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mathematics
calculus with applications
Calculus With Applications 12th Edition Margaret L. Lial - Solutions
Find the area between the x-axis and the upper half of the ellipse 4x2 + 9y2 = 36 by using n = 12 with the following methods.(a) The trapezoidal rule (b) Simpson’s rule(c) Compare the results with the actual area, 3π ≈ 9.4248 (which can be found by methods not considered in this text).
Explain the differences between an indefinite integral and a definite integral.Determine whether each of the following statements is true or false, and explain why.
Find the area under the semicircle y = √4 - x2 and above the x-axis by using n = 8 with the following methods.(a) The trapezoidal rule (b) Simpson’s rule(c) Compare the results with the area found by the formula for the area of a circle. Which of the two approximation techniques was more
Simpson’s rule usually gives a better approximation than the trapezoidal rule.Determine whether each of the following statements is true or false, and explain why.
In Exercises, use n = 4 to approximate the value of the given integrals by the following methods: (a) The trapezoidal rule, and (b) Simpson’s rule. (c) Find the exact value by integration. 4 J2 3 x³ dx
In the trapezoidal rule, the number of subintervals must be even.Determine whether each of the following statements is true or false, and explain why.
The consumers’ surplus and the producers’ surplus equal each other.Determine whether each of the following statements is true or false, and explain why.
In Exercises, use n = 4 to approximate the value of the given integrals by the following methods: (a) The trapezoidal rule, and (b) Simpson’s rule. (c) Find the exact value by integration. .3 S (2x³ + 1) dx
In Exercises, use n = 4 to approximate the value of the given integrals by the following methods: (a) The trapezoidal rule, and (b) Simpson’s rule. (c) Find the exact value by integration. dx x² X
In Exercises, use n = 4 to approximate the value of the given integrals by the following methods: (a) The trapezoidal rule, and (b) Simpson’s rule. (c) Find the exact value by integration. -2 J-1 (2x³ + 1) dx
The area between two distinct curves is always a positive quantity.Determine whether each of the following statements is true or false, and explain why.
The definite integral of a function is always a positive quantity.Determine whether each of the following statements is true or false, and explain why.
The Fundamental Theorem of Calculus gives a relationship between the definite integral and an antiderivative of a function.Determine whether each of the following statements is true or false, and explain why.
In Exercises, use n = 4 to approximate the value of the given integrals by the following methods: (a) The trapezoidal rule, and (b) Simpson’s rule. (c) Find the exact value by integration. -5 6 2x + 1 dx
The definite integral of a positive function is the limit of the sum of the areas of rectangles.Determine whether each of the following statements is true or false, and explain why.
In Exercises, use n = 4 to approximate the value of the given integrals by the following methods: (a) The trapezoidal rule, and (b) Simpson’s rule. (c) Find the exact value by integration. 3 -1 3 - dx 5-x
In Exercises, use n = 4 to approximate the value of the given integrals by the following methods: (a) The trapezoidal rule, and (b) Simpson’s rule. (c) Find the exact value by integration. -2 (2x² + 1) dx
The definite integral gives an approximation to the area under a curve.Determine whether each of the following statements is true or false, and explain why.
The definite integral gives the instantaneous rate of change of a function.Determine whether each of the following statements is true or false, and explain why.
The indefinite integral is another term for the family of all antiderivatives of a function.Determine whether each of the following statements is true or false, and explain why.
In Exercises, use n = 4 to approximate the value of the given integrals by the following methods: (a) The trapezoidal rule, and (b) Simpson’s rule. (c) Find the exact value by integration. -2 (3x² + 2) dx
Substitution can often be used to turn a complicated integral into a simpler one.Determine whether each of the following statements is true or false, and explain why.
In order to use Simpson’s rule, an antiderivative for the integrand must first be found.Determine whether each statement is true or false, and explain why.
The velocity function is an antiderivative of the acceleration function.Determine whether each of the following statements is true or false, and explain why.
The trapezoidal rule always gives a better approximation of a definite integral than Simpson’s rule.Determine whether each statement is true or false, and explain why.
The indefinite integral ∫ xƒ(x) dx is equal to ∫ ƒ(x) dx.Determine whether each of the following statements is true or false, and explain why.
In Simpson’s rule, the number of subintervals (n) must be even.Determine whether each statement is true or false, and explain why.
The indefinite integral of xn is xn+1/(n + 1) + C for all real numbers n.Determine whether each of the following statements is true or false, and explain why.
Simpson’s rule approximates the definite integral using trapezoidal approximations.Determine whether each statement is true or false, and explain why.
1. Find the number of years that the estimated petroleum reserves would last if used at the same rate as in the base year.2. How long would the estimated petroleum reserves last if the growth constant was only 2% instead of 4.7%?In Exercises, estimate the length of time until depletion for each
On March 13, 2020, President Trump signed a proclamation declaring that the COVID-19 outbreak in the U. S. constituted a national emergency. Sweeping changes were made throughout the nation, including social distancing, in an effort to control the spread of the virus. The following graph gives the
In Exercises, find each indefinite integral. n- -U 2 − u² du
The following table shows the results from a chemical experiment.Repeat parts (a) and (b) of Exercise 23 for these data.Exercise 23The daily milk consumption (in kilograms) for calves can be approximated by the function y = b0wb1e-b2 w, where w is the age of the calf (in weeks) and b0 , b1 , and b2
In Exercises, find each indefinite integral. [(₁². (x² - 5x)4(2x - 5) dx
In Exercises, find each indefinite integral. (3 In z + 2)4 Z dz
Find ∫40 (2x + 3) dx by using the formula for the area of a trapezoid: A =(1/2)(B + b)h, where B and b are the lengths of the parallel sides and h is the distance between them. Compare with Exercise 43.Exercise 43Approximate the area under the graph of ƒ(x) = 2x + 3 and above the x-axis from x =
Even when money flow is not continuous, it is often useful to treat it as if it were.Determine whether each statement is true or false, and explain why.
Integration by parts should be used to evaluateDetermine whether each of the following statements is true or false, and explain why. x² x³ + 1 S3 dx.
A particle is moving along a straight line with velocity v(t) = t2 - 2t. Its distance from the starting point after 3 seconds is 8 cm. Find s(t), the distance of the particle from the starting point after t seconds.
In Exercises, find each indefinite integral. -3e²x dx
In the study of bioavailability in pharmacy, a drug is given to a patient. The level of concentration of the drug is then measured periodically, producing blood level curves such as the ones shown in the figure.The areas under the curves give the total amount of the drug available to the patient
In the study of bioavailability in pharmacy, a drug is given to a patient. The level of concentration of the drug is then measured periodically, producing blood level curves such as the ones shown in the figure.The areas under the curves give the total amount of the drug available to the patient
In Exercises, find each indefinite integral. Зи u² - 1 np. du
In Exercises, find each indefinite integral. 5e* dx
In the study of bioavailability in pharmacy, a drug is given to a patient. The level of concentration of the drug is then measured periodically, producing blood level curves such as the ones shown in the figure.The areas under the curves give the total amount of the drug available to the patient
In Exercises, find each indefinite integral. 5 4 dy
In Exercises, find each indefinite integral. 2xe* dx
In Exercises, find each indefinite integral. 3x fre²r² xe dx
The daily milk consumption (in kilograms) for calves can be approximated by the function y = b0wb1e-b2 w, where w is the age of the calf (in weeks) and b0, b1, and b2 are constants.(a) The age in days is given by t = 7w. Use this fact to convert the function above to a function in terms of t.(b)
The growth rate of a certain tree (in feet) is given bywhere t is time (in years). Find the total growth from t = 1 to t = 7 by using n = 12 with the following methods.(a) The trapezoidal rule (b) Simpson’s rule y = 2 t + 2 +e-1²/12
In Exercises, find each indefinite integral. -4 23 dy
The reaction rate to a new drug is given bywhere t is time (in hours) after the drug is administered. Find the total reaction to the drug from t = 1 to t = 9 by letting n = 8 and using the following methods.(a) The trapezoidal rule (b) Simpson’s rule = e-7² y = + 1 t + 1'
In Exercises, find each indefinite integral. [(2x-4/13 (2x4/3 + x-1/2) dx
In Exercises, find each indefinite integral. [(x¹/2 J (x1/2 + 3x-2/3) dx
A company’s marginal costs (in thousands of dollars per year) were as follows over a certain period.Repeat parts (a) and (b) of Exercise 23 for these data to find the total cost over the given period.Exercise 23In Exercise , we estimated the total U.S. wind energy consumption (in trillion BTUs)
In Exercises, find each indefinite integral. Vt 2 dt
As we saw in an earlier chapter, the average individual daily milk consumption for herds of Charolais, Angus, and Hereford calves can be described by a mathematical function. Here we write the consumption in kg/day as a function of the age of the calf in days (t) as M(t) =
An electronics company analyst has determined that the rate per month at which revenue (in hundreds of dollars) comes in from the calculator division is given by R(x) = 105e0.01x + 32, where x is the number of months the division has been in operation. Find the total revenue between the 12th and
In Exercises, find each indefinite integral. V5 In z + 3 Z dz
Let ƒ(x) = 3x + 1, x1 = -1, x2 = 0, x3 = 1, x4 = 2, and x5 = 3. Find 5 Σ f(x;). i=1
In Exercises, find each indefinite integral. ₂3x²+4x dx
Find ∫04 ƒ(x) dx for each graph of y = ƒ(x).(a)(b) 1 y 0 X
Find each definite integral. [²(3x² + 5) dx
Find each definite integral. -5 1 (3x¹ + x³) dx
Find each definite integral. -6 (2x² + x) dx
Approximate the area under the graph of ƒ(x) = 2x + 3 and above the x-axis from x = 0 to x = 4 using four rectangles. Let the height of each rectangle be the function value on the left side.
Find each definite integral. 2 3e-2w dw 0
In Exercises 35 and 36 of the section on Area and the Definite Integral, you calculated the distance that a car traveled by estimating the integral ∫0T v(t) dt.(a) Let s(t) represent the mileage reading on the odometer. Express the distance traveled between t = 0 and t = T using the function
Find each definite integral. -3 [²(2x²¹ + x^²) dx 1
What does the Fundamental Theorem of Calculus state?
Find each definite integral. 0 xV5x² + 4 dx
Find each definite integral. 2 0 x²(3x³ + 1)¹/3 dx
Use the substitution u = x2 and the equation of a semicircle to evaluate V₂ Jo 4xV4 - x4 dx.
In Exercises, use substitution to change the integral into one that can be evaluated by a formula from geometry, and then find the value of the integral. es V25 - (In x)² X -dx
Find each definite integral. 2 e0.4w dw
Use the substitution u = 4x2 and the equation of a semicircle to evaluate - 1/2 Jo XVI - 16x dx.
In Exercises, find the area between the x-axis and f(x) over each of the given intervals. f(x) = √4x − 3; [1,3] -
In Exercises, use substitution to change the integral into one that can be evaluated by a formula from geometry, and then find the value of the integral. Vi ľ 2x√36 (x² - 1)² dx
Use the trapezoidal rule with n = 4 to approximate the value of each integral. Then find the exact value and compare the two answers. -3 In x X dx
In Exercises, find the area between the x-axis and f(x) over each of the given intervals.f(x) = (3x + 2)6; [-2,0]
In Exercises, find the area between the x-axis and f(x) over each of the given intervals.ƒ(x) = xex2; [0, 2]
Use Simpson’s rule with n = 4 to approximate the value of each integral. Compare your answers with the answers to Exercises -3 In x - dx J1 X
In Exercises, find the area between the x-axis and f(x) over each of the given intervals.ƒ(x) = 1 + e-x; [0, 4]
Use the trapezoidal rule with n = 4 to approximate the value of each integral. Then find the exact value and compare the two answers. -10 12 x dx x - 1
Find the area of the region enclosed by each group of curves.ƒ(x) = 5 - x2, g(x) = x2 - 3
Use the trapezoidal rule with n = 4 to approximate the value of each integral. Then find the exact value and compare the two answers. Leve 0 e*ve* + 4 dx et
Find the area of the region enclosed by each group of curves.ƒ(x) = x2 - 4x, g(x) = x - 6
Use the trapezoidal rule with n = 4 to approximate the value of each integral. Then find the exact value and compare the two answers. 2 0 -x² xe dx
Use Simpson’s rule with n = 4 to approximate the value of each integral. Compare your answers with the answers to Exercises 2 10 xe - dx
Find the area of the region enclosed by each group of curves.ƒ(x) = x2 - 4x, g(x) = x + 6, x = -2, x = 4
Find the area of the region enclosed by each group of curves.ƒ(t) = 5 - t2, g(t) = t2 - 3, t = 0, t = 4
Use Simpson’s rule with n = 4 to approximate the value of each integral. Compare your answers with the answers to Exercises 10 J2 x dx x - 1
Use Simpson’s rule with n = 4 to approximate the value of each integral. Compare your answers with the answers to Exercises e* Ve* + 4 dx et 0
GivenandcalculateChoose one of the following.(a) 3/2 (b) 3 (c) 4(d) 6 (e) 8 f(x) dx = 3
Find the area of the region between the graphs of y = √x - 1 and 2y = x - 1 from x = 1 to x = 5 in three ways.(a) Use antidifferentiation.(b) Use the trapezoidal rule with n = 4.(c) Use Simpson’s rule with n = 4.
Find the cost function for each of the marginal cost functions in Exercises. C'(x) 8 2x + 1' -; fixed cost is $18.
The graph below shows recent values (in billions of dollars per year) of the U.S. deficit, which gives the rate that the U.S. debt accumulates. Use rectangles of width 2 years and height determined by the function value at the midpoint to estimate the total amount of debt accumulated from mid-2009
Find the area of the region between the graphs of y = 1/x + 1 and y = x + 2/2 from x = 0 to x = 4 in three ways.(a) Use antidifferentiation.(b) Use the trapezoidal rule with n = 4.(c) Use Simpson’s rule with n = 4.
Let ƒ(x) = [x(x - 1)(x + 1)(x - 2)(x + 2)]2.(a) Find ∫2-2 ƒ(x) dx using the trapezoidal rule with n = 4.(b) Find ∫2-2 ƒ(x) dx using Simpson’s rule with n = 4.
The rate of change of sales of a new brand of tomato soup (in thousands of dollars per month) is given bywhere t is the time (in months) that the new product has been on the market. Find the total sales after 4 months. S' (t) = 3√2t + 1 + 3,
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