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mathematics
applied calculus
Applied Calculus 6th Edition Deborah Hughes Hallett, Patti Frazer Lock, Andrew M. Gleason, Daniel E. Flath, Sheldon P. Gordon, David O. Lomen, David Lovelock, William G. McCallum, Brad G. Osgood, Andrew Pasquale - Solutions
For Problems find the derivative. Assume a, b, c, k are constants.y = 5x + 13
For Problems use the definition of the derivative to obtain the following results.If f(x) = x4, then f'(x) = 4x3.
Find the derivative. Assume that a, b, c, and k are constants.y = 5xex2
Differentiate the functions in Problems. Assume that A and B are constants.R(q) = q2 − 2 cos q
Match f' with the corresponding f in Figure 2.30. (1) € + (V) 3 1 + x € (IV) -1 X + -1 1 Figure 2.30 X + x 1 X
The following table gives the percent of the US population living in urban areas as a function of year.(a) Find the average rate of change of the percent of the population living in urban areas between 1890 and 1990.(b) Estimate the rate at which this percent is increasing for the year 1990.(c)
Are the functions in Problem continuous on the given intervals?f(x) = 2x on 0 ≤ x ≤ 10
For three minutes the temperature of a feverish person has had positive first derivative and negative second derivative. Which of the following is correct?(a) The temperature rose in the last minute more than it rose in the minute before.(b) The temperature rose in the last minute, but less than it
Let G be annual US government purchases, T be annual US tax revenues, and Y be annual US output of all goods and services. All three quantities are given in dollars. Interpret the statements about the two derivatives, called fiscal policy multipliers.(a) dY ∕dG = 0.60 (b) dY ∕dT = −0.26
Match f' with the corresponding f in Figure 2.30. (1) € + (V) 3 1 + x € (IV) -1 X + -1 1 Figure 2.30 X + x 1 X
Are the functions in Problem continuous on the given intervals?f(x) = x2 + 2 on 0 ≤ x ≤ 5
Yesterday’s temperature at t hours past midnight was f(t) ◦C. At noon the temperature was 20◦C. The first derivative, f'(t), decreased all morning, reaching a low of 2◦C∕hour at noon, then increased for the rest of the day. Which one of the following must be correct?(a) The temperature
The weight, W, in lbs, of a child is a function of its age, a, in years, so W = f(a).(a) Do you expect f'(a) to be positive or negative? Why?(b) What does f(8) = 45 tell you? Give units for the numbers 8 and 45.(c) What are the units of f'(a)? Explain what f'(a) tells you in terms of age and
Are the functions in Problem continuous on the given intervals? f(x) = 1 x-1 on 2 ≤ x ≤ 3
A city grew in population throughout the 1980s and into the early 1990s. The population was at its largest in 1995, and then shrank until 2010. Let P = f(t) represent the population of the city t years since 1980. Sketch graphs of f(t) and f'(t), labeling the units on the axes.
A function f has f(5) = 20, f'(5) = 2, and f"(x) < 0, for x ≥ 5. Which of the following are possible values for f(7) and which are impossible?(a) 26 (b) 24 (c) 22
The height, in feet, of a tomato, t seconds after it is dropped from a 200-ft balcony, is s = f(t) = −16t2 + 200.(a) Find the average velocity between t = 2 and t = 3.(b) Find the average velocity between t = 2 and t = 2 + ℎ if:(i) ℎ = 0.1, (ii) ℎ = 0.01, (iii) ℎ = 0.001.(c) Use
Figure 2.17 shows f(t) and g(t), the positions of two cars with respect to time, t, in minutes.(a) Describe how the velocity of each car is changing during the time shown.(b) Find an interval over which the cars have the same average velocity.(c) Which of the following statements are true?(i)
(a) Graph f(x) = x2 and g(x) = x2 + 3 on the same axes. What can you say about the slopes of the tangent lines to the two graphs at the point x = 0? x = 1? x = 2? x = a, where a is any value?(b) Explain why adding a constant to any function will not change the value of the derivative at any point.
The following table shows the number of hours worked in a week, f(t), hourly earnings, g(t), in dollars, and weekly earnings, ℎ(t), in dollars, of production workers as functions of t, the year.(a) Indicate whether each of the following derivatives is positive, negative, or zero: f'(t), g'(t),
Find the derivative of the functions in Problems.f(x) = (x + 1)99
Differentiate the functions in Problems. Assume that A, B, and C are constants.P = 3t3 + 2et
If f(x) = x2(x3 + 5), find f'(x) two ways: by using the product rule and by multiplying out before taking the derivative. Do you get the same result? Should you?
Coroners estimate time of death using the rule of thumb that a body cools about 2◦F during the first hour after death and about 1◦F for each additional hour. Assuming an air temperature of 68◦F and a living body temperature of 98.6◦F, the temperature T (t) in ◦F of a body at a time t
Use the definition of the derivative to obtain the following results.If f(x) = 2x + 1, then f'(x) = 2.
Find the derivative. Assume a, b, c, k are constants.y = 3x
Find the derivative. Assume a, b, c, k are constants.y = 5
Derivative of the functions in Problems.g(x) = (4x2 + 1)7
Differentiate the functions in Problems. Assume that A, B, and C are constants.f(x) = 2ex + x2
The cost, C (in dollars), to produce g gallons of a chemical can be expressed as C = f(g). Using units, explain the meaning of the following statements in terms of the chemical:(a) f(200) = 1300 (b) f'(200) = 6
At exactly two of the labeled points in Figure 2.48, the derivative f' is 0; the second derivative f" is not zero at any of the labeled points. On a copy of the table, give the signs of f, f' , f" at each marked point. A B C Figure 2.48 D Point fff" A B C D
The table gives P = f(t), the number of households, in millions, in the US with cable television t years since 1998.(a) Does f'(2) appear to be positive or negative? What does this tell you about the number of households with cable television?(b) Estimate f'(2). Estimate f'(10). Explain what each
Match f' with the corresponding f in Figure 2.30. (1) € + (V) 3 1 + x € (IV) -1 X + -1 1 Figure 2.30 X + x 1 X
Are the functions in Problem continuous on the given intervals?f(x) = x + 2 on −3 ≤ x ≤ 3
The average weight, W, in pounds, of an adult is a function, W = f(c), of the average number of Calories per day, c, consumed.(a) Interpret the statements f(1800) = 155 and f'(2000) = 0 in terms of diet and weight.(b) What are the units of f'(c) = dW ∕dc?
Sketch the graph of a function f such that f(2) = 5, f'(2) = 1∕2, and f"(2) > 0.
Match f' with the corresponding f in Figure 2.30. (1) € + (V) 3 1 + x € (IV) -1 X + -1 1 Figure 2.30 X + x 1 X
For the function f(x) = 3x, estimate f'(1). From the graph of f(x), would you expect your estimate to be greater than or less than the true value of f'(1)?
Find the limit by simplifying the expression. lim h→0 (h-2)²-4 h
Table 2.7 shows world gold production, G = f(t), as a function of year, t.(a) Does f'(t) appear to be positive or negative? What does this mean in terms of gold production?(b) In which time interval does f'(t) appear to be greatest?(c) Estimate f'(2015). Give units and interpret your answer in
Sketch a graph of a continuous function f with the following properties:• f'(x) > 0 for all x• f"(x) < 0 for x < 2 and f"(x) > 0 for x > 2.
Match the points labeled on the curve in Figure 2.16 with the given slopes. Slope Point -3 -1 0 1/2 1 2 BE B Figure 2.16 E F
Find the limit by simplifying the expression. lim h→0 (h+ 1)² - 1 h
Match the functions in Problem with one of the derivatives in Figure 2.29. (III) E 3 (M) (VII) -5 15+ 2 -5+ 5+ f'(x) ƒ'(x) x x (10) (M) (VI) (MI) -5 Figure 2.29 -5+ 5+ -5+ 5 f'(x) -5+ 5+ f'(x) x f'(x) f'(x) x
The table gives the number of passenger cars, C = f(t), in millions, in the US in the year t.(a) Do f'(t) and f"(t) appear to be positive or negative during the period 1975–1990?(b) Do z'(t) and f"(t) appear to be positive or negative during the period 1990–2000?(c) Estimate f'(2005). Using
In April 2015 in the US, there was one birth every 8 seconds, one death every 12 seconds, and one new international migrant every 32 seconds.(a) Let f(t) be the population of the US, where t is time in seconds measured from the start of April 2015. Find f'(0). Give units.(b) To the nearest second,
For the function shown in Figure 2.15, at what labeled points is the slope of the graph positive? Negative? At which labeled point does the graph have the greatest (i.e., most positive) slope? The least slope (i.e., negative and with the largest magnitude)? A B C D Figure 2.15 E F
Match the functions in Problem with one of the derivatives in Figure 2.29. (III) E 3 (M) (VII) -5 15+ 2 -5+ 5+ f'(x) ƒ'(x) x x (10) (M) (VI) (MI) -5 Figure 2.29 -5+ 5+ -5+ 5 f'(x) -5+ 5+ f'(x) x f'(x) f'(x) x
Find the limit by simplifying the expression. lim h→0 (3h-7)+7 h
Values of f(t) are given in the following table. (a) Does this function appear to have a positive or negative first derivative? Second derivative? Explain.(b) Estimate f'(2) and f'(8). 1 02 4 6 8 10 f(t) 150 145 137 122 98 56
Investing $1000 at an annual interest rate of r%, compounded continuously, for 10 years gives you a balance of $B, where B = g(r). Give a financial interpretation of the statements:(a) g(2) ≈ 1221.(b) g'(2) ≈ 122. What are the units of g'(2)?
Table 2.13 shows the cost, C(q), and revenue, R(q), in terms of quantity q. Estimate the marginal cost, MC(q), and marginal revenue, MR(q), for q between 0 and 6. Table 2.13 1 2 3 4 5 6 60 120 200 300 420 450 480 9 C(q) 20 R(q) 100 220 330 410
(a) Let g(t) = (0.8)t. Use a graph to determine whether g'(2) is positive, negative, or zero.(b) Use a small interval to estimate g'(2).
Match the functions in Problem with one of the derivatives in Figure 2.29. (III) E 3 (M) (VII) -5 15+ 2 -5+ 5+ f'(x) ƒ'(x) x x (10) (M) (VI) (MI) -5 Figure 2.29 -5+ 5+ -5+ 5 f'(x) -5+ 5+ f'(x) x f'(x) f'(x) x
Find the limit by simplifying the expression. lim h→0 (5+2h) - 5 h
The cost, C = f(w), in dollars of buying a chemical is a function of the weight bought, w, in pounds.(a) In the statement f(12) = 5, what are the units of the 12? What are the units of the 5? Explain what this is saying about the cost of buying the chemical.(b) Do you expect the derivative f' to be
Values of f(x, y) and its derivatives are given in Table 8.13.(a) Identify which points in Table 8.13 are critical points of f.(b) For each critical point, determine whether it is a local minimum, maximum, or saddle point. Table 8.13 Values of f and its derivatives (x, y) yy (0,0) 0 0 0 -2 -3 (1,0)
Refer to Table 8.2 which gives a person’s body mass index, BMI, in terms of their weight w (in lbs) and height ℎ (in inches).Estimate the BMI of a man who weighs 90 kilograms and is 1.9 meters tall. Table 8.2 Body mass index (BMI) Height h (inches) 120 60 23.4 63 66 69 72 75 Weight w (lbs) 140
Using the contour diagram for f(x, y) in Figure 8.45, decide whether each of these partial derivatives is positive, negative, or approximately zero.(a) fx(4, 1) (b) fy(4, 1)(c) fx(5, 2) (d) fy(5, 2) 3 برا 2 1 y 2 5 10 3 1 2 3 4 5 6 Figure 8.45 X
Use the contour diagram for the function z = f(x, y) in Figure 8.26.Find a value of x for which(a) f(x, −1) = 2 (b) f(x, 2) = 0 2 -1 0 642 -2 -1 y Figure 8.26 -6 -2 2 X
According to the contour diagram for f(x, y) in Figure 8.45, which is larger: fx(3, 1) or fx(5, 2)? Explain. 3 برا 2 1 y 2 5 10 3 1 2 3 4 5 6 Figure 8.45 X
Use the contour diagram for the function z = f(x, y) in Figure 8.26.Find a value of y for which(a) f(1, y) = −4 (b) f(0, y) = 0 2 -1 0 642 -2 -1 y Figure 8.26 -6 -2 2 X
Refer to Table 8.3, which shows the weekly beef consumption, C, (in lbs) of an average household as a function of p, the price of beef (in $/lb) and I, annual household income (in $1000s).Give tables for beef consumption as a function of p, with I fixed at I = 20 and I = 100. Give tables for beef
Use Lagrange multipliers to find the maximum or minimum values of f(x, y) subject to the constraint.f(x, y) = x2 + y, x2 − y2 = 1
Find the partial derivatives in Problems. The variables are restricted to a domain on which the function is defined.ft if f(t, a) = 5a2t3
Figure 8.27 shows the contours of the temperature H in a room near a recently opened window. Label the three contours with reasonable values of H if the house is in the following locations.(a) Minnesota in winter (where winters are harsh).(b) San Francisco in winter (where winters are mild).(c)
Refer to Table 8.3, which shows the weekly beef consumption, C, (in lbs) of an average household as a function of p, the price of beef (in $/lb) and I, annual household income (in $1000s).How does beef consumption vary as a function of household income if the price of beef is held constant? Table
The function has a critical point at (0, 0). What sort of critical point is it?f(x, y) = xy
Refer to Table 8.5 shown below giving the heat index, I, in ◦F, as a function f(H, T) of the relative humidity, H, and the temperature, T, in ◦F. The heat index is a temperature which tells you how hot it feels as a result of the combination of humidity and temperature.Estimate ∂I∕∂H and
Use Lagrange multipliers to find the maximum or minimum values of f(x, y) subject to the constraint.f(x, y) = xy, 4x2 + y2 = 8
Find the partial derivatives in Problems. The variables are restricted to a domain on which the function is defined.fx and fy if f(x, y) = 5x2y3 + 8xy2 − 3x2
Refer to Table 8.3, which shows the weekly beef consumption, C, (in lbs) of an average household as a function of p, the price of beef (in $/lb) and I, annual household income (in $1000s).Make a table showing the amount of money, M, that the average household spends on beef (in dollars per
The function has a critical point at (0, 0). What sort of critical point is it?f(x, y) = x2 + y2
Refer to Table 8.5 on page 350 giving the heat index, I, in ◦F, as a function f(H, T) of the relative humidity, H, and the temperature, T , in ◦F. The heat index is a temperature which tells you how hot it feels as a result of the combination of humidity and temperature.Answer the question in
Figure 8.28 shows the contour map of z = f(B, t) which gives the balance in a bank account (in thousands of dollars) t years after an initial investment of B thousand dollars.(a) Label the contours with their values.(b) Determine the initial investment necessary so that the account has $4, 000 in
Use Lagrange multipliers to find the maximum or minimum values of f(x, y) subject to the constraint.f(x, y) = x2 + y2, 4x − 2y = 15
Find the partial derivatives in Problems. The variables are restricted to a domain on which the function is defined.fx and fy if f(x, y) = 10x2e3y
The function has a critical point at (0, 0). What sort of critical point is it?g(x, y) = x4 + y3
Figure 8.29 shows the contours of the amount, C, in mg, of medication in the blood stream, as a function of the time t since an initial dose D is administered to the patient. Which axis corresponds to the initial dosage? Figure 8.29 2 8 10 15 20 25 30
Find the partial derivatives in Problems. The variables are restricted to a domain on which the function is defined. ;(;mı²) д ????m
Refer to Table 8.3, which shows the weekly beef consumption, C, (in lbs) of an average household as a function of p, the price of beef (in $/lb) and I, annual household income (in $1000s).Make a table of the proportion, P, of household income spent on beef per week as a function of price and
Use Lagrange multipliers to find the maximum or minimum values of f(x, y) subject to the constraint.f(x, y) = x2 + y2, x4 + y4 = 2
Figure 8.59 shows contours labeled with values of f(x, y) and a constraint g(x, y) = c. Mark the approximate points at which:(a) f has a maximum(b) f has a maximum on the constraint g = c. g=c 10 13 12. Figure 8.59
Refer to Table 8.3, which shows the weekly beef consumption, C, (in lbs) of an average household as a function of p, the price of beef (in $/lb) and I, annual household income (in $1000s).Express P, the proportion of household income spent on beef per week, in terms of the original function f(I, p)
Find the partial derivatives in Problems. The variables are restricted to a domain on which the function is defined.zx if z = x2y + 2x5y
The function has a critical point at (0, 0). What sort of critical point is it?f(x, y) = x6 + y6
A topographic map is given in Figure 8.30. How many hills are there? Estimate the x- and y-coordinates of the tops of the hills. Which hill is the highest? A river runs through the valley; in which direction is it flowing? y 8 7 6 5 4 3 2 in 43 210 224 in 1 2 3 4 5 6 7 8 Figure 8.30 Η North x
Find the partial derivatives in Problems. The variables are restricted to a domain on which the function is defined. ар др if P = 100e"
The demand for coffee, Q, in pounds sold per week, is a function of the price of coffee, c, in dollars per pound and the price of tea, t, in dollars per pound, so Q = f(c, t).(a) Do you expect fc to be positive or negative? What about ft? Explain.(b) Interpret each of the following statements in
Figure 8.60 shows contours of f(x, y) and the constraint g(x, y) = c. Approximately what values of x and y maximize f(x, y) subject to the constraint? What is the approximate value of f at this maximum? g(x, y) = c 16 14 12 10 f = 100 8 6 4 2 y 2 4 6 8 10 12 14 16 Figure 8.60 f = 700 f = 600 f =
The function has a critical point at (0, 0). What sort of critical point is it?f(x, y) = xey − x
Find the partial derivatives in Problems. The variables are restricted to a domain on which the function is defined. JA dh if A = 1/(a+b)h
A drug is injected into a patient’s blood vessel. The function c = f(x, t) represents the concentration of the drug at a distance x mm in the direction of the blood flow measured from the point of injection and at time t seconds since the injection. What are the units of the following partial
Graph the bank-account function f in Example 1(a) on page 340, holding B fixed at B = 10, 20, 30 and letting t vary. Then graph f, holding t fixed at t = 0, 5, 10 and letting B vary. Explain what you see.Example 1 Give a formula for the function M = f(B,t) where M is the amount of money in a bank
The function has a critical point at (0, 0). What sort of critical point is it?f(x, y) = x − ex − y2
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) = x + y
The total sales of a product, S, can be expressed as a function of the price p charged for the product and the amount, a, spent on advertising, so S = f(p, a).Do you expect f to be an increasing or decreasing function of p? Do you expect f to be an increasing or decreasing function of a? Why?
The quantity Q (in pounds) of beef that a certain community buys during a week is a function Q = f(b, c) of the prices of beef, b, and chicken, c, during the week. Do you expect ∂Q∕∂b to be positive or negative? What about ∂Q∕∂c?
The quantity, Q, of a good produced depends on the quantities x1 and x2 of two raw materials used:Q = x0.31 x0.72.A unit of x1 costs $10, and a unit of x2 costs $25. We want to maximize production with a budget of $50 thousand for raw materials.(a) What is the objective function?(b) What is the
Table 8.7 gives the number of calories burned per minute, B = f(s,w), for someone roller-blading, as a function of the person’s weight, w, and speed, s.(a) Is fw positive or negative? Is fs positive or negative?What do your answers tell us about the effect of weight and speed on calories burned
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