Date modified: Sat 2023-07-15 . 06:38 PM
Date created: Sat 2023-10-07 . 09:02 PM
# Week 4 Monday Problems ## Reading. Please read sections 7.7 and 7.8. Do try the exercises for additional practice. Do the exam correction by next Monday. ## Problems 1. This is a thinking problem about the error bound theorems of these approximations. Really think about them. 1. Suppose $f(x)$ is a linear polynomial, that is $f(x)=Ax+B$ for some $A,B$. Suppose we use midpoint rule or trapezoidal rule to estimate $\int_{a}^{b}f(x)dx$. What you can say about the error from $M_{N}$ or $T_{N}$? 2. Suppose $f(x)$ is a cubic polynomial, that is, $f(x)=Ax^{3}+Bx^{2}+Cx+D$. What can we say about the error from $S_{N}$, using Simpson's rule with $N$ pieces? 2. Consider the function $f(x)=\displaystyle\frac{\sin(x)}{x}$, estimate $\displaystyle\int_{1}^{3} \frac{\sin(x)}{x}dx$ with the following methods 1. Left end point Riemann estimation with $N=4$, namely find $L_{4}$. 2. Right end point Riemann estimation with $N = 4$, namely find $R_{4}$. 3. Midpoint rule estimation with $N=4$, namely find $M_{4}$. 4. Trapezoidal rule estimation with $N=4$, namely find $T_{4}$. 5. Simpson's rule estimation with $N=4$, namely find $S_{4}$. 6. Try if you can: Redo everything above with a computer program, and use $N=1000$ for each of the cases. 3. Consider the function $f(x)=e^{x^{2}}$ on the interval $[0,2]$. 1. Calculate $f'(x),f''(x)$, and $f^{(4)}(x)$. Factoring will help with the mess. 2. Give a crude upper bound $K_{1}$ for $|f'(x)|$ on $[0,2]$. 3. Give a crude upper bound $K_{2}$ for $|f''(x)|$ on $[0,2]$. 4. Give a crude upper bound $K_{4}$ for $|f^{(4)}(x)|$ on $[0,2]$. 5. Use your estimate to find what $N$ we need so that $L_{N}$ is within $0.0001$ of the true value of $\int_{0}^{2}e^{x^{2}}dx$. (Left end point estimation) 6. Use your estimate to find what $N$ we need so that $M_{N}$ is within $0.0001$ of the true value of $\int_{0}^{2}e^{x^{2}}dx$. (Midpoint rule estimation) 7. Use your estimate to find what $N$ we need so that $T_{N}$ is within $0.0001$ of the true value of $\int_{0}^{2}e^{x^{2}}dx$. (Trapezoidal rule estimation) 8. Use your estimate to find what $N$ we need so that $S_{N}$ is within $0.0001$ of the true value of $\int_{0}^{2}e^{x^{2}}dx$. (Simpson's rule estimation) ///