(a) Experiment with a calculator to find an interval 0 < u 1 , where...

(a) Experiment with a calculator to find an interval 0 ≤ θ < u₁, where θ is measured in radians, for which you think sin θ ≈ θ is a fairly good estimate. Then use a graphing utility to plot the graphs of y = x and y = sin x on the same coordinate axes for 0 ≤ x ≤ π/2. Do the graphs confirm your observations with the calculator?

(b) Use a numerical solver to plot the solution curves of the initial-value problems

for several values of θ₀ in the interval 0 ≤ θ < θ₁ found in part (a). Then plot solution curves of the initial value problems for several values of θ₀ for which θ₀ > θ₁.

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+ sin 0 = 0, 0(0) = 6, 0'(0) = 0 dt? %3D + 0 = 0, dr? and 0(0) = 60, 0'(0) = 0