The diagram shows a river,500m wide between the straight parallel banks where AD = 500m and DB = 2000. A man swims across the river from A to C at 0.5 m/s and then he runs along the banks from C to B at 0.8 m/s (a)If he swims in the direction AC making an […]

Given y = 4/√(2x-3), x > 1.5 (a)Find dy/dx (b)Explain why the graph of y = 4/√(2x – 3) does not have a stationary point (c)State whether y = 4√(2x – 3) is an increasing or decreasing function. Explain your answer clearly. y = 4(2x – 3)-1/2 dy/dx = -2(2x – 3)-3/2(2) = -4(2x […]

Find the minimum gradient of the curve y = 2×3 – 9×2 + 5x + 3 and the value of x when the minimum gradient occurs. y = 2×3 – 9×2 + 5x + 3 gradient, m = 6×2 – 18x + 5 dm/dx = 12x – 18 minimum gradient, dm/dx = 0 12x – […]

A curve with equation in the form of y = ax + b/x y = ax + b/x2 has a stationary point at (3 , 4), where a and b are constants. Find the value of a and of b. X = 3 y = 4 4 = 3a + b/a —–(1) Y = ax […]

Solution: (a) A=2, B=-5, C=0 (b) 2x/(x2+3) (c) ln(2x-1) – (5/2)ln(x2+3) + C (a)6 + 5x – 8×2 = A(x2 + 3) + (Bx + c)(2x – 1) Sub x = ½ 6 + 5/2 – 8(1/2)2 = A(1/4 + 3) + 0 13/2 = (13/4)A —— A=2 When x=0, 6 = 2(3) + c(-1) —— […]

Integrate (3-sin22x)/cos22x w.r.t x ∫3/cos22x – sin2x/cos22x dx =∫3sec22x – (sec22x-1) dx =∫2sec22x +1 dx =2tan2x/2 + x + c =tan2x + x + c

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