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mathematics
calculus with applications
Calculus With Applications 12th Edition Margaret L. Lial - Solutions
Find the cost function for each of the marginal cost functions in Exercises.C'(x) = 3√2x - 1; 13 units cost $270.
The following table shows the amount of crude oil (in billions of barrels) produced in the United States in recent years.In this exercise we are interested in the total amount of crude oil produced over the 9-year period from mid-2010 to mid-2019, using the data for the 10 years above.(a) One
A manufacturer of electronic equipment requires a certain rare metal. He has a reserve supply of 4,000,000 units that he will not be able to replace. If the rate at which the metal is used is given by ƒ(t) = 100,000e0.03t, where t is time (in years), how long will it be before he uses up the
Suppose that the supply function for some commodity is S(q) = q2 + 5q + 100 and the demand function for the commodity is D(q) = 350 - q2.(a) Find the producers’ surplus.(b) Find the consumers’ surplus.
A company has installed new machinery that will produce a savings rate (in thousands of dollars per year) of S′(x) = 225 - x2, where x is the number of years the machinery is to be used. The rate of additional costs (in thousands of dollars per year) to the company due to the new machinery is
The rate of change of the population of a rare species of Australian spider for one year is given bywhere ƒ(t) is the number of spiders present at time t (in months). Find the total number of additional spiders in the first 10 months. f'(t) = 100tV0.4t² + 1,
At time t = 0, a store has 19 units of a product in inventory. The cumulative number of units sold is given by S(t) = e3t - 1, where t is measured in weeks. The inventory will be replenished when it drops to 1 unit. The cost of carrying inventory until then is 15 per unit per week (prorated for a
The rate of infection of a disease (in people per month) is given by the functionwhere t is the time (in months) since the disease broke out. Find the total number of infected people over the first four months of the disease. I' (t) 100t 1² + 1'
In certain species of flour beetles, the larvae cannibalize the unhatched eggs. In calculating the population cannibalism rate per egg, researchers needed to evaluate the integralwhere A is the length of the larval stage and c(x) is the cannibalism rate per egg per larva of age x. The minimum value
Researchers report that the average amount of milk produced (in kilograms per day) by a 4- to 5-year-old cow weighing 700 kg can be approximated by y = 1.87t1.49e-0.189 (ln t)2, where t is the number of days into lactation.(a) Approximate the total amount of milk produced from t = 1 to t = 321
A research group studied the effect of a large injection of glucose in sheep fed a normal diet compared with sheep that were fasting. A graph of the plasma insulin levels (in pM—pico molars, or 10-12 of a molar) for both groups is shown below. The red graph designates the fasting sheep and the
If F(x) and G(x) are both antiderivatives of f(x), then F(x) = G(x).Determine whether each statement is true or false, and explain why.
If ƒ and g are continuous and ƒ(x) > g(x) on [a, b], then the area between the two curves from x = a to x = b is ∫ab[ƒ[x2 - g(x)] dx.Determine whether each statement is true or false, and explain why.
A definite integral is a family of antiderivatives of a function and an indefinite integral is a real number.Determine whether each statement is true or false, and explain why.
To find the area between ƒ and g on an interval, both functions must be positive on the interval.Determine whether each statement is true or false, and explain why.
As Δx = (b - a)/n increases, the sum ∑ni=1ƒ(xi) Δx always gives a better approximation to the area under ƒ(x) from a to b.
Integration by substitution is related to the chain rule for derivatives.Determine whether each statement is true or false, and explain why.
The Fundamental Theorem of Calculus shows the connection between the definite integral and an antiderivative of a function.Determine whether each statement is true or false, and explain why.
When using rectangles to approximate the area under the curve, the left sum always gives a more accurate estimate than the midpoint rule.
In order to use substitution, a function and its derivative (disregarding any constant multiplier) must be present in the integral.Determine whether each statement is true or false, and explain why.
By the power rule, ∫ x³ dx = 3x² + C.Determine whether each statement is true or false, and explain why.
Consumers’ surplus occurs when the price that consumers pay for a product or service is less than the price they are willing to pay.Determine whether each statement is true or false, and explain why.
If ƒ(x) ≥ 0 on the interval [a, b], the definite integral gives the exact area under the curve between x = a and x = b.
The following integrals may be solved using substitution. Choose a function u that may be used to solve each problem. Then find du. [ (3x²- (3x² - 5)4 2x dx
To use substitution to determine ∫4x³(x4 + 12)5 dx, let u = (x4 + 12)5.Determine whether each statement is true or false, and explain why.
For any function ƒ(x), the area between the graph of ƒ and the x-axis from x = a to x = b is ∫abƒ(x) dx.
The indefinite integral ∫ ax dx is equal to ax + C for all a > 0.Determine whether each statement is true or false, and explain why.
The producers’ surplus is the additional income the producer gains when the price they receive is more than the lowest price they would be willing to accept.Determine whether each statement is true or false, and explain why.
The area between the graph of ƒ and the x-axis, bounded by x = a and x = a, is ∫aaƒ(x) dx.
Find the area between the curves in Exercises.x = -2, x = 1, y = 2x2 + 5, y = 0
The function for the position of a particle, s(t), is an antiderivative of the velocity function, v(t).Determine whether each statement is true or false, and explain why.
To use substitution to determine ∫ 2x√x2 - 5 dx, let u = √x2 - 5.Determine whether each statement is true or false, and explain why.
Complete the following statement. fo is (x² + 3) dx = lim 11-0 where Ax= and xi
The area under the graph of the marginal cost function and above the x-axis, between x = a and x = b, gives the total change in the cost as x goes from a to b.
The following integrals may be solved using substitution. Choose a function u that may be used to solve each problem. Then find du. [vi V1 - x dx
Explain the difference between an indefinite integral and a definite integral.
What must be true of F(x) and G(x) if both are antiderivatives of ƒ(x)?
Find the area between the curves in Exercises.x = 1, x = 2, y = 3x3 + 2, y = 0
The following integrals may be solved using substitution. Choose a function u that may be used to solve each problem. Then find du. x² X - dx 2x³ + 1
How is the antiderivative of a function related to the function?
Find the area between the curves in Exercises.x = -3, x = 1, y = x3 + 1, y = 0
Let ƒ(x) = 2x + 5, x1 = 0, x2 = 2, x3 = 4, x4 = 6, and Δx = 2(a) Find(b) The sum in part (a) approximates a definite integral using rectangles. The height of each rectangle is given by the value of the function at the left endpoint. Write the definite integral that the sum approximates.
The following integrals may be solved using substitution. Choose a function u that may be used to solve each problem. Then find du. [4x³ex dx
Let ƒ(x) = 1/x, x1 = 1/2, x2 = 1, x3 = 3/2, x4 = 2, andΔx = 1/2.(a) Find(b) The sum in part (a) approximates a definite integral using rectangles. The height of each rectangle is given by the value of the function at the left endpoint. Write the definite integral that the sum approximates.
Explain what is wrong with the following use of the power rule: 5 dx = 5 x³/3 + C.
Use substitution to find each indefinite integral. [4(2x 4(2x + 3) dx
Explain why the restriction n ≠ -1 is necessary in the rule [x" d x" dx = xn+1 n+ 1 + C.
Find the area between the curves in Exercises.x = -3, x = 0, y = 1 - x2, y = 0
Use substitution to find each indefinite integral. √(-41 + (-4t + 1)³ dt
Find the area between the curves in Exercises.x = -2, x = 1, y = 2x, y = x2 - 3
Use substitution to find each indefinite integral. 2 dm (2m + 1)³
Use substitution to find each indefinite integral. 3 du V3u 5
In Exercises, approximate the area under the graph of f(x) and above the x-axis using the following methods with n = 4. (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts (a) and (b). (d) Use midpoints.ƒ(x) = 2x + 5 from x = 2 to x = 4
Find the area between the curves in Exercises.x = 0, x = 6, y = 5x, y = 3x + 10
In Exercises, approximate the area under the graph of f(x) and above the x-axis using the following methods with n = 4. (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts (a) and (b). (d) Use midpoints.ƒ(x) = 3x + 2 from x = 1 to x = 3
Find the area between the curves in Exercises.y = x2 - 30, y = 10 - 3x
Find the area between the curves in Exercises.y = x2 - 18, y = x - 6
In Exercises, approximate the area under the graph of f(x) and above the x-axis using the following methods with n = 4. (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts (a) and (b). (d) Use midpoints.ƒ(x) = -x2 + 4 from x = -2 to x = 2
Suppose the supply function for oil is given (in dollars) by S(q) = q2 + 10q, and the demand function is given (in dollars) by D(q) = 900 - 20q - q2.(a) Graph the supply and demand curves.(b) Find the point at which supply and demand are in equilibrium.(c) Find the consumers’ surplus.(d) Find the
Two cars start from rest at a traffic light and accelerate for several minutes. The graph on the next page shows their velocities (in feet per second) as a function of time (in seconds). Car A is the one that initially has greater velocity.(a) How far has car A traveled after 2 seconds?(b) When is
Use substitution to find each indefinite integral. J 2x + 2 (x² + 2x − 4)4 - dx
Find the area between the curves in Exercises.y = x2, y = 2x
Use substitution to find each indefinite integral. | 6x2 dx (2x³ + 7)3/2
Use substitution to find each indefinite integral. J 2x + 2 (x² + 2x − 4)4 - dx
In Exercises, approximate the area under the graph of f(x) and above the x-axis using the following methods with n = 4. (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts (a) and (b). (d) Use midpoints.ƒ(x) = x2 from x = 1 to x = 5
Use substitution to find each indefinite integral. [₂√42²- zV4z² - 5 dz
Find the area between the curves in Exercises. - x = 1, x = 6, y = 1 X y - 2
In Exercises, approximate the area under the graph of f(x) and above the x-axis using the following methods with n = 4. (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts (a) and (b). (d) Use midpoints.ƒ(x) = ex + 1 from x = -2 to x = 2
Find the area between the curves in Exercises.y = x2, y = x3
In Exercises, approximate the area under the graph of f(x) and above the x-axis using the following methods with n = 4. (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts (a) and (b). (d) Use midpoints.ƒ(x) = ex - 1 from x = 0 to x = 4
Find the area between the curves in Exercises. x = 0, x = 4, y = 1 x + 1 X-1 2
In Exercises, approximate the area under the graph of f(x) and above the x-axis using the following methods with n = 4. (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts (a) and (b). (d) Use midpoints. f(x) = 2 from x = 1 to x = 9
Use substitution to find each indefinite integral. fr√5P² + 2 5r² + 2 dr
Use substitution to find each indefinite integral. за 3x²e2r dx
In Exercises, approximate the area under the graph of f(x) and above the x-axis using the following methods with n = 4. (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts (a) and (b). (d) Use midpoints. f(x) X from x = 1 to x = 3
Use substitution to find each indefinite integral. fret dr
Use substitution to find each indefinite integral. (1 – t)e²¹-²² dt
Find ∫04 ƒ(x) dx for each graph of y = ƒ(x).(a)(b) y 1 H 1 X
Find the area between the curves in Exercises.x = -1, x = 1, y = ex, y = 3 - ex
Consider the region below ƒ(x) = x/2, above the x-axis, and between x = 0 and x = 4. Let xi be the midpoint of the ith subinterval.(a) Approximate the area of the region using four rectangles.(b) Find ∫04 ƒ(x) dx by using the formula for the area of a triangle.
Use substitution to find each indefinite integral. (x² − 1)ex³- 1)ex³-3x dx
Consider the region below ƒ(x) = 5 - x, above the x-axis, and between x = 0 and x = 5. Let xi be the midpoint of the ith subinterval.(a) Approximate the area of the region using five rectangles.(b) Find ∫05 (5 - x) dx by using the formula for the area of atriangle.
Find the area between the curves in Exercises.x = -1, x = 2, y = e-x, y = ex
Find ∫06 ƒ(x) dx for each graph of y = ƒ(x), where ƒ(x) consists of line segments and circular arcs.(a)(b) y |||| X
Find the area between the curves in Exercises. x = 2, x = 4, y = - 4 y = 1 x = 1 -
Find the area between the curves in Exercises.x = -1, x = 2, y = 2e2x, y = e2x + 1
Find the exact value of each integral using formulas from geometry. V16 - x² dx
Use substitution to find each indefinite integral. 1/z -2 Z dz
The booklet All About Lawns, published by Ortho Books, gives instructions for approximating the area of an irregularly shaped region. (See figure.) The booklet suggests measuring a long axis of the area, such as the vertical line in the figure. Then, every 10 feet along this axis, measure the width
Use substitution to find each indefinite integral. eVy 2Vy S dy
Find the area between the curves in Exercises.y = x3 - x2 + x + 1, y = 2x2 - x + 1
Find the exact value of each integral using formulas from geometry. √9 - x² dx J-3
Use substitution to find each indefinite integral. t 1² + 2 +2 dt
Find the area between the curves in Exercises.y = 2x3 + x2 + x + 5, y = x3 + x2 + 2x + 5
Find the exact value of each integral using formulas from geometry. -5 2 (1 + 2x) dx
Use substitution to find each indefinite integral. -4x x² + 3 X dx
Find the area between the curves in Exercises.y = x4 + ln (x + 10), y = x3 + ln(x + 10)
Find the exact value of each integral using formulas from geometry. -3 (5 + x) dx
Use substitution to find each indefinite integral. x³ + 2x 2 4 X x² + 4x² + 7 dx
Find the area between the curves in Exercises.y = x5 - 2 ln (x + 5), y = x3 - 2 ln (x + 5)
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