1. The diagram marks point P, the root of z4 = 16 with argument 3π/4. Which exact Cartesian form is P?
Each root has modulus 161/4 = 2. At argument 3π/4, P = 2(cos 3π/4 + i sin 3π/4) = -√2 + √2i.
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Practice proof, vectors, complex numbers, advanced calculus, mechanics, differential equations, and probability.
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Each root has modulus 161/4 = 2. At argument 3π/4, P = 2(cos 3π/4 + i sin 3π/4) = -√2 + √2i.
The direction vector is AB = (4 - 1, -1 - 2, 2 - 0) = (3, -3, 2). A valid line equation is r = (1, 2, 0) + λ(3, -3, 2).
Acceleration is the derivative of velocity. Since v'(t) = 6t - 12, a(3) = 18 - 12 = 6 m/s2.
The equilibrium is y = 10. When y < 10, dy/dt is positive, so the solution increases and approaches the stable equilibrium y = 10.
Use u = sin x, so du = cos x dx. The integral becomes ∫ from 0 to 1 of u du = 1/2.
For f(x) = x3, f'(x) = 3x2, so f'(0) = 0. But x = 0 is a stationary point of inflection, not a local maximum, so the claim is false.
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