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mathematics
applied calculus

Questions and Answers of Applied Calculus

Find all functions f(t) that satisfy the given condition. f'(t) 4 6 + t
For the Riemann sums on an interval [a, b], in Exercises, determine n, b, and f (x).[(5 + e5) + (6 + e6) + (7 + e7)] (1); a = 4
Find all functions f(t) that satisfy the given condition.f′(t) = 0
Use a Riemann sum to approximate the area under the graph of f(x) in Fig. 14 on the given interval, with selected points as specified. Draw the approximating rectangles.0 ≤ x ≤ 8, n = 4,
Find the area of the region between the curves y = 2x2 + x and y = x2 + 2 from x = 0 to x = 2.
A property with an appraised value of $200,000 in 2015 is depreciating at the rate R(t) = -8e-0.04t, where t is in years since 2015 and R(t) is in thousands of dollars per year. Estimate the loss in
Find all functions f(t) that satisfy the given condition. f'(t) = t³/2
Suppose that the velocity of a car at time t is y(t) = 40 + 8/(t + 1)2 kilometers per hour.(a) Compute the area under the velocity curve from t = 1 to t = 9.(b) What does the area in part (a)
Find the area of the region bounded by the curves y = x3 - 3x + 1 and y = x + 1.
In Exercises, find the value of k that makes the antidifferentiation formula true. 5 2 - 3x S dx = k ln |2 - 3x + C
An investment grew at an exponential rate R(t) = 700e0.07t + 1000, where t is in years and R(t) is in dollars per year. Approximate the net increase in value of the investment after the first 10
For the Riemann sums on an interval [a, b], in Exercises, determine n, b, and f (x).[(8.25)3 + (8.75)3 + (9.25)3 + (9.75)3](.5); a = 8
Some food is placed in a freezer. After t hours, the temperature of the food is dropping at the rate of r(t) degrees Fahrenheit per hour, where r(t) = 12 + 4/(t + 3)2.(a) Compute the area under the
Use a Riemann sum to approximate the area under the graph of f(x) on the given interval, with selected points as specified.f (x) = ln x; 2 ≤ x ≤ 4, n = 5, left endpoints
A company’s marginal cost function is given by C′(x) = 32 + x/ 20, where x denotes the number of items produced in 1 day and C(x) is in thousands of dollars. Determine the increase in cost if the
Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves.y = e-x from x = 0 to x = 1
In Exercises, find the value of k that makes the antidifferentiation formula true. S 3 - dx = k ln |2 + x] + C 2 + x
Let M(x) be a company’s marginal cost at production level x. Give an economic interpretation of the number ∫0100 M(x)dx.
Use a Riemann sum to approximate the area under the graph of f(x) on the given interval, with selected points as specified.f (x) = e-x; 2 ≤ x ≤ 3, n = 5, right endpoints
A company’s marginal cost function is .1x2 - x + 12 dollars, where x denotes the number of units produced in 1 day.(a) Determine the increase in cost if the production level is raised from x = 1 to
Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves.y = 2x + 1 from x = 0 to x = 1
In Exercises, find the value of k that makes the antidifferentiation formula true. (2x − 1)³ dx = k(2x − 1)ª + C -
Let M(x) be a company’s marginal profit at production level x. Give an economic interpretation of the number ∫4448 M(x)dx.
Use a Riemann sum to approximate the area under the graph of f(x) on the given interval, with selected points as specified.f (x) = x3; 0 ≤ x ≤ 1, n = 5, right endpoints
In Exercises, find the value of k that makes the antidifferentiation formula true. f(3x (3x + 2) dx = k(3x + 2)5 + C
The velocity of a skydiver at time t seconds is y(t) = 45 - 45e-0.2t meters per second. Find the distance traveled by the skydiver the first 9 seconds.
Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves.y = 2x - x2 from x = 0 to x = 2
Suppose that the marginal profit function for a company is P′(x) = 100 + 50x - 3x2 at production level x.(a) Find the extra profit earned from the sale of 3 additional units if 5 units are
Suppose that the marginal cost function of a handbag manufacturer isdollars per unit at production level x (where x is measured in units of 100 handbags).(a) Find the total cost of producing 6
Use a Riemann sum to approximate the area under the graph of f(x) on the given interval, with selected points as specified.f (x) = x3; 1 ≤ x ≤ 3, n = 5, left endpoints
The velocity at time t seconds of a ball thrown up into the air is y(t) = -32t + 75 feet per second.(a) Compute the displacement of the ball during the time interval 1 ≤ t ≤ 3.(b) Is the position
Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curvesy = √x from x = 0 to x = 4
In Exercises, find the value of k that makes the antidifferentiation formula true. √ 7 (8 - x)4 dx = k (8 - x)³ 3 + C
Use a Riemann sum to approximate the area under the graph of f(x) on the given interval, with selected points as specified.f (x) = x2; -2 ≤ x ≤ 2, n = 4, midpoints of subintervals
The velocity at time t seconds of a ball thrown up into the air is y(t) = -32t + 75 feet per second.(a) Find the displacement of the ball during the time interval 0 ≤ t ≤ 3.(b) Given that the
Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves.y = kx from x = 0 to x = h
After t hours of operation, an assembly line is producing lawn mowers at the rate of r(t) = 21 - 4/5 t mowers per hour.(a) How many mowers are produced during the time from t = 2 to t = 5 hours?(b)
Use a Riemann sum to approximate the area under the graph of f(x) on the given interval, with selected points as specified.f(x) = x2; 1 ≤ x ≤ 3, n = 4, midpoints of subintervals
Refer to Fig. 7 and evaluate p2 -1 f(t)dt.
In Exercises, find the value of k that makes the antidifferentiation formula true. [(4-x)-¹α - x) dx = k ln |4 = x + C -
In Exercises, find the value of k that makes the antidifferentiation formula true. S √x + 1 dx = k(x + 1)³/2 + C
A rock is dropped from the top of a 400-foot cliff. Its velocity at time t seconds is y(t) = -32t feet per second. Find the displacement of the rock during the time interval 2 ≤ t ≤ 4.
Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves.y = x2 from x = 1 to x = 2
Refer to Fig. 6 and evaluate L -1 f(t)dt.
A helicopter is rising straight up in the air. Its velocity at time t is v(t) = 2t + 1 feet per second.(a) How high does the helicopter rise during the first 5 seconds?(b) Represent the answer to
Determine Δx and the midpoints of the subintervals formed by partitioning the given interval into n subintervals.3 ≤ x ≤ 5; n = 5
Find the area in Fig. 27 of the region bounded by y = 1/x2, y = x, and y = 8x, for x ≥ 0. y Figure 27 y = 8x y y = x X
Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves.y = √r2 - x2 from x = -r to x = r
In Exercises, find the value of k that makes the antidifferentiation formula true. Jes - - 7)-² dx = k(5x −7)¹ + C -
Find the area of the region bounded by y = 1/x, y = 4x, and y = x/2, for x ≥ 0.
Refer to Fig. 5 and evaluate .3 S f(x)dx.
Determine Δx and the midpoints of the subintervals formed by partitioning the given interval into n subintervals.1 ≤ x ≤ 4; n = 5
In Exercises, find the value of k that makes the antidifferentiation formula true. [264x=1dx=kets=1+ C
Refer to Fig. 4 and evaluate 2 f(x)dx.
Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves.y = √4 - x2 from x = -2 to x = 2
In Exercises, find the value of k that makes the antidifferentiation formula true. Sa 4 e³x+1 dx = k + C e³x+1
Find the area under the curve y = 1 + √x from x = 1 to x = 9.
Determine Δx and the midpoints of the subintervals formed by partitioning the given interval into n subintervals.0 ≤ x ≤ 3; n = 6
Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves.y = -x2 + 1 from x = 0 to x = 1.
Find the area of the region between y = x2 and y = 1/x2(a) From x = 1 to x = 4,(b) From x = 1/2 to x = 4.
Find the area of the region between y = x2 - 3x and the x-axis(a) from x = 0 to x = 3,(b) from x = 0 to x = 4,(c) from x = -2 to x = 3.
Find the area under the curve y = (3x - 2)-3 from x = 1 to x = 2.
Find the real number b > 0 so that the area under the graph of y = x2 from 0 to b is equal to the area under the graph of y = x3 from 0 to b.
Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves.y = x + 1 from x = 0 to x = 2.
Determine Δx and the midpoints of the subintervals formed by partitioning the given interval into n subintervals.0 ≤ x ≤ 2; n = 4
Calculate the following integrals. S 0 2x 3 + ²x et dx
Calculate the following integrals. In 3 S et + e e²x X dx
In Exercises, find the value of k that makes the antidifferentiation formula true. 3e¹/10 dt = ket/10 + C
In Exercises, find the value of k that makes the antidifferentiation formula true. [5e-21 5e-2t dt = ke-2t + C
Find the area under each of the given curves. y = @³x; x = − to x = 0
A savings account pays 4.25% interest compounded continuously. At what rate per year must money be deposited steadily in the account to accumulate a balance of $100,000 after 10 years?
Find the real number b > 0 so that the area under the graph of y = x3 from 0 to b is equal to 4.
Calculate the following integrals. In 2 S 0 (e² - edx
Calculate the following integrals. ln 3 In 2 (ex + e)dx
An investment pays 10% interest compounded continuously. If money is invested steadily so that $5000 is deposited each year, how much time is required until the value of the investment reaches
Calculate the following integrals. 2 L 3 -22e³x dx
Suppose that money is deposited steadily in a savings account so that $14,000 is deposited each year. Determine the balance at the end of 6 years if the account pays 4.5% interest compounded
Find the area under each of the given curves.y = (x - 3)4; x = 1 to x = 4
Combine the integrals into one integral, then evaluate the integral. L'o (7x + 4)dx + Lo (7x + 5)dx
Suppose that money is deposited steadily in a savings account so that $16,000 is deposited each year. Determine the balance at the end of 4 years if the account pays 8% interest compounded
Find the area under each of the given curves.y = √x; x = 0 to x = 4
Calculate the following integrals. 5 fo 0 -1 (5 + 3x)-¹ dx
Combine the integrals into one integral, then evaluate the integral. .0 (x³ + x²)dx -L'0x³. + (x³ + x²) dx
Calculate the following integrals. 3 6 e2-(x/3) dx
Suppose that money is deposited daily in a savings account at an annual rate of $2000. If the account pays 6% interest compounded continuously, approximately how much will be in the account at the
Find the area under each of the given curves.y = 3x2 + x + 2ex/2; x = 0 to x = 1
Combine the integrals into one integral, then evaluate the integral. √ (4x- L' (4x - 2)dx + 3 (x - 1)dx
For a particular commodity, the quantity produced and the unit price are given by the coordinates of the point where the supply and demand curves intersect. For each pair of supply and demand curves,
Combine the integrals into one integral, then evaluate the integral. 2 ² √₁² (³x + 1² x ² − x ³ ) dx + 3 √ ₁² (x² - 2 3x - (x² - 2x + 7)dx
Suppose that money is deposited daily in a savings account at an annual rate of $1000. If the account pays 5% interest compounded continuously, estimate the balance in the account at the end of 3
For a particular commodity, the quantity produced and the unit price are given by the coordinates of the point where the supply and demand curves intersect. For each pair of supply and demand curves,
Find the area under each of the given curves.y = 3x2; x = -1 to x = 1
Draw the region whose area is given by the definite integral. 5. Vx dx
Calculate the following integrals. 4 -| 1 X dx
Calculate the following integrals. 2|3 .8 √x + 1 dx
Find the area under each of the given curves.y = 4x; x = 2 to x = 3
Figure 8 shows a supply curve for a commodity. It gives the relationship between the selling price of the commodity and the quantity that producers will manufacture. At a higher selling price, a
Figure 8 shows a supply curve for a commodity. It gives the relationship between the selling price of the commodity and the quantity that producers will manufacture. At a higher selling price, a
Draw the region whose area is given by the definite integral. √² (8- (8 - 2x)dx
Calculate the following integrals. S 4 X dx

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