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applied calculus

Calculus And Its Applications 14th Edition Larry Goldstein, David Lay, David Schneider, Nakhle Asmar - Solutions

Each of the graphs in Figure 2.50 shows the position of a particle moving along the x-axis as a function of time, 0 ≤ t ≤ 5. The vertical scales of the graphs are the same. During this time interval, which particle has(a) Constant velocity?(b) The greatest initial velocity?(c) The greatest
Which of the functions described in Problems are continuous?The distance traveled by a car in stop-and-go traffic as a function of time.
For Problem estimate the change in y for the given change in x.y = f(x), f'(100) = 0.4, x increases from 100 to 101
In Problems use the tangent line approximation.Given f(4) = 5, f'(4) = 7, approximate f(4.02).
Which of the functions described in Problems are continuous?The weight of a baby as a function of time during the second month of the baby’s life.
Let P(t) represent the price of a share of stock of a corporation at time t. What does each of the following statements tell us about the signs of the first and second derivatives of P(t)?(a) “The price of the stock is rising faster and faster.”(b) “The price of the stock is close to
For Problem estimate the change in y for the given change in x.y = f(x), f'(12) = 30, x increases from 12 to 12.2
In Problems use the tangent line approximation.Given f(4) = 5, f'(4) = 7, approximate f(3.92).
Which of the functions described in Problems are continuous?The number of pairs of pants as a function of the number of yards of cloth from which they are made. Each pair requires 3 yards.
For Problem estimate the change in y for the given change in x.y = g(x), g'(250) = −0.5, x increases from 250 to 251.5
In Problems use the tangent line approximation.Given f(5) = 3, f'(5) = −2, approximate f(5.03).
Which of the functions described in Problems are continuous?You start in North Carolina and go westward on Interstate 40 toward California. Consider the function giving the local time of day as a function of your distance from your starting point.
Match each property (a)–(d) with one or more of graphs (I)–(IV) of functions.(a) f'(x) = 1 for all 0 ≤ x ≤ 4(b) f'(x) > 0 for all 0 ≤ x ≤ 4(c) f'(2) = 1(d) f'(1) = 2 (1) 4 (III) 4 4 4 X x (11) (IV) 4 4 x X
Suppose f(x) = 1/3 x3. Estimate f'(2), f'(3), and f'(4). What do you notice? Can you guess a formula for f'(x)?
A child inflates a balloon, admires it for a while and then lets the air out at a constant rate. If V (t) gives the volume of the balloon at time t, then Figure 2.32 shows V'(t) as a function of t. At what time does the child:(a) Begin to inflate the balloon?(b) Finish inflating the balloon?(c)
(a) Using Table 2.10, calculate the average rate of change of the number of Facebook users, N (in millions), per month for each of the 3-month intervals.(b) What can you say about the sign of d2N∕dt2 during the period September 2014–September 2015? Table 2.10 Month, 1 Sep 2014 N 1350 Dec 2014
Show how to represent the following on Figure 2.20.(a) f(4)(b) f(4) − f(2)(c) f(5) − f(2) /5 − 2(d) f'(3) f(x) 12345 Figure 2.20 X
The Arctic Sea ice extent, the area of the sea covered by ice, grows seasonally over the winter months each year, typically from November to March, and is modeled by G(t), in millions of square kilometers, t months after November 1, 2014.(a) What is the sign of G¨(t) for 0 < t < 4?(b)
For Problem estimate the change in y for the given change in x.y = p(x), p'(400) = 2, x decreases from 400 to 398
In Problems use the tangent line approximation.Given f(2) = −4, f'(2) = −3, approximate f(1.95).
Use the definition of the derivative to show how the formulas in Problem are obtained.If f(x) = 5x, then f¨(x) = 5.
In 1913, Carlson conducted the classic experiment in which he grew yeast, Saccharomyces cerevisiae, in laboratory cultures and collected data every hour for 18 hours. Table 2.11 shows the yeast population, P, at representative times t in hours.(a) Calculate the average rate of change of P per hour
In 2009, a study was done on the impact of sea-level rise in the mid-Atlantic states. Let a(t) be the depth of the sea in millimeters (mm) at a typical point on the Atlantic Coast, and let m(t) be the depth of the sea in mm at a typical point on the Gulf of Mexico, with time t in years since data
For each of the following pairs of numbers, use Figure 2.20 to decide which is larger. Explain your answer.(a) f(3) or f(4)?(b) f(3) − f(2) or f(2) − f(1)?(c)(d) f'(1) or f'(4)? f(2)-f(1) 2-1 or ƒ(3) - ƒ(1) ₂ 3-1
Let(a) For small Δr, write an approximate equation relating ΔV and Δr near r = 2.(b) Estimate ΔV if Δr = 0.1.(c) Let V = 32 when r = 2. Estimate V when r = 2.1. dᏙ dr r=2 = 16.
In Problems use the tangent line approximation.Given f(−3) = −4, f'(−3) = 2, approximate f(−2.99).
Use the definition of the derivative to show how the formulas in Problem are obtained.If f(x) = 3x − 2, then f¨(x) = 3.
In Problems use the tangent line approximation.Given f(3) = −4, f'(3) = −2 approximate f(2.99).
Use the definition of the derivative to show how the formulas in Problem are obtained.If f(x) = x2 + 4, then f¨(x) = 2x.
Use the definition of the derivative to show how the formulas in Problem are obtained.If f(x) = 3x2, then f¨(x) = 6x.
Use the definition of the derivative to show how the formulas in Problem are obtained.If f(x) = x − x2, then f¨(x) = 1 − 2x.
The population, P(t), of China, in billions, can be approximated bywhere t is the number of years since the start of 2014. According to this model, how fast was the population growing at the start of 2014 and at the start of 2015? Give your answers in millions of people per year. P(t) =
Estimate the instantaneous rate of change of the function f(x) = x ln x at x = 1 and at x = 2. What do these values suggest about the concavity of the graph between 1 and 2?
Problems concern g(t) in Figure 2.35, which gives the weight of a human fetus as a function of its age.(a) What are the units of g¨(24)?(b) What is the biological meaning of g¨(24) = 0.096? weight (kg) 3 2 1 8 16 24 g(1) 1, age of fetus 40 (weeks after last menstruation) 32 Figure 2.35
Problems concern g(t) in Figure 2.35, which gives the weight of a human fetus as a function of its age.Is the instantaneous weight growth rate greater or less than the average rate of change of weight over the 40- week period(a) At week 16? (b) At week 36? weight (kg) 3 2 1 8 16 24 g(1) 1, age
Problems concern g(t) in Figure 2.35, which gives the weight of a human fetus as a function of its age.(a) Which is greater, g'(20) or g'(36)?(b) What does your answer say about fetal growth? weight (kg) 3 2 1 8 16 24 g(1) 1, age of fetus 40 (weeks after last menstruation) 32 Figure 2.35
Problems concern g(t) in Figure 2.35, which gives the weight of a human fetus as a function of its age.Estimate (a) g'(20) (b) g'(36)(c) The average rate of change of weight for the entire 40-week gestation. weight (kg) 3 2 1 8 16 24 g(1) 1, age of fetus 40 (weeks after last
Let R = f(S) and f'(10) = 3.(a) For small ΔS, write an approximate equation relating ΔR and ΔS near S = 10.(b) Estimate the change in R if S changes from S = 10 to S = 10.2.(c) Let f(10) = 13. Estimate f(10.2).
Use the definition of the derivative to show how the formulas in Problem are obtained.If f(x) = 5x2 + 1, then f¨(x) = 10x.
The US population officially reached 300 million on October 17, 2006 and was gaining 1 person each 11 seconds. If f(t) is the US population in millions t years after October 17, 2006, find f(0) and f'(0).
Use the definition of the derivative to show how the formulas in Problem are obtained.If f(x) = 2x2 + x, then f¨(x) = 4x + 1.
Use the definition of the derivative to show how the formulas in Problem are obtained.If f(x) = −2x3, then f¨(x) = −6x2.
Use the definition of the derivative to show how the formulas in Problem are obtained.If f(x) = 1 − x3, then f¨(x) = −3x2.
Use the definition of the derivative to show how the formulas in Problem are obtained.If f(x) = 1∕x, then f¨(x) = −1∕x2.
Annual net sales, in billion of dollars, for the Hershey Company, the largest US producer of chocolate, is a function S = f(t) of time, t, in years since 2010.(a) Interpret the statements f(5) = 7.39 and f¨(5) = −0.03 in terms of Hershey sales.(b) Estimate f(8) and interpret it in terms of
US beef production, M = f(t), in billion pounds, is a function of t, years since 2010.(a) Interpret f(5) = 23.7 and f'(5) = −0.1 in terms of beef production. (b) Estimate f(8) and interpret it in terms of beef production.
For some painkillers, the size of the dose, D, given depends on the weight of the patient, W. Thus, D = f(W), where D is in milligrams and W is in pounds.(a) Interpret the statements f(140) = 120 and f'(140) = 3 in terms of this painkiller.(b) Use the information in the statements in part (a) to
The quantity, Q mg, of nicotine in the body t minutes after a cigarette is smoked is given by Q = f(t).(a) Interpret the statements f(20) = 0.36 and f'(20) = −0.002 in terms of nicotine. What are the units of the numbers 20, 0.36, and −0.002?(b) Use the information given in part (a) to estimate
A mutual fund is currently valued at $80 per share and its value per share is increasing at a rate of $0.50 a day. Let V = f(t) be the value of the share t days from now.(a) Express the information given about the mutual fund in term of f and f'.(b) Assuming that the rate of growth stays constant,
Figure 2.36 shows how the contraction velocity, v(x), of a muscle changes as the load on it changes.(a) Find the slope of the line tangent to the graph of contraction velocity at a load of 2 kg. Give units.(b) Using your answer to part (a), estimate the change in the contraction velocity if the
Problems refer to Figure 2.38, which shows the depletion of food stores in the human body during starvation.Which is being consumed at a greater rate, fat or protein, during the(a) Third week? (b) Seventh week? quantity of stored food (kg) 12 10 8 6 4 2 Fat 1 2 3 3 Protein 4 5 6 7 8 Figure
Problems refer to Figure 2.38, which shows the depletion of food stores in the human body during starvation.Estimate the rate of fat consumption after(a) 3 weeks (b) 6 weeks (c) 8 weeks quantity of stored food (kg) 12 10 8 6 4 2 Fat 1 2 3 3 Protein 4 5 6 7 8 Figure 2.38 weeks of starvation
Problems refer to Figure 2.38, which shows the depletion of food stores in the human body during starvation.The fat storage graph is linear for the first four weeks. What does this tell you about the use of stored fat? quantity of stored food (kg) 12 10 8 6 4 2 Fat 1 2 3 3 Protein 4 5 6 7 8 Figure
Problems refer to Figure 2.38, which shows the depletion of food stores in the human body during starvation.What seems to happen during the sixth week? Why do you think this happens? quantity of stored food (kg) 12 10 8 6 4 2 Fat 1 2 3 3 Protein 4 5 6 7 8 Figure 2.38 weeks of starvation
Problems refer to Figure 2.38, which shows the depletion of food stores in the human body during starvation.Figure 2.39 shows the derivatives of the protein and fat storage functions. Which graph is which? quantity of stored food (kg) 12 10 8 6 4 2 Fat 1 2 3 3 Protein 4 5 6 7 8 Figure 2.38 weeks of
Suppose C(r) is the total cost of paying off a car loan borrowed at an annual interest rate of r%. What are the units of C'(r)? What is the practical meaning of C'(r)? What is its sign?
A person with a certain liver disease first exhibits larger and larger concentrations of certain enzymes (called SGOT and SGPT) in the blood. As the disease progresses, the concentration of these enzymes drops, first to the pre-disease level and eventually to zero (when almost all of the liver
A company’s revenue from car sales, C (in thousands of dollars), is a function of advertising expenditure, a, in thousands of dollars, so C = f(a).(a) What does the company hope is true about the sign of f¨?(b) What does the statement f'(100) = 2 mean in practical terms? How about f'(100) =
A company making solar panels spends x dollars on materials, and the revenue from the sale of the solar panels is f(x) dollars.(a) What does the statement f'(80,000) = 2 mean in practical terms? How about f'(80,000) = 0.5?(b) Suppose the company plans to spend about $80,000 on materials. If
For a new type of biofuel, scientists estimate that it takes A = f(g) gallons of gasoline to produce the raw materials to generate g gallons of biofuel. Assume the biofuel is equal in efficiency to the gasoline.(a) At a certain level of production, we have dA∕dg = 1.3. Interpret this in practical
The area of the Amazon’s rain forest, R = f(t), in thousand square kilometers, is a function of the number of years, t, since 2010.(a) Interpret f(5) = 5500 and f'(5) = −10.9 in terms of the Amazon’s rain forests.(b) Find and interpret the relative rate of change of f(t) when t = 5.
The number of active Facebook users hit 1.55 billion at the end of September 2015 and 1.59 billion at the end of December. With t in months since the start of 2015, let f(t) be the number of active users in billions. Estimate f(12) and f'(12) and the relative rate of change of f at t = 12.
Estimate the relative rate of change of f(t) = t2 at t = 4. Use Δt = 0.01.
Estimate the relative rate of change of f(t) = t2 at t = 10. Use Δt = 0.01.
The world population in billions is approximately P = 7.4e0.0107t where t is in years since 2016. Estimate the relative rate of change of population in 2020 using this model and(a) Δt = 1 (b) Δt = 0.1 (c) Δt = 0.01
The weight, w, in kilograms, of a baby is a function f(t) of her age, t, in months.(a) What does f(2.5) = 5.67 tell you?(b) What does f'(2.5)∕f(2.5) = 0.13 tell you?
The number of barrels of oil produced from North Dakota oil wells since January 2014 is estimated to be B = 29.05e0.0214t million barrels, where t is in months since January 2014.28 Estimate the relative rate of change of oil production in March 2015 using(a) Δt = 1 (b) Δt = 0.1 (c) Δt
Downloads of Apple Apps, D = g(t), in billions of downloads from iTunes, is a function of t months since its inception in June, 2008.(a) Interpret the statements g(36) = 15 and g'(36) = 0.93 in terms of App downloads.(b) Calculate the relative rate of change of D at t = 36; interpret it in terms of
During the 1970s and 1980s, the buildup of chlorofluorocarbons (CFCs) created a hole in the ozone layer over Antarctica. After the 1987 Montreal Protocol, an agreement to phase out CFC production, the ozone hole has shrunk. The ODGI (ozone depleting gas index) shows the level of CFCs present. Let
Refer to Exercise 31.(a) From midnight to noon, which 2-hour time intervals have the same rate of sales and what is this rate?(b) What is the total amount of sales between midnight and 8 a m.? Compare this amount to the total sales in the period between 8 a m. and 10 a m.Exercise 31The graph in
Use limits to compute the following derivatives.f′(0), where f(x) = x2 + 2x + 2
A supermarket finds that its average daily volume of business, V (in thousands of dollars), and the number of hours, t, that the store is open for business each day are approximately related by the formula Find dV dt V = 20 (1 t=10 100 72). 100+ t 0≤t≤ 24.
In an 8-second test run, a vehicle accelerates for several seconds and then decelerates. The function s(t) gives the number of feet traveled after t seconds and is graphed in Fig. 9.(a) How far has the vehicle traveled after 3.5 seconds?(b) What is the velocity after 2 seconds?(c) What is the
If f(x) = x3/2, compute f (16) and f′(16).
In Exercises, find an equation of the tangent line to the graph of y = f (x) at the given x. Do not apply formula (6), but proceed as we did in Example 4.f (x) = x3, x = -2Example 4.Finding the Equation of the Tangent Line at a Given x Find the point–slope equation of the tangent line to the
Write the equation of the line tangent to the graph of y = x3 at the point where x = -1.
In Exercises, determine the value of a that makes the function f(x) continuous at x = 0. for x ≥ 0 x) = { x + a forx < 0 f(x)
In Exercises, you are shown the tangent line to the graph of f (x) = x2 at the point (a, f (a)). Find a, f(a), and the slope of the parabola at (a, f (a)). y = x² 0 Y a y = 2x - 1 X
In Exercises , refer to a line of slope m. If you begin at a point on the line and move h units in the x-direction, how many units must you move in the y-direction to return to the line? m = 3, h = 3
Differentiate.f(t) = t10 - 10t9
The line through the points (-1, 2) and (3, b) is parallel to x + 2y = 0. Find b.
In Exercises, apply the three-step method to compute f′(x) for the given function. Follow the steps that we used in Example 6. Make sure to simplify the difference quotient as much as possible before taking limits.Example 6Computing a Derivative from the Limit Definition Use limits to compute the
In Exercises, find an equation of the tangent line to the graph of y = f (x) at the given x. Do not apply formula (6), but proceed as we did in Example 4.Example 4.Finding the Equation of the Tangent Line at a Given x Find the point–slope equation of the tangent line to the graph of 2 _f(x) =
In Exercises, you are shown the tangent line to the graph of f (x) = x2 at the point (a, f (a)). Find a, f (a), and the slope of the parabola at (a, f (a)). y = -x a 111 Y 0 y = x² X
If s = PT, find(a) ds/dp(b) ds/dp
In a psychology experiment, people improved their ability to recognize common verbal and semantic information with practice. Their judgment time after t days of practice was f (t) = .36 + .77(t - .5)-0.36 seconds.(a) Display the graphs of f (t) and f′(t) in the window [.5, 6] by [-3, 3]. Use
In Exercises, determine the value of a that makes the function f(x) continuous at x = 0. [2(x-a) for x ≥ 0 for x < 0 f(x) = √x² + 1 2
In Exercises, apply the three-step method to compute f′(x) for the given function. Follow the steps that we used in Example 6. Make sure to simplify the difference quotient as much as possible before taking limits.Example 6Computing a Derivative from the Limit Definition Use limits to compute the
Find the point(s) on the graph in Fig. 15 where the slope is equal to 3/2. Y y = x³ (x, y) Slope is 3² X Figure 15 Slope of tangent line to y = x³.
In Exercises, find an equation of the tangent line to the graph of y = f (x) at the given x. Do not apply formula (6), but proceed as we did in Example 4.f (x) = 3x + 1, x = 4Example 4.Finding the Equation of the Tangent Line at a Given x Find the point–slope equation of the tangent line to the
If s = P2T, find(a) d2s/dP2,(b) d2s/dT2,
A ball thrown straight up into the air has height s(t) = 102t - 16t2 feet after t seconds.(a) Display the graphs of s(t) and s′(t) in the window [0, 7] by [-100, 200]. Use these graphs to answer the remaining questions.(b) How high is the ball after 2 seconds?(c) When, during descent, is the
In Exercises, find an equation of the tangent line to the graph of y = f (x) at the given x. Do not apply formula (6), but proceed as we did in Example 4.f (x) = 5, x = -2 Example 4.Finding the Equation of the Tangent Line at a Given x Find the point–slope equation of the tangent
Differentiate.h(t) = 3√2
Find the points on the graph in Fig. 15 where the tangent line is parallel to y = 2x. Y y = x³ (x, y) Slope is 3² X Figure 15 Slope of tangent line to y = x³.
In Exercises , refer to a line of slope m. If you begin at a point on the line and move h units in the x-direction, how many units must you move in the y-direction to return to the line?m = 2, h = 1/2
In Exercises, apply the three-step method to compute f′(x) for the given function. Follow the steps that we used in Example 6. Make sure to simplify the difference quotient as much as possible before taking limits.Example 6Computing a Derivative from the Limit Definition Use limits to compute the
If s = Tx2 + 3xP + T2, find:(a) ds/dx(b) ds/dP(c) ds/dT
In Exercises , refer to a line of slope m. If you begin at a point on the line and move h units in the x-direction, how many units must you move in the y-direction to return to the line?m = -3, h = .25
Let l be the line through the points P and Q in Fig. 16.(a) If P = (2, 4) and Q = (5, 13), find the slope of the line l and the length of the line segment d.(b) Suppose that the point Q moves toward P along the graph. Does the slope of the line l increase or decrease? Y Р. Q d 1 X

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