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Q: 1) Compute y (0.5) with y!!! xy¹ = (1-x) ex by @Taylor series y (o)= y) (o)=y" (a)=1₂h=0-5
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- Prove the conjecture made in the previous exercise.Show that F (x, y) = x2 + 3y is not uniformly continuous on the whole plane.Hint: You must prove that there are pairs of points, arbitrarily close together, on which thevariation of F is large, for example, (n, 0) and (n + 1/n, 0).Use Simpson rule’ 1/3 Method with (n = 8) to estimate the integral: f(x) = 5x*((13x - x²)³)½ , where a= 0.5 & b= 3.5: Give answer with (3) decimals places. *
- Define f:[0, 1]→ℝ by f(x) = 5x if x ∈ ℚ f(x) = 0 if x ∉ ℚ Show that U(f) = 1/2 and L(f) = 0, so that f is not integrable on [0, 1].Use the definition of the convergence of an improper integral to determine whether the followingintegral converges or diverges:(a). ∫infinity 3 dx/(x-3)^(3/2) b)∫1 0 dx/xsqrtxEstimate the area under f(x)=x2 on the interval [0,10] using the midpoint Riemann Sum for n=5.