Calculate the following integrals. 1∫24/x5 dx
> Suppose that the interval 0 ≤ t ≤ 3 is divided into 1000 subintervals of width Δt. Let t1, t2, … , t1000 denote the right endpoints of these subintervals. If we need to estimate the
> For each number x satisfying -1 … x … 1, define h(x) by h(x) = -1∫x √ (1 - t2) dt. (a) Give a geometric interpretation of the values h(0) and h(1). (b) Find the derivative h ‘(x).
> Let x be any positive number, and define g(x) to be the number determined by the definite integral g(x) = 0∫x 1/(1 + t2) dt. (a) Give a geometric interpretation of the number g(3). (b) Find the derivative g ‘ (x).
> A retail store sells a certain product at the rate of g(t) units per week at time t, where g(t) = rt. At time t = 0, the store has Q units of the product in inventory. (a) Find a formula f (t) for the amount of product in inventory at time t. (b) Determi
> A store has an inventory of Q units of a certain product at time t = 0. The store sells the product at the steady rate of Q/A units per week and exhausts the inventory in A weeks. (a) Find a formula f (t) for the amount of product in inventory at time t.
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under the curve y = 1 - x2 from x = 0 to x = 1.
> In Fig. 2, three regions are labeled with their areas. Determine a∫c f (x) dx and determine a∫d f (x) dx. Figure 2: y a .68 b .42 1.7 d
> Suppose that the interval 0 ≤ x ≤ 1 is divided into 100 subintervals with a width of Δx = .01. Show that the sum [3e-0.01] Δx + [3e-0.02] Δx + [3e-0.03] Δx + … + [3e-1] Δx is close to 3(1 - e-1).
> Find the value of k that makes the antidifferentiation formula true. ∫7/(8 - x)4 dx = k/(8 - x)3 + C
> Find the average value of f (x) = 1/x3 from x = 1/3 to x = 1/2.
> Three thousand dollars is deposited in the bank at 4% interest compounded continuously. What will be the average value of the money in the account during the next 10 years?
> Find the consumers’ surplus for the demand curve p = √(25 - .04x) at the sales level x = 400.
> Use a Riemann sum with n = 5 and midpoints to estimate the area under the graph of f (x) = e2x on the interval 0 ≤ x ≤ 1. Then, use a definite integral to find the exact value of the area to five decimal places.
> Use a Riemann sum with n = 2 and midpoints to estimate the area under the graph of f (x) = 1/(x + 2) on the interval 0 ≤ x ≤ 2. Then, use a definite integral to find the exact value of the area to five decimal places.
> Redo Exercise 47 using right endpoints. Exercise 47: Use a Riemann sum with n = 4 and left endpoints to estimate the area under the graph in Fig. 1 for 0 ≤ x ≤ 2. Figure 1: 20 10 ม 0 (.5, 14) .5 (1, 10) 1 (1.
> Use a Riemann sum with n = 4 and left endpoints to estimate the area under the graph in Fig. 1 for 0 ≤ x ≤ 2. Figure 1: 20 10 ม 0 (.5, 14) .5 (1, 10) 1 (1.5, 6) 1.5 (2,4) 2
> A rock thrown straight up into the air has a velocity of υ(t) = -9.8t + 20 meters per second after t seconds. (a) Determine the distance the rock travels during the first 2 seconds. (b) Represent the answer to part (a) as an area.
> A drug is injected into a patient at the rate of f (t) cubic centimeters per minute at time t. What does the area under the graph of y = f (t) from t = 0 to t = 4 represent?
> If the marginal revenue function for a company is 400 - 3x2, find the additional revenue received from doubling production if 10 units are currently being produced.
> Find the value of k that makes the antidifferentiation formula true. ∫ (4 - x)-1 dx = k ln |4 – x| + C
> An airplane tire plant finds that its marginal cost of producing tires is .04x + $150 at a production level of x tires per day. If fixed costs are $500 per day, find the cost of producing x tires per day.
> Let k be a constant, and let y = f (t) be a function such that y ’ = kty. Show that y = Cekt2/2, for some constant C.
> Describe all solutions of the following differential equations, where y represents a function of t. (a) y ’ = 4t (b) y ’ = 4y (c) y ’ = e4t
> Find the function f (x) for which f ‘(x) = e-5x, f (0) = 1.
> Find the function f (x) for which f ‘ (x) = (x - 5)2, f (8) = 2.
> Find the area of the region between the curves y = 2x2 + x and y = x2 + 2 from x = 0 to x = 2.
> Find the area of the region bounded by the curves y = x3 - 3x + 1 and y = x + 1.
> Find the area of the shaded region. -2 ม y = 2 – 2 – in 3
> Find the area of the shaded region. Y 0 y=e² - ex 1
> Find the area of the shaded region. Y 0 1/2 y=x²-x-1 y = 1-1/x 1 2
> Find the value of k that makes the antidifferentiation formula true. ∫√(x + 1) dx = k(x + 1)3/2 + C
> Find all antiderivatives of each following function: f (x) = e3x
> Find the area of the shaded region. 4 0 y=4-2² 1 y=1-x² 2
> Find the area of the shaded region. 1.21 3 = fi X} = f 0 "
> Find the area of the shaded region. y=e-z Y 0 1 y=e² In 2 1
> Find the area of the shaded region. -2 =R Y 0 y = x³ + 2x 2
> Find the area of the shaded region. x x=k xt = f 0 fi
> Find the area under the curve y = 1 + √x from x = 1 to x = 9.
> Find the area under the curve y = (3x - 2)-3 from x = 1 to x = 2.
> Calculate the following integrals. 0∫1 (3 + e2x)/ex dx
> Calculate the following integrals. 0∫ln 3 (ex + e-x)/ e2x dx
> Calculate the following integrals. ln 2∫ln 3 (ex + e-x) dx
> Find the value of k that makes the antidifferentiation formula true. ∫ (5x - 7)-2 dx = k(5x - 7)-1 + C
> Calculate the following integrals. 0∫ln 2 (ex - e-x) dx
> Calculate the following integrals. -2∫2 3/2e3x dx
> Calculate the following integrals. 0∫5 (5 + 3x)-1 dx
> Calculate the following integrals. 3∫6 e2-(x/3) dx
> Calculate the following integrals. 1∫4 1/x2 dx
> Calculate the following integrals. 2/3 0∫8 √ (x + 1) dx
> Calculate the following integrals. 0∫1 [2/(x + 1) – 1/(x + 4)] dx
> Calculate the following integrals. -1∫2 √ (2x + 4) dx
> Calculate the following integrals. 0∫1/8 5√x dx
> Find the value of k that makes the antidifferentiation formula true. ∫4 e3x+1 dx = k e3x+1 + C
> Calculate the following integrals. -1∫1 (x + 1)2 dx
> Calculate the following integrals. ∫ (5/x – x/5) dx
> Calculate the following integrals. ∫ √(4 – x) dx
> Calculate the following integrals. ∫ (2x + 3)7 dx
> Calculate the following integrals. ∫ (3x4 - 4x3) dx
> Calculate the following integrals. ∫5/√(x – 7) dx
> Calculate the following integrals. ∫ e-x/2 dx
> Calculate the following integrals. ∫ 5√(x + 3) dx
> What is a Riemann sum?
> In the formula Δx = (b – a)/n, what do a, b, n, and Δx denote?
> Find the value of k that makes the antidifferentiation formula true. ∫2e4x-1 dx = ke4x-1 + C
> State the formula for 1h(x)dx for each of the following functions. (a) h(x)xr, r ≠ -1 (b) h(x) = ekx (c) h(x) = 1/x (d) h(x) = f (x) + g(x) (e) h(x) = kf (x)
> What does it mean to anti-differentiate a function?
> State the formula for each of the following quantities: (a) average value of a function (b) consumers’ surplus (c) future value of an income stream (d) volume of a solid of revolution
> Outline a procedure for finding the area of a region bounded by two curves.
> How is F (x) |ba calculated, and what is it called?
> State the fundamental theorem of calculus.
> What is the difference between a definite integral and an indefinite integral?
> What is a definite integral?
> Give an interpretation of the area under a rate of change function. Give a concrete example.
> Determine the average value of f (x) over the interval from x = a to x = b, where f (x) = 100e-0.5x; a = 0, b = 4.
> Find the value of k that makes the antidifferentiation formula true. ∫3et/10 dt = ket/10 + C
> Determine the average value of f (x) over the interval from x = a to x = b, where f (x) = 1 - x; a = -1, b = 1.
> Determine the average value of f (x) over the interval from x = a to x = b, where f (x) = x2; a = 0, b = 3.
> The following exercises ask for an unknown quantity x. After setting up the appropriate formula involving a definite integral, use the fundamental theorem to evaluate the definite integral as an expression in x. Because the resulting equation will be too
> The following exercises ask for an unknown quantity x. After setting up the appropriate formula involving a definite integral, use the fundamental theorem to evaluate the definite integral as an expression in x. Because the resulting equation will be too
> The following exercises ask for an unknown quantity x. After setting up the appropriate formula involving a definite integral, use the fundamental theorem to evaluate the definite integral as an expression in x. Because the resulting equation will be too
> The following exercises ask for an unknown quantity x. After setting up the appropriate formula involving a definite integral, use the fundamental theorem to evaluate the definite integral as an expression in x. Because the resulting equation will be too
> For the Riemann sum on an interval [a, b], determine n, b, and f (x). Suppose that the interval 0 ≤ x ≤ 1 is divided into 100 subintervals of width Δx = .01. Show that the following sum is close to 5/4. [2(.01) + (.01)3]Δx + [2(.02) + (.02)3] Δx + … + [2
> For the Riemann sum on an interval [a, b], determine n, b, and f (x). Suppose that the interval 0 ≤ x ≤ 3 is divided into 100 subintervals of width Δx = .03. Let x1, x2, … , x100 be points in these subintervals. Suppose that in a particular application w
> For the Riemann sum on an interval [a, b], determine n, b, and f (x). [3(.3)2 + 3(.9)2 + 3(1.5)2 + 3(2.1)2 + 3(2.7)2] (.6); a = 0
> For the Riemann sum on an interval [a, b], determine n, b, and f (x). [(5 + e5) + (6 + e6) + (7 + e7)] (1); a = 4
> Find the value of k that makes the antidifferentiation formula true. ∫5e-2t dt = ke-2t + C
> For the Riemann sum on an interval [a, b], determine n, b, and f (x). [3/1 + 3/1.5 + 3/2 + 3/2.5 + 3/3 + 3/3.5] (.5); a = 1
> For the Riemann sum on an interval [a, b], determine n, b, and f (x). [(8.25)3 + (8.75)3 + (9.25)3 + (9.75)3](.5); a = 8
> 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
> 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 (The solid generated is called a truncated cone.)
> 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
> 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 from x = 0 to x = 4 (The solid generated is called a paraboloid.)
> 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 (generates a cone)
> 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
> 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 (generates a sphere of radius r)
> 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 (generates a sphere of radius 2)
> Determine the following: ∫ (-3e-x + 2x - e0.5x/2) 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 = -x2 + 1 from x = 0 to x = 1.
> 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.
> 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?
