Question

The approximation ex = 1+x + x^2/2 is used when x is small. Use the Remainder Estimation Theorem to estimate the error when |x| < 1/10 Select the correct choice below and fill in the answer box to complete your choice. (Use scientific notation. Round to two decimal places as needed.) A. The maximum error is approximately __ for M=1/10 B. The maximum error is approximately __ for M =1/e

C. The maximum error is approximately __ for M=1 D. The maximum error is approximately __ for M=2

Answer 1

The maximum error in approximating
`\(e^x\)` by the expression
`\(1+x+(x^2)/(2)\)` when
`\(|x| < (1)/(10)\)` is approximately
`\(1.67 * 10^(-4)\) for \(M = (1)/(10)\)`. This estimation is based on the Remainder Estimation Theorem, which considers the third derivative of
`\(e^x\)` and utilizes a value of
`\((1)/(10)\)` to represent the interval of interest. Thus the correct option is A. The maximum error is approximately
`\(1.67 * 10^(-4)\) for \(M=(1)/(10)\)`.

Step-by-step explanation:

The Remainder Estimation Theorem states that if f(x) is approximated by the Taylor series
`\(P_n(x)\)` up to the nth term, then the remainder
`\(R_n(x)\)` is given by:

`\[R_n(x) = (f^((n+1))(c))/((n+1)!) \cdot (x-a)^(n+1)\]`

where (c) is between (a) and (x). In this case, the approximation is
`\(e^x \approx 1+x+(x^2)/(2)\)`, and we want to estimate the error when \
`(|x| < (1)/(10)\).`

The Remainder Estimation Theorem helps us bound the error term by finding the maximum value of
`\(|R_n(x)|\)` for (x) in the given interval. For our approximation, the third derivative of
`\(e^x\)` is
`\(e^x\)`, and we take (M) as
`\((1)/(10)\)` because we're interested in the interval where
`\(|x| < (1)/(10)\)`.

Plugging these values into the Remainder Estimation Theorem formula, we get:

`\[|R_2(x)| = (e^c)/(6) \cdot \left((1)/(10)\right)^3\]`

To find the maximum error, we maximize
`\(|R_2(x)|\) by setting \(c = (1)/(10)\)`. Calculating this expression gives us the final answer:
`\(1.67 * 10^(-4)\)`.

This means that the maximum error in the given interval is approximately
`\(1.67 * 10^(-4)\).`

Complete Question:

The approximation ex = 1+x + x^2/2 is used when x is small. Use the Remainder Estimation Theorem to estimate the error when |x| < 1/10. Select the correct choice below and fill in the answer box to complete your choice. (Use scientific notation. Round to two decimal places as needed.)

A. The maximum error is approximately __ for M=1/10

B. The maximum error is approximately __ for M =1/e

C. The maximum error is approximately __ for M=1

D. The maximum error is approximately __ for M=2

Answer 2

The maximum error is approximately __ for M=1/10. Option A

We can solve this problem using the Remainder Estimation Theorem:

Function: f(x) =
`e^x`

Interval: |x| < 1/10

We are given the approximation:

`f(x) = 1 + x + x^2/2`

The third-degree derivative of f(x) =
`e^x`

If we have that;

`|R_n(x)| \leq |M|(|x-a|^(n+1))/(n+1)!`

where:

• R_n(x) is the remainder after the nth-degree term
• M is the upper bound of the absolute value of the third-degree derivative on the interval
• a is the center of the Taylor series expansion (here, a = 0)

Since we want to estimate the maximum error when |x| < 1/10, we can simply use the maximum value of |x| in the interval, which is |x| = 1/10.

M =
`|e^(^1^/^1^0^)^|` ≈ 1.1052 (upper bound of the third-degree derivative on the interval)

`R_2(1/10) \leq (1.1052)|((1/10) - 0)^3| / 3!`

`R_2(1/10) \leq 1.1052 * (1/1000) / 6`

`= 1.84 * 10^-6`