Calculus 2 Quiz: Approximation Error And Bounds

The trapezoidal rule with $n = 4$ subintervals is used to approximate $\int_0^2 e^{x^2} dx$. Given that $|f''(x)| \leq 400$ on $[0,2]$, which statement about the approximation error is correct?

Simpson's rule is applied to approximate $\int_1^3 \frac{1}{x} dx$ with $n = 6$ subintervals. If the fourth derivative satisfies $|f^{(4)}(x)| \leq \frac{24}{x^5}$ for $x \in [1,3]$, what is the error bound for this approximation?

A function $f(x)$ is approximated using its third-degree Taylor polynomial $P_3(x)$ centered at $a = 2$. If $|f^{(4)}(x)| \leq 120$ for all $x$ in $[1.5, 2.5]$, what is the maximum possible error when using $P_3(x)$ to approximate $f(1.8)$?

The power series $\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$ converges to $\sin(x)$ for all real $x$. When this series is truncated after the first 4 terms to approximate $\sin(1.2)$, what type of error bound applies, and what is the bound?

The alternating series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^4}$ converges to a sum $S$. What is the smallest integer $N$ such that the partial sum $S_N$ is guaranteed by the Alternating Series Estimation Theorem to approximate $S$ with an error less than $10^{-3}$?

The sum of the series $\sum_{n=1}^{\infty} \frac{1}{n^3}$ is approximated by its 10th partial sum, $S_{10}$. Let $R_{10}$ be the remainder $S - S_{10}$. According to the Integral Test remainder estimate, which of the following is a correct statement about $R_{10}$?

The function $f(x) = e^{-3x}$ is approximated by its second-degree Taylor polynomial, $P_2(x)$, centered at $a=0$. Using Taylor's inequality, what is the tightest upper bound for the error $|f(x) - P_2(x)|$ on the interval $[0, 0.2]$?

To approximate $\ln(1.5)$ using a Taylor polynomial for $f(x) = \ln(x)$ centered at $a=1$, what is the minimum degree $n$ required to guarantee an absolute error of no more than $\frac{1}{3000}$?

Let $B_T$ be the error bound for the Trapezoidal Rule and $B_S$ be the error bound for Simpson's Rule when approximating $\int_0^1 (x^4+x) dx$ with $n=4$. What is the value of the ratio $B_T / B_S$?

The sum of a convergent alternating series $\sum_{n=1}^{\infty} (-1)^{n+1} a_n$ (with $a_n > 0$) is approximated by the partial sum $S_N$. The error bound is given by $|R_N| \le a_{N+1}$. Which of the following conditions is necessary for this error bound to be valid?

The sum of the convergent alternating series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2 + 1}$ is $S$. This sum is approximated by its fourth partial sum, $S_4$. Which of the following statements is true about the error, $R_4 = S - S_4$?

The error bound for approximating $\int_a^b f(x) dx$ with the Trapezoidal Rule is $|E_T| \le \frac{K(b-a)^3}{12n^2}$, where $K$ is an upper bound for $|f''(x)|$. If $f(x)$ is a linear function, what is the value of this error bound?

The function $f(x) = \sqrt{1+x}$ is approximated by its first-degree Taylor polynomial $P_1(x)$ centered at $a=0$. What is the Lagrange error bound for this approximation on the interval $[0, 0.2]$?

The series $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$ is a telescoping series that sums to 1. If we approximate this sum with $S_N = \sum_{n=1}^{N} \frac{1}{n(n+1)}$, the exact error is $R_N = 1 - S_N$. By using partial fractions, $S_N = 1 - \frac{1}{N+1}$. Thus, the exact error is $R_N = \frac{1}{N+1}$. What upper bound does the Integral Test provide for this remainder?

For the approximation $\sin(x) \approx x - x^3/6$ on the interval $[-0.5, 0.5]$, the Lagrange error bound involves finding the maximum of a derivative of $\sin(x)$. Which derivative should be used and what is its maximum absolute value on the interval?

A function $f(x)$ has a fourth derivative $f^{(4)}(x) = \frac{\cos(x^2)}{10}$. We approximate $\int_{-1}^{1} f(x) dx$ using Simpson's rule with $n$ subintervals. What is the smallest even integer $n$ that guarantees an error no more than $\frac{1}{9000}$?

The function $f(x)$ is approximated by its $n$-th degree Taylor polynomial $P_n(x)$ centered at $a$. The Lagrange error bound for $|f(x) - P_n(x)|$ is given by $\frac{M}{(n+1)!}|x-a|^{n+1}$. How is the value of $M$ determined?

A student uses $n=10$ subintervals to approximate $\int_1^2 \ln(x) dx$. They find the error bound for the Trapezoidal Rule is $B_T$ and the error bound for Simpson's Rule is $B_S$. Which statement correctly compares the bounds?

An integral is approximated using Simpson's Rule with $n$ subintervals, resulting in a theoretical error bound $B$. If the number of subintervals is tripled (to $3n$), what will the new theoretical error bound be in terms of $B$?

Consider approximating $e$ using the Maclaurin series for $e^x$ evaluated at $x=1$, which is $\sum_{n=0}^\infty \frac{1}{n!}$. We use the partial sum $S_N = \sum_{n=0}^N \frac{1}{n!}$. The Lagrange error bound for this approximation states that $|R_N| \le \frac{M}{(N+1)!}$, where $M$ is the maximum of $e^c$ on $[0,1]$. To use this bound, we need a value for $M$, but $e$ is the number we are approximating. Which is the most effective and valid approach to establish a numerical error bound?