[SOLVED] Evaluate the given integral. 1∫4 (x2 -

> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. f (x, y) = x2 - 3y2 + 4x + 6y + 8

> A company manufactures and sells two competing products, I and II, that cost $pI and $pII per unit, respectively, to produce. Let R(x, y) be the revenue from marketing x units of product I and y units of product II. Show that if the company’s profit is m

> A company manufactures and sells two products, I and II, that sell for $p1 and $p2 per unit, respectively. Let C(x, y) be the cost of producing x units of product I and y units of product II. Show that if the companyâs profit is maximiz

> A monopolist manufactures and sells two competing products, I and II, that cost $30 and $20 per unit, respectively, to produce. The revenue from marketing x units of product I and y units of product II is 98x + 112y - .04xy - .1x2 - .2y2. Find the values

> A company manufactures and sells two products, I and II, that sell for $10 and $9 per unit, respectively. The cost of producing x units of product I and y units of product II is 400 + 2x + 3y + .01(3x2 + xy + 3y2). Find the values of x and y that maxim

> Find the dimensions of the rectangular box of least surface area that has a volume of 1000 cubic inches.

> U.S. postal rules require that the length plus the girth of a package cannot exceed 84 inches. Find the dimensions of the rectangular package of greatest volume that can be mailed. [Note: From Fig. 5 we see that 84 = (length) + (girth) = l + (2x + 2y).]

> Find the possible values of x, y, z at which f (x, y, z) = 5 + 8x - 4y + x2 + y2 + z2 assumes its minimum value.

> The present value of A dollars to be paid t years in the future (assuming a 5% continuous interest rate) is P(A, t) = Ae-0.05t. Find and interpret P(100, 13.8).

> Find the possible values of x, y, z at which f (x, y, z) = 2x2 + 3y2 + z2 - 2x - y – z assumes its minimum value.

> Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of f (x, y) at each of these points. If the second-derivative test is inconclusive, so state. f (

> Let f (x, y) = 10x2/5y3/5. Show that f (3a, 3b) = 3f (a, b).

> Let f (x, y) = x2 - 3xy - y2. Compute f (5, 0), f (5,-2), and f (a, b).

> The velocity of a skydiver at time t seconds is υ(t) = 45 - 45e-0.2t meters per second. Find the distance traveled by the skydiver the first 9 seconds.  

> The velocity at time t seconds of a ball thrown up into the air is υ(t) = -32t + 75 feet per second. (a) Compute the displacement of the ball during the time interval 1 ≤ t ≤ 3. (b) Is the position of the ball at time t = 3 higher than its position at ti

> The velocity at time t seconds of a ball thrown up into the air is υ(t) = -32t + 75 feet per second. (a) Find the displacement of the ball during the time interval 0 ≤ t ≤ 3. (b) Given that the initial position of the ball is s(0) = 6 feet, use (a) to de

> A rock is dropped from the top of a 400-foot cliff. Its velocity at time t seconds is υ(t) = -32t feet per second. Find the displacement of the rock during the time interval 2 ≤ t ≤ 4.

> Refer to Fig. 7 and evaluate -1â«2 f (t)dt. Figure 7: 0 f(t) y=t-1 S || =²-1 2

> Refer to Fig. 6 and evaluate -1â«1 f (t)dt. Figure 6: I 7-1=k 0 I- 7+T=; DL

> Refer to Fig. 5 and evaluate 0â«3 f (x)dx. Figure 5: y y¹ 0 y=1-x² y = (1-x)(x − 3) 3 8 -

> Refer to Fig. 4 and evaluate 0â«2 f (x)dx. Figure 4: 2 4 0 ม y 1 บ y=r T

> Use formula (8) to help you answer the question. Given f (t) = -12t â 1/et, compute f (3) - f (0). Formula (8): [ F'(x)dx= F(b) - F(a).

> Use formula (8) to help you answer the question. Given f â(t) = -.5t + e-2t, compute f (1) - f (-1). Formula (8): [ F'(x)dx= F(b) - F(a).

> Determine the following: ∫7 dx

> Use formula (8) to help you answer the question. Given f â(x) = 73, compute f (4) - f (2). Formula (8): [ F'(x)dx= F(b) - F(a).

> Use formula (8) to help you answer the question. Given f â(x) = -2x + 3, compute f (3) - f (1). Formula (8): [ F'(x)dx= F(b) - F(a).

> Combine the integrals into one integral, then evaluate the integral. 0∫1 (7x + 4) dx + 1∫2 (7x + 5) dx

> Combine the integrals into one integral, then evaluate the integral. -1∫0 (x3 + x2) dx + 0∫1 (x3 + x2) dx

> Combine the integrals into one integral, then evaluate the integral. 0∫1 (4x - 2) dx + 3 0∫1 (x - 1) dx

> Combine the integrals into one integral, then evaluate the integral. 2 1∫2 (3x + 1/2 x2 - x3) dx + 3 1∫2 (x2 - 2x + 7) dx

> Given -0.5∫3 f (x)dx = 0 and -0.5∫3 (2g(x) + f (x))dx = -4, find -0.5∫3 g(x)dx.

> Given 1∫3f (x)dx = 3 and 1∫3 g(x)dx = -1, find 1∫3 (2f (x) - 3g(x)) dx.

> Given -1∫1 f (x)dx = 0 and -1∫10 f (x)dx = 4, find 1∫10 f (x)dx.

> Given 0∫1 f (x)dx = 3.5 and 1∫4 f (x)dx = 5, find 0∫4 f (x)dx.

> Determine the following: ∫x/3 dx

> Evaluate the given integral. 0∫ln 2 (ex + e-x)/2 dx

> Evaluate the given integral. 0∫1 (ex + e0.5x)/e2x dx

> Evaluate the given integral. -2∫-1 (1 + x)/x dx

> Evaluate the given integral. 1∫2 2/x dx

> Evaluate the given integral. -2∫2 2/e2t dt

> Evaluate the given integral. -1∫0 (3e3t + t) dt

> Evaluate the given integral. 1∫2 (5 - 2x3)/x6 dx

> Evaluate the given integral. 1∫8 (-x + 3√x) dx

> Evaluate the given integral. 1∫2 -3/x2 dx

> Determine the following: ∫4x3 dx

> Evaluate the given integral. 1∫9 1/√x dx

> Plot the graph of the solution of the differential equation y = e-x2, y(0) = 0. Observe that the graph approaches the value √π/2 ≈ .9 as x increases.

> Find an antiderivative of f (x), call it F (x), and compare the graphs of F (x) and f (x) in the given window to check that the expression for F (x) is reasonable. [That is, determine whether the two graphs are consistent. When F (x) has a relative extre

> Drilling of an oil well has a fixed cost of $10,000 and a marginal cost of C ‘(x) = 1000 + 50x dollars per foot, where x is the depth in feet. Find the expression for C(x), the total cost of drilling x feet.

> Since 1987, the rate of production of natural gas in the United States has been approximately R(t) quadrillion British thermal units per year at time t, with t = 0 corresponding to 1987 and R(t) = 17.04e0.016t. Find a formula for the total U.S. productio

> The United States has been consuming iron ore at the rate of R(t) million metric tons per year at time t, where t = 0 corresponds to 1980 and R(t) = 94e0.016t. Find a formula for the total U.S. consumption of iron ore from 1980 until time t.

> A soap manufacturer estimates that its marginal cost of producing soap powder is C ‘(x) = .2x + 100 dollars per ton at a production level of x tons per day. Fixed costs are $200 per day. Find the cost of producing x tons of soap powder per day.

> A small tie shop finds that at a sales level of x ties per day, its marginal profit is MP(x) dollars per tie, where MP(x) = 1.30 + .06x - .0018x2. Also, the shop will lose $95 per day at a sales level of x = 0. Find the profit from operating the shop at

> A flu epidemic hits a town. Let P(t) be the number of persons sick with the flu at time t, where time is measured in days from the beginning of the epidemic and P(0) = 100. After t days, if the flu is spreading at the rate of P ‘(t) = 120t - 3t2 people

> Find all antiderivatives of each following function: f (x) = -4x

> A package of frozen strawberries is taken from a freezer at -5˚C into a room at 20˚C. At time t, the average temperature of the strawberries is increasing at the rate of T ‘(t) = 10e-0.4t degrees Celsius per hour. Find the temperature of the strawberrie

> After t hours of operation, a coal mine is producing coal at the rate of C ’(t) = 40 + 2t – 1/5 t2 tons of coal per hour. Find a formula for the total output of the coal mine after t hours of operation.

> Let P(t) be the total output of a factory assembly line after t hours of work. If the rate of production at time t is P’(t) = 60 + 2t – 1/4t2 units per hour, find the formula for P(t).

> 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. (a) Find s(t), the height of the rock above the ground at time t. (b) How long will the rock take to reach the ground? (c) What will be its

> A ball is thrown upward from a height of 256 feet above the ground, with an initial velocity of 96 feet per second. From physics it is known that the velocity at time t is y(t) = 96 - 32t feet per second. (a) Find s(t), the function giving the height abo

> The function g(x) in Fig. 9 resulted from shifting the graph of f (x) up 2 units. What is the derivative of h(x) = g(x) - f (x)? Figure 9: Y + 2 y = g(x) y = f(x) X

> The function g(x) in Fig. 8 resulted from shifting the graph of f (x) up 3 units. If f â(5) = 1/4 , what is g(5)? Figure 8: 3 ↓ 5 y = g(x) y = f(x) x

> Figure 7 contains an antiderivative of the function f (x). Draw the graph of another antiderivative of f (x). Figure 7: TH Y [T I

> Figure 6 contains the graph of a function F (x). On the same coordinate system, draw the graph of the function G(x) having the properties G (0) = 0 and G â(x) = F â(x) for each x. Figure 6: 2 -2 y H 4 y = F(x) 8

> Which of the following is ∫x√(x + 1) dx? (a) 2/5 (x + 1)5/2 – 2/3 (x + 1)3/2 + C (b) ½ x2 * 2/3 (x + 1)3/2 + C

> Find all antiderivatives of each following function: f (x) = 3

> Which of the following is ∫ln x dx? (a) 1/x + C (b) x * ln x - x + C (c) ½ * (ln x)2 + C

> Figure 5 shows the graphs of several functions f (x) for which f â(x) = 1/3. Find the expression for the function f (x) whose graph passes through (6, 3). Figure 5: ∞ 6 4 Y + 2 (6,3) 6 8 8 10 -x 12

> Figure 4 shows the graphs of several functions f (x) for which f (x) = 2/x. Find the expression for the function f â(x) whose graph passes through (1, 2). Figure 4: (1, 2) 6 4 2 y -2. -4- 4 6 + 8 | | | 10 12 X

> Find all functions f (x) that satisfy the given conditions. f ‘(x) = x2 + √x, f (1) = 3

> Find all functions f (x) that satisfy the given conditions. f ‘(x) = √x + 1, f (4) = 0

> Find all functions f (x) that satisfy the given conditions. f ‘(x) = 8x1/3, f (1) = 4

> Find all functions f (x) that satisfy the given conditions. f ‘(x) = x, f (0) = 3

> Find all functions f (x) that satisfy the given conditions. f ‘(x) = 2x - e-x, f (0) = 1

> Find all functions f (x) that satisfy the given conditions. f ‘(x) = .5e-0.2x, f (0) = 0

> Find all functions f (t) that satisfy the given condition. f ‘(t) = t2 - 5t - 7

> Find all antiderivatives of each following function: f (x) = e-3x

> Find all functions f (t) that satisfy the given condition. f ‘(t) = 0

> Find all functions f (t) that satisfy the given condition. f ‘(t) = 4/(6 + t)

> Find all functions f (t) that satisfy the given condition. f ‘(t) = t3/2

> Find the value of k that makes the antidifferentiation formula true. ∫5/(2 - 3x) dx = k ln |2 - 3x| + C

> Find the value of k that makes the antidifferentiation formula true. ∫3/(2 + x) dx = k ln |2 + x| + C

> Calculate the following integrals. ∫ (x3 + 3x2 - 1)dx

> Calculate the following integrals. ∫ 2/(x + 4) dx

> Calculate the following integrals. ∫ 2x + 1 dx

> Calculate the following integrals. ∫ (x2 - 3x + 2) dx

> Calculate the following integrals. ∫32 dx

> Generalize the result of Exercise 73 as follows: Let n be a positive integer. Show that 0∫1 (n√x – xn) dx = (n-1)/(n+1).

> Find the value of k that makes the antidifferentiation formula true. ∫ (2x - 1)3 dx = k(2x - 1)4 + C

> Show that 0∫1 (√x - x2) dx = 1/3.

> Generalize the result of Exercise 71 as follows: Let n be a positive integer. Show that for any positive number b we have 0∫bn n√x dx + 0∫b xn dx = bn+1

Question: Evaluate the given integral. 1∫4 (x2 -