It comes from integrating by parts twice. Let
Recall the IBP formula,
Let
Then
Apply IBP once more, with
Notice that the ∫ v du term contains the original integral, so that
Basically, you just move every individual point up, down, left, or right by the amount indicated.
For example, point G is graphed on the point (-3, -1). Moving it right 5 and up 1 will give you G’, which is (2,0)
T is graphed on the point (-1, -1). Moving it right 5 and up 1 will give you T’, which is (4,0)
B is graphed on the point (-3, -5). Moving it right 5 and up 1 will give you B’, which is
(2,4)
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Answer:
Step-by-step explanation:
The ratios all have ...
first number : second number = 1 : 4
Using first numbers of 1, 2, 3, the second numbers can be found by multiplying these by 4. (1, 4), (2, 8), (3, 12)
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You plot these (x, y) points the same way you plot <em>any</em> point on a coordinate grid. The first (x) value is the horizontal distance from the vertical axis. Positive is to the right. The second (y) value is the vertical distance from the horizontal axis. Positive is up.
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Of course, the origin is where the horizontal and vertical axes meet. It can be convenient to find one of the coordinates on its respective axis, then use the other coordinate to find the point at the desired distance from that axis.
Usually, you would choose the axis on the basis of how easy it is to determine exactly where the coordinate lies. If the y-axis is marked every 5, for example, it might be hard to determine where a multiple of 4 will lie. Locating the x-coordinate on the x-axis may be an easier way to start.
Answer:
the rate of change of the water depth when the water depth is 10 ft is;
Step-by-step explanation:
Given that:
the inverted conical water tank with a height of 20 ft and a radius of 8 ft is drained through a hole in the vertex (bottom) at a rate of 4 ft^3/sec.
We are meant to find the rate of change of the water depth when the water depth is 10 ft.
The diagrammatic expression below clearly interprets the question.
From the image below, assuming h = the depth of the tank at a time t and r = radius of the cone shaped at a time t
Then the similar triangles ΔOCD and ΔOAB is as follows:
( similar triangle property)
h = 2.5r
The volume of the water in the tank is represented by the equation:
The rate of change of the water depth is :
Since the water is drained through a hole in the vertex (bottom) at a rate of 4 ft^3/sec
Then,
Therefore,
the rate of change of the water at depth h = 10 ft is:
Thus, the rate of change of the water depth when the water depth is 10 ft is;