Find the values of the constants A, B and C.
A hollow hemispherical bowl is shown in Figure 1. Water is flowing into the bowl. When the depth of the water is h m, the volume V m3 is given byV = 1/12 π h2(3 - 4h), 0 ≤ h ≤ 0.25
(a) Find, in terms of π, dV/dh when h = 0.1
Water flows into the bowl at a rate of π/800 m3s-1.
(b) Find the rate of change of h, in m s-1, when h = 0.1
The finite region R, shown shaded in Figure 2, is bounded by the curve, the x-axis and the line x = √2
The table below shows corresponding values of x and y for y = x3ln(x2 + 2)
(a) Complete the table above giving the missing values of y to 4 decimal places.
(b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to 2 decimal places.
(c) Use the substitution u = x2 + 2 to show that the area of R is
(d) Hence, or otherwise, find the exact area of R.
5. Find the gradient of the curve with equation
ln y = 2x ln x, x > 0, y > 0
at the point on the curve where x = 2. Give your answer as an exact value.
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