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Condition |
free/or 0.5$ |
| m97569 | Verify the values of sin t and cos t in the table used to construct Figure 6. |
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| m97572 | Water is being pumped into a cylindrical tank at a constant rate of 5 gallons per minute, as shown in Figure 21.The tank has diameter 3 feet and length 9.5 feet. The volume of the tank is πr2l = π × 1.52 × 9.5 ≈ 67.152 cubic feet ≈ 500 gallons. With-out doing any calculations, sketch a graph of the height h of the water as a function of time t (see Example 6). Where is h concave up? Concave down? |
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| m97573 | Water leaks out of a 200-gallon storage tank (initially full) at the rate V (t) = 20 - t, where t is measured in hours and V in gallons. How much water leaked out between 10 and 20 hours? How long will it take the tank to drain completely? |
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| m97576 | We assume that an oil spill is being cleaned up by deploying bacteria that consume the oil at 4 cubic feet per hour. The oil spill itself is modeled in the form of a very thin cylinder whose height is the thickness of the oil slick. When the thickness of the slick is 0.001 foot, the cylinder is 500 feet in diameter. If the height is decreasing at 0.0005 foot per hour, at what rate is the area of the slick changing? |
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| m97577 | We claim that
(a) Use Figure 9 to justify this by a geometric argument.
(b) Prove the result using the Second Fundamental Theorem of Calculus.
(c) Show that An = nBn |
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| m97579 | We have defined the Laplacian of a scalar field by
Show that if Dnf is the directional derivative in the direction of the unit normal vector n, then |
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| m97582 | We normally consider the gravitational force of the earth on an object of mass m to be given by the constant F = - g mk, but, of course, this is valid only in regions near the earth s surface. Find the potential function f for F and use it to show that the work done by F when an object is moved from (x1, y1, z1) to a nearby point (x2, y2, z2) is mg (z1 - z2). |
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| m97583 | We now explore the relationship between A sin(wt) + B cos(ωt) and C sin(ωt + ϕ).
(a) By expanding sin(wt + ϕ) using the sum of the angles formula, show that the two expressions are equivalent if A = C cos ϕ and B = C sin ϕ.
(b) Consequently, show that A2 + B2 = C2 and that 4, then satisfies the equation tan ϕ = B/A.
(c) Generalize your result to state a proposition about A1 sin(ωt + ϕ1) A2 sin(ωt + ϕ2) + A3 sin(wt + ϕ3).
(d) Write an essay, in your own words, that expresses the importance of the identity between A sin(wt) + B cos(wt) and C sin(wt + ϕ). Be sure to note that |C| ≥ max(|A|. |B|) and that the identity holds only when you are forming a linear combination (adding andlor subtracting multiples of single powers) of sine and cosine of the same frequency. |
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| m97586 | What are the dimensions of the rectangular box, open at the top, that has maximum volume when the surface area is 48? |
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| m97589 | What are you able to deduce about the shape of a vase based on each of the following tables, which give measurements of the volume of the water as a function of the depth.
(a)
(b) |
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| m97591 | What conclusions can you draw about f from the information that f (c) = f"(c) = 0 and f" (c) > 0? |
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| m97592 | What condition leads to a graph that is symmetric with respect to the following?
a. xz-plane
b. y-axis
c. x-axis |
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| m97593 | What condition on a, b, and c will make f(x) = ax3 + bx2 + cx + d always increasing? |
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| m97594 | What do all members of the family of linear functions f(x) = c - x have in common Sketch several members of the family. |
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| m97595 | What heading and airspeed are required for an airplane to fly 450 miles per hour due north if a wind of 100 miles per hour is blowing in the direction N 60° E? |
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| m97596 | What heading and airspeed are required for an airplane to fly 837 miles per hour due north if a wind of 63 miles per hour is blowing in the direction S 11.5° E? |
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| m97598 | What is the distance along the 45o parallel between St. Paul and Turin? |
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| m97607 | What is the maximum height that the object of Problem 23 reaches?
On the surface of the moon, the acceleration of gravity is -5.28 feet per second per second. If an object is thrown upward from an initial height of 1000 feet with a velocity of 56 feet per second, find its velocity and height 4.5 seconds later. |
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| m97616 | What number exceeds its square by the maximum amount? Begin by convincing yourself that this number is on the interval [0, 1]. |
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| m97618 | What relationship between a, b, and c must hold if x2 + ax + y2 + by + c = 0 is the equation of a circle? |
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