Final Thoughts: Question 1-3.2 & Solution

Question 2

The exact solution of $y'=y$ , $y(0)=1$ , is supplied as $y=e^x$ . A polynomial approximation is $P(x)=1+x+\frac{x^2}{2}+\frac{x^3}{6}.$ You may use Taylor’s theorem with a fourth-derivative remainder.

Tasks

Calculate the residual $P'-P$ and determine whether $P$ solves the IVP exactly on any open interval containing $0$ .
Let $E(x)=e^x-P(x)$ . Prove that for $0\le x\le 1$ , $\frac{x^4}{24}\le E(x)\le\frac{e x^4}{24}.$
Give exact lower and upper bounds for $E(1)$ , and identify whether $P$ overestimates or underestimates the solution for $0<x\le 1$ .
Plot the error and its two bounds in your solution. Explain why matching the initial value and several derivatives at $0$ does not make the polynomial an exact interval solution.

Original worksheet page 1: question and worked solution for 1-3-002

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Question 2 – Solution

Strategy. Use a residual to test exactness and Taylor’s theorem to assess the approximation. These answer different questions.

Step 1: Exactness test. Since $P'=1+x+x^2/2$ , $\boxed{P'-P=-\frac{x^3}{6}.}$ The residual is nonzero whenever $x\ne 0$ . Although $P(0)=1$ , the polynomial does not solve the equation on any open interval containing $0$ .

Step 2: Error bounds. For $0<x\le 1$ , Taylor’s theorem supplies a number $\xi$ with $0<\xi<x$ such that $E(x)=\frac{e^\xi x^4}{24}.$ Because $1\le e^\xi\le e$ , the requested bounds follow; at $x=0$ all three expressions are zero.

See the diagram in the original worksheet below.

Step 3: Size and sign. At $x=1$ , $P(1)=8/3$ , so $\boxed{\frac 1{24}\le E(1)=e-\frac 83\le\frac e{24}.}$ For $x>0$ the lower bound is positive, so $P$ underestimates the exact solution. The two endpoint bounds are in fact strict at $1$ .

Step 4: Local agreement is limited evidence. The values of $P$ and its first three derivatives agree with those of $e^x$ at $0$ . That is local Taylor information. Solving the ODE requires $P'(x)=P(x)$ at every point of an interval, which the nonzero residual disproves.

Original worksheet page 2: question and worked solution for 1-3-002

Final Thoughts — Question 2

Question 2

Question 2 – Solution