Date modified: Fri 2023-11-10 . 11:30 AM
Date created: Fri 2023-11-10 . 10:33 AM
# Week 11 Problem Set A. ## Reading. Chapter 4.4 Concavity and curve sketching Chapter 4.5 Applied optimization Chapter 4.6 Newton's method This problem set is on curve sketching (4.4). ## Problems. ### Graphing functions. Graph the following functions. Be sure to (0) identify the domain of the function, (1) identify intervals of increasing and decreasing, (2) identify intervals of concavity, (3) identify locations of interesting points (local extrema, inflection points, $x-$ and $y-$intercepts), and (4) asymptotes if any. 1. $y = x^{2} - 4x + 3$ 2. $y = x^{3} - 3x + 3$ 3. $y = -2x^{3} + 6x^{2} - 3$ 4. $y = (x-2)^{3} + 1$ 5. $\displaystyle y = \frac{x}{\sqrt{x^{2}+1}}$ 6. $y = x \sqrt{8-x^{2}}$ 7. $y = \sqrt{16 - x^{2}}$ 8. $\displaystyle y = \frac{x^{2}-3}{x-2}$ 9. $\displaystyle y = \frac{8x}{x^{2}+4}$ ### Graphing rational functions Graph the following rational functions. Again (0) identify the domain of the function, (1) identify intervals of increasing and decreasing, (2) identify intervals of concavity, (3) identify locations of interesting points (local extrema, inflection points, $x-$ and $y-$intercepts), and (4) asymptotes if any. Don't forget, long division can help you deduce long term asymptotic behaviors. 1. $\displaystyle y= \frac{2x^{2}+x-1}{x^{2}-1}$ 2. $\displaystyle y=\frac{x^{4}+1}{x^{2}}$ 3. $\displaystyle y = \frac{1}{x^{2}-1}$ 4. $\displaystyle y = \frac{x^{2}-2}{x^{2}-1}$ 5. $\displaystyle y = \frac{x^{2}}{x+1}$ 6. $\displaystyle y = \frac{x^{2}-x+1}{x-1}$ 7. $\displaystyle y = \frac{x^{3}-3x^{2}+3x-1}{x^{2}+x-2}$ ////