a
- Determine whether each of the following tables of values could correspond to a linear function, an exponential function, or neither. For each table of values that could correspond to a linear or an
- Figure 1.46 shows the position of an object at time t. (a) Draw a line on the graph whose slope represents the average velocity between t = 2 and t = 8. (b) Is average velocity greater
- For the functions f in Problem graph: (a) f(x + 2)(b) f(x − 1) (c) f(x) − 4 (d) f(x + 1) + 3 (e) 3f(x) (f) −f(x) + 1 4 3 نیا 2+ -2 -1 0 1 2 f(x) x
- For the functions f in Problem graph: (a) f(x + 2)(b) f(x − 1) (c) f(x) − 4 (d) f(x + 1) + 3 (e) 3f(x) (f) −f(x) + 1 4 3 ليا 2 + f(x) -2 -1 0 1 2 X
- For the functions f in Problem graph: (a) f(x + 2)(b) f(x − 1) (c) f(x) − 4 (d) f(x + 1) + 3 (e) 3f(x) (f) −f(x) + 1 -2 + 4 2 -2 f(x) 2 X
- 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
- The demand for a product is given in Problem 42. Find the revenue and the derivative of revenue with respect to price at a price of $10. Interpret your answers in economic terms.In Problem 42 If p is
- Use the graph in Figure 2.7 to decide if each of the following quantities is positive, negative or approximately zero. Illustrate your answers graphically.(a) The average rate of change of f(x)
- A dose-response curve is given by R = f(x), where R is percent of maximum response and x is the dose of the drug in mg. The curve has the shape shown in Figure 4.79. The inflection point is at (15,
- In Problems use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local
- What is the elasticity for potatoes in Table 4.5? Explain what this number tells you about the effect of price increases on the demand for potatoes. Is the demand for potatoes elastic or inelastic?
- Let f(x) = x3 - 6x2 - 15x + 20. Find f'(x) and all values of x for which f'(x) = 0. Explain the relationship between these values of and the graph of f(x)
- If the demand curve is a line, we can write p = b + mq, where p is the price of the product, q is the quantity sold at that price, and b and m are constants.(a) Write the revenue as a function of
- In Problems use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local
- If appropriate, evaluate the following integrals by substitution. If substitution is not appropriate, say so, and do not evaluate. (a) x sin(x²) dx x² 1+x2 dx (c) /₁ (e) [x²³x² dx (b) /
- For each of the following integrals, indicate whether integration by substitution or integration by parts is more appropriate. Do not evaluate the integrals. (a) x² 1 + x³ /₁ 3 dx (b)
- Match solutions and differential equations. (a) (b) (c) (d) (e) aaaaa dx dy - 32 = dx X dy dx dy y x dx dy dx = 3x = y 3y (1) y = x³ (II) y = 3x (III) y = e³x (IV) y = 3e* (V) y = x
- Marmots are large squirrels that hibernate in the winter and come out in the spring. Figure 1.28 shows the date (days after Jan 1) that they are first sighted each year in Colorado as a function of
- Let C(q) represent the cost, R(q) the revenue, and (q) the total profit, in dollars, of producing q items.(a) If C'(50) = 75 and R'(50) = 84, approximately how much profit is earned by the 51st
- The demand for a product is given by p = 90−10q. Find the elasticity of demand when p = 50. If this price rises by 2%, calculate the corresponding percentage change in demand.
- Linear supply and demand curves are shown in Figure 1.66, with price on the vertical axis.(a) Label the equilibrium price p0 and the equilibrium quantity q0 on the axes.(b) Explain the effect on
- Problem are about chikungunya, a disease that arrived in the Americas in 2013 and spread rapidly in 2014. While seldom fatal, the disease causes debilitating joint pain and a high fever. In August
- Figure 2.14 shows the cost, y = f(x), of manufacturing x kilograms of a chemical.(a) Is the average rate of change of the cost greater between x = 0 and x = 3, or between x = 3 and x = 5? Explain
- The maximum heart rate (MHR), which is the maximum number of times a person’s heart can safely beat in one minute. If MHR is in beats per minute and a is age in years, the formulas used to estimate
- Use Table 1.36 to find: (a) f(g(1)) (b) g(f(1)) (c) f(g(4)) (d) g(f(4)) (e) f(g(6)) (f) g(f(6)) Table 1.36 X 1 2 3 f(x) 5 4 3 g(x) 4 5 5 3 4 5 6 5 4 3 2 1 6 لنا
- Use Table 1.37 to find: (a) f(g(0)) (b) f(g(1)) (c) f(g(2)) (d) g(f(2)) (e) g(f(3)) Table 1.37 x f(x) g(x) 3 4 5 3 4 7 11 5 8 12 15 012 10 6 3 6 2 3
- Indicate all critical points of the function f. How many critical points are there? Identify each critical point as a local maximum, a local minimum, or neither. f(x) x
- Indicate the approximate locations of all inflection points. How many inflection points are there? f(x) X
- If f(x) = x2 + 1, find and simplify: (a) f(t + 1) (b) f(t2 + 1) (c) f(2) (d) 2f(t) (e) (f(t))2 + 1
- For the functions f and g in problem find (a) f(g(1)) (b) g(f(1)) (c) f(g(x)) (d) g(f(x)) (e) f(t)g(t)f(x) = x2, g(x) = x + 1
- For the functions f and g in problem find (a) f(g(1)) (b) g(f(1)) (c) f(g(x)) (d) g(f(x)) (e) f(t)g(t)f(x) = √ x + 4, g(x) = x2
- For f(x) in Figure 4.39, find the x-values of the global maximum and global minimum on the given domain.Domain: −3 ≤ x ≤ 0 60 40 20 -3 -2 -1 1 2 Figure 4.39 f(x) 3 4 X
- For the functions f and g in problem find (a) f(g(1)) (b) g(f(1)) (c) f(g(x)) (d) g(f(x)) (e) f(t)g(t) f(x) = ex, g(x) = x2
- For the functions f and g in problem find (a) f(g(1)) (b) g(f(1)) (c) f(g(x)) (d) g(f(x)) (e) f(t)g(t)f(x) = 1∕x, g(x) = 3x + 4
- (a) Write an equation for a graph obtained by vertically stretching the graph of y = x2 by a factor of 2, followed by a vertical upward shift of 1 unit. Sketch it. (b) What is the equation if
- Using the cost and revenue graphs in Figure 4.52, sketch the following functions. Label the points q1 and q2.(a) Total profit(b) Marginal cost(c) Marginal revenue $ 91 C(q) Figure 4.52 92 R(q) 9
- Absorption of different forms of the antibiotic erythromycin may be increased, decreased, delayed or not affected by food. Figure 4.96 shows the drug concentration levels of erythromycin in healthy,
- Problem are about chikungunya, a disease that arrived in the Americas in 2013 and spread rapidly in 2014. While seldom fatal, the disease causes debilitating joint pain and a high fever. In August
- World population was 6.13 billion in 2000 and was 7.32 billion in 2015, which means the population grew, on average, by 0.079 billion people per year during that time. The percent rate of growth of
- Figure 4.48 in Section 4.4 shows the points, q1 and q2, where marginal revenue equals marginal cost.(a) On the graph of the corresponding total cost and total revenue functions in Figure 4.53, label
- The cost function is C(q) = 1000 + 20q. Find the marginal cost to produce the 200th unit and the average cost of producing 200 units.
- Graph the average cost function corresponding to the total cost function shown in Figure 4.67. cost 500 C 1000 Figure 4.67 q (quantity)
- For f(x) in Figure 4.39, find the x-values of the global maximum and global minimum on the given domain.Domain: −2 ≤ x ≤ 3 60 40 20 -3 -2 -1 1 2 Figure 4.39 f(x) 3 4 X
- For f(x) in Figure 4.39, find the x-values of the global maximum and global minimum on the given domain.Domain: −2 ≤ x ≤ 1 60 40 20 -3 -2 -1 1 2 Figure 4.39 f(x) 3 4 X
- Hydrocodone bitartrate is a cough suppressant usually administered in a 10 mg oral dose. The peak concentration of the drug in the blood occurs 1.3 hours after consumption and the peak concentration
- The elasticity of coffee is E = 1.4. If the price of coffee drops from $12.00 to $11.50, does the demand for coffee increase or decrease? Estimate the percent change in the quantity demanded.
- Graph a function with only one critical point (at x = 5) and one inflection point (at x = 10). Label the critical point and the inflection point on your graph.
- Graph two continuous functions f and g, each of which has exactly five critical points, the points A–E in Figure 4.13, and that satisfy the following conditions:(a) f(x) → ∞ as x → −∞ and
- What are the units of elasticity if:(a) Price p is in dollars and quantity q is in tons?(b) Price p is in yen and quantity q is in liters?(c) What can you conclude in general?
- The following table shows the total sales, in thousands, since a new game was brought to market.(a) Plot this data and mark on your plot the point of diminishing returns.(b) Predict total possible
- There are many brands of laundry detergent. Would you expect the elasticity of demand for any particular brand to be high or low? Explain.
- Figure 4.85 shows the concentration of nicotine in the blood during and after smoking a cigarette. Figure 4.97 shows the concentration of nicotine in the blood during and after using chewing tobacco
- The cost of producing q units of a good is C(q) = 0.1q2 + 1000 million dollars.(a) What are the fixed costs? The marginal cost? The average cost?(b) At what production level does the average cost
- Let C(q) be the total cost of producing a quantity q of a certain product. See Figure 4.54.(a) What is the meaning of C(0)?(b) Describe in words how the marginal cost changes as the quantity produced
- For f(x) in Figure 4.39, find the x-values of the global maximum and global minimum on the given domain.Domain: 0 ≤ x ≤ 3 60 40 20 -3 -2 -1 1 2 Figure 4.39 f(x) 3 4 X
- (a) Graph a polynomial with two local maxima and two local minima.(b) What is the least number of inflection points this function must have? Label the inflection points.
- If t is in minutes since the drug was administered, the concentration, C(t) in ng/ml, of a drug in a patient’s bloodstream is given byC(t) = 20te−0.03t.(a) How long does it take for the drug to
- Would you expect the demand for high-definition television sets to be elastic or inelastic? Explain.
- During an illness a person ran a fever. His temperature rose steadily for eighteen hours, then went steadily down for twenty hours. When was there a critical point for his temperature as a function
- Write a paragraph explaining why sales of a new product often follow a logistic curve. Explain the benefit to the company of watching for the point of diminishing returns.
- For time t ≥ 0, the function C = ate−bt with positive constants a and b gives the concentration, C, of a drug in the body. Figure 4.98 shows the maximum concentration reached (in nanograms per
- The following table gives the percentage, P, of households with cable television between 1977 and 2003.(a) Explain why a logistic model is reasonable for this data.(b) Estimate the point of
- Shows how a surge can be modeled with a difference of exponential decay functions.(a) Using graphs of e−t and e−2t, explain why the graph of f(t) = e−t − e−2t has the shape of a surge.(b)
- For f(x) in Figure 4.39, find the x-values of the global maximum and global minimum on the given domain.Domain: −3 ≤ x ≤ 4 60 40 20 -3 -2 -1 1 2 Figure 4.39 f(x) 3 4 X
- The total cost of production, in thousands of dollars, is C(q) = q3 − 12q2 + 60q, where q is in thousands and 0 ≤ q ≤ 8.(a) Graph C(q). Estimate visually the quantity at which average cost is
- Graph a function which has a critical point and an inflection point at the same place.
- For each interval, use Figure 4.40 to choose the statement that gives the location of the global maximum and global minimum of f on the interval.(a) 4 ≤ x ≤ 12(b) 11 ≤ x ≤ 16(c) 4 ≤ x ≤
- The Tojolobal Mayan Indian community in Southern Mexico has available a fixed amount of land. The proportion, P, of land in use for farming t years after 1935 is modeled with the logistic function(a)
- Table 4.2 shows cost, C(q), and revenue, R(q).(a) At approximately what production level, q, is profit maximized? Explain your reasoning.(b) What is the price of the product?(c) What are the fixed
- In the spring of 2003, SARS (Severe Acute Respiratory Syndrome) spread rapidly in several Asian countries and Canada. Table 4.9 gives the total number, P, of SARS cases reported in Hong Kong by day
- Graph a function with the given properties.Has local minimum at x = 3, local maximum at x = 8, but no global maximum or minimum.
- (a) Use the derivative to find all critical points.(b) Use a graph to classify each critical point as a local minimum, a local maximum, or neither.f(x) = 5x − x2 + 8
- (a) Draw a logistic curve. Label the carrying capacity L and the point of diminishing returns t0.(b) Draw the derivative of the logistic curve. Mark the point t0 on the horizontal axis.(c) A company
- There is only one company offering wireless service in a town. Would you expect the elasticity of demand for wireless service to be high or low? Explain.
- Figure 4.55 shows graphs of marginal cost and marginal revenue. Estimate the production levels that could maximize profit. Explain your reasoning. $/unit 1000 3000 Figure 4.55 MC MR 9
- School organizations raise money by selling candy door to door. The table shows p, the price of the candy, and q, the quantity sold at that price.(a) Estimate the elasticity of demand at a price of
- Table 4.3 shows marginal cost, MC, and marginal revenue, MR.(a) Use the marginal cost and marginal revenue at a production of q = 5000 to determine whether production should be increased or decreased
- Figure 4.99 shows the plasma levels of canrenone in a healthy volunteer after a single oral dose of spironolactone given on a fasting stomach and together with a standardized breakfast. Discuss the
- (a) Use the derivative to find all critical points.(b) Use a graph to classify each critical point as a local minimum, a local maximum, or neither.f(x) = x3 − 75x
- During a flood, the water level in a river first rose faster and faster, then rose more and more slowly until it reached its highest point, then went back down to its pre-flood level. Consider water
- You are the manager of a firm that sells slippers for $20 a pair. You are producing 1200 pairs of slippers each month, at an average cost of $2 each. The marginal cost at a production level of 1200
- As I left home in the morning, I put on a light jacket because, although the temperature was dropping, it seemed that the temperature would not go much lower. But I was wrong. Around noon a northerly
- The average cost per item to produce q items is given by a(q) = 0.01q2 − 0.6q + 13, for q > 0.(a) What is the total cost, C(q), of producing q goods?(b) What is the minimum marginal cost? What
- Graph a function with the given properties.Has local minimum and global minimum at x = 3 but no local or global maximum.
- (a) Use the derivative to find all critical points.(b) Use a graph to classify each critical point as a local minimum, a local maximum, or neither.f(x) = 9x2 − x3
- The demand for a product is given by q = 200 − 2p2. Find the elasticity of demand when the price is $5. Is the demand inelastic or elastic, or neither?
- For f(x) = x3 − 18x2 − 10x + 6, find the inflection point algebraically. Graph the function with a calculator or computer and confirm your answer.
- The method of administering a drug can have a strong influence on the drug concentration curve. Figure 4.100 shows drug concentration curves for penicillin following various routes of administration.
- (a) Use the derivative to find all critical points.(b) Use a graph to classify each critical point as a local minimum, a local maximum, or neither.f(x) = x4 − 8x2
- (a) Find the average cost at each of the production levels q in the table.(b) At which q-value shown does the average cost have a critical point?(c) Is the critical value a local maximum, local
- The marginal cost at a production level of 2000 units of an item is $10 per unit and the average cost of producing 2000 units is $15 per unit. If the production level were increased slightly above
- Figure 4.101 shows drug concentration curves after oral administration of 0.5 mg of four digoxin products. All the tablets met current USP standards of potency, disintegration time, and dissolution
- A manufacturing process has marginal costs given in the table; the item sells for $30 per unit. At how many quantities, q, does the profit appear to be a maximum? In what intervals do these
- Graph a function with the given properties.Has no local or global maxima or minima.
- An agricultural worker in Uganda is planting clover to increase the number of bees making their home in the region. There are 100 bees in the region naturally, and or every acre put under clover, 20
- (a) Use the derivative to find all critical points.(b) Use a graph to classify each critical point as a local minimum, a local maximum, or neither.f(x) = x4 − 4x3