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mathematics
applied calculus
Calculus And Its Applications 14th Edition Larry Goldstein, David Lay, David Schneider, Nakhle Asmar - Solutions
Use limits to compute the following derivatives.f′(2), where f(x) = x3
In the next section, we shall see that the tangent line to the graph of y = x3 at the point (x, y) has slope 3x2. See Fig. 15. Using this result, find the slope of the curve at the points in Exercises.Figure 15 Slope of tangent line to y = x3.(2, 8) Y y = x³ (x, y) Slope is 3x2 X
In Exercises, we specify a line by giving the slope and one point on the line. Start at the given point and use plotting a line using a slope and a point to sketch the graph of the line.m = - 1/3, (1, -1) on line
Differentiate.f(x) = 5
Compute the following. g'(0) and g"(0), when g(T) = (T + 2)³
Consider the cost function C(x) = 6x2 + 14x + 18 (thousand dollars).(a) What is the marginal cost at production level x = 5?(b) Estimate the cost of raising the production level from x = 5 to x = 5.25.(c) Let R(x) = -x2 + 37x + 38 denote the revenue in thousands of dollars generated from the
Refer to Fig. 14. Estimate the price of one barrel of crude oil on January 12, 2015, and the rate at which it was rising on that day. $/Barrel $29 $ 28 $27 $26 $25 5 10 15 20 January 2016 Figure 14 Price of crude oil. 25 30
The owner of a photocopy store charges 7 cents per copy for the first 100 copies and 4 cents per copy for each copy exceeding 100. In addition, there is a setup fee of $2.50 for each photocopying job.(a) Determine R(x), the revenue from selling x copies.(b) If it costs the store owner 3 cents per
Refer to Exercise 27.(a) Find a formula for T(x) for all taxable income x.(b) Plot T(x).(c) Determine the maximum amount of tax that you will pay on the portion of your income in the fourth tax bracket. Exercise 27The tax that you pay to the federal government is a percentage of your taxable
If f(x) = x1/3, compute f(8) and f′(8).
Use limits to compute the following derivatives.f′(3), where f(x) = x2 + 1
In Exercises, we specify a line by giving the slope and one point on the line. Start at the given point and use plotting a line using a slope and a point to sketch the graph of the line.m = 1/2, (-1, 1) on line
Differentiate.f(x) = (2x + 1)3
The tax that you pay to the federal government is a percentage of your taxable income, which is what remains of your gross income after you subtract your allowed deductions. In a recent year, there were five rates or brackets for a single taxpayer, as shown in Table 1.So, if you are single and your
Let f (x) be the value in dollars of one share of a company x days since the company went public.(a) Interpret the statements f(100) = 16 and f′(100) = .25.(b) Estimate the value of one share on the 101st day since the company went public.
Refer to Fig. 14, which shows an enlarged version of one portion of the curve in Fig. 13. Estimate the price of one barrel of crude oil on January 20, 2016, and the rate at which the price was rising on that day. (Your answer for the rate should be in dollars per day.)Figure 14 Price of crude oil.
Compute the following. ƒ'(1) and ƒ"(1), when f(t) = = 1 2+1
If f(x) = 2x + 6, compute f(0) and f′(0).
Compute the following. dy dx dx $12 x=1 3 where y = x³ + 2x - 11
Use the limit definition of the derivative to show that if f(x) = mx + b, then f′(x) = m.
In Exercises, we specify a line by giving the slope and one point on the line. Start at the given point and use plotting a line using a slope and a point to sketch the graph of the line.m = 1, (1, 0) on line
Differentiate.f(x) = 1/4√x
Compute the limits that exist, given that(a)(b)(c)(d) lim f(x) x →0 = 1 2 and lim g(x) x-0 = 1 2
Determine which of the following limits exist. Compute the limits that exist. 1 lim x→9 (x − 9)²
Let P(x) be the profit (in dollars) from manufacturing and selling x cars. Interpret P(100) = 90,000 and P′(100) = 1200. Estimate the profit from manufacturing and selling 99 cars.
The functions in Exercises are defined for all x except for one value of x. If possible, define f (x) at the exceptional point in a way that makes f (x) continuous for all x. f(x)= √9+x - √9 V9 X ; x = 0
If f(x) = x3, compute f(-5) and f′(-5).
Refer to Fig. 13. Do you agree with the statement that the price of one barrel of crude oil fell on July 1, 2015, and on December 1, 2015, at approximately the same rate? Justify your answer. $/Barrel y $95 $85 $ 75 $ 65 $ 55 $45 $35 $25 March May July Sep. 2015 2015 2015 2015 Nov. Jan. March 2015
Find an equation of the given line.Perpendicular to y = -5x + 1; (1, 5) on line
Let C(x) be the cost (in dollars) of manufacturing x items. Interpret the statements C(2000) = 50,000 and C′(2000) = 10. Estimate the cost of manufacturing 1998 items.
Compute the following. d² dx2 (3x³ − x² + 7x − 1) – - x=2
Determine which of the following limits exist. Compute the limits that exist. 8 X 8 X + 64 ex X lim
Find the slope of the curve y = x5 at x = 1/3.
Find an equation of the given line.Perpendicular to y + x = 0; (2, 0) on line
Differentiate.y = √x2 + 1
Let f(x) be the number (in thousands) of computers sold when the price is x hundred dollars per computer. Interpret the statements f(12) = 60 and f′(12) = -2. Then, estimate the number of computers sold if the price is set at $1250 per computer.
Figure 13 shows the price of 1 barrel of crude oil on the New York Stock Exchange from March 1, 2015, to March 1, 2016. Determine the price decrease from March 1, 2015, to January 1, 2016. Also determine whether the price was rising, falling, or holding steady on these days.Figure 13 Price of crude
The functions in Exercises are defined for all x except for one value of x. If possible, define f (x) at the exceptional point in a way that makes f (x) continuous for all x. f(x) = (6 + x)² - 36 X , x 0
Compute the following. d² dx2 (3x² + 4x²) x=2
Determine which of the following limits exist. Compute the limits that exist. lim x-7 x³ - 2x² + 3x .2
In Exercises, find the derivative of f(x) at the designated value of x. f(x) at x = 32
Find the slope of the curve y = x4 at x = 2.
The functions in Exercises are defined for all x except for one value of x. If possible, define f (x) at the exceptional point in a way that makes f (x) continuous for all x. f(x) = = x² + 25 2 x - 5 x = 5
Find an equation of the given line.Parallel to y - x = 13; y-intercept is 0
Differentiate.y = (x3 + x2 + 1)5
Compute the following. de+2z+ | + 2z + 1) z=-1
Determine which of the following limits exist. Compute the limits that exist. 9- x6x²5x6 x9 - zx er lim
Let f (x) be the number of toys sold when x dollars are spent on advertising. Interpret the statements f(100,000) = 3,000,000 and f′(100,000) = 30.
Find the point on the graph of y = x2 where the tangent line is parallel to the line 3x - 2y = 2.
The functions in Exercises are defined for all x except for one value of x. If possible, define f (x) at the exceptional point in a way that makes f (x) continuous for all x. f(x): r3 – 5x2+4 2 +3 -₂x = 0
Find an equation of the given line.Parallel to y = 3x + 7; x-intercept is 2
Let f (p) be the number of cars sold when the price is p dollars per car. Interpret the statements f (10,000) = 200,000 and f′(10,000) = -3.
In Exercises, find the derivative of f(x) at the designated value of x. f(x)=√x at x = 1/6
Find the point on the graph of y = x2 where the tangent line is parallel to the line 2x + 3y = 4.
Find an equation of the given line.Parallel to x + 2y = 0; (1, 2) on line
Table 2 gives a car’s trip odometer reading (in miles) at 1 hour into a trip and at several nearby times. What is the average speed during the time interval from 1 to 1.05 hours? Estimate the speed at time 1 hour into the trip.A particle is moving in a straight line in such a way that its
In Exercises, find the derivative of f(x) at the designated value of x. f(x) = x³ at x tx = 1/2
Determine whether each of the following functions is continuous and/or differentiable at x = 1. f(x) = {² 1 2x1 for 0 ≤ x ≤ 1 for 1 < x
Find an equation of the given line.x-intercept is 1; y-intercept is -3
Determine which of the following limits exist. Compute the limits that exist. lim x-0 1-² + 3x X
In Exercises, find the slope of the tangent line to the graph of y = x2 at the point indicated and then write the corresponding equation of the tangent line.Find the slope of the tangent line to the graph of y = x2 at the point where x =-1/4.
A car is traveling from New York to Boston and is partway between the two cities. Let s(t) be the distance from New York during the next minute. Match each behavior with the corresponding graph of s(t) in Fig. 7.(a) The car travels at a positive steady speed.(b) The car is stopped.(c) The car is
In Exercises, find the derivative of f(x) at the designated value of x. 5 f(x) = x³ at x = ³/
A particle is moving in a straight line in such a way that its position at time t (in seconds) is s(t) = t2 + 3t + 2 feet to the right of a reference point, for t ≥ 0.(a) What is the velocity of the object when the time is 6 seconds?(b) Is the object moving toward the reference point when t = 6?
What is meant by the average rate of change of a function over an interval?
Determine whether each of the following functions is continuous and/or differentiable at x = 1. f(x) = {2 for x for x = 1 = 1
Differentiate.y = 6√x
Find an equation of the given line.x-intercept is -p; y-intercept is 1
Determine which of the following limits exist. Compute the limits that exist. [ - X [←x uu
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