1. **State the problem:** We are given the differential equation $$\frac{dy}{dx} = \sin x$$ with the initial condition $$y(0) = 0$$. We want to find the function $$y(x)$$ and calculate its values at $$x = 0, 0.25, 0.5, 0.75, 1.0$$. 2. **Formula and rules:** To solve this, we integrate both sides with respect to $$x$$: $$ y = \int \sin x \, dx + C $$ where $$C$$ is the constant of integration. 3. **Integration:** The integral of $$\sin x$$ is $$-\cos x$$, so: $$ y = -\cos x + C $$ 4. **Apply initial condition:** Using $$y(0) = 0$$: $$ 0 = -\cos 0 + C = -1 + C \implies C = 1 $$ 5. **Final solution:** $$ y = 1 - \cos x $$ 6. **Calculate values:** - At $$x=0$$: $$y = 1 - \cos 0 = 1 - 1 = 0$$ - At $$x=0.25$$: $$y = 1 - \cos 0.25 \approx 1 - 0.9689 = 0.0311$$ - At $$x=0.5$$: $$y = 1 - \cos 0.5 \approx 1 - 0.8776 = 0.1224$$ - At $$x=0.75$$: $$y = 1 - \cos 0.75 \approx 1 - 0.7317 = 0.2683$$ - At $$x=1.0$$: $$y = 1 - \cos 1.0 \approx 1 - 0.5403 = 0.4597$$ These values correspond to the increments $$\Delta y$$ and the function values $$y$$ at the given $$x$$ points. **Final answer:** $$y = 1 - \cos x$$ with values as above.