## Hand drawn Character Fonts

Photoshop Elements Photoshop Elements comes in a version for novice users and a version for more advanced users. You access Elements from the same Creative Suite application environment as Photoshop, Illustrator, and InDesign. It is a simple version of Photoshop in which you can perform simple photo editing tasks. You can edit images within the application. You can flatten them out, which enables you to work with multiple layers, and you can add or remove the background by using the background eraser tool. You also have options for adding text. Figure 12-9 shows the primary tools that are available in Elements. Photoshop Elements is a great place for beginners to start their foray into digital photography and editing. It provides basic photo management and editing tools for any kind of editing or presentation needs. Photoshop Elements is available with Creative Suite. **Figure 12-8:** You use the basic drawing tools to draw in Illustrator. **Figure 12-9:** You can use the primary drawing tools in Elements. ## Watching Video The most common file format for online video is the QuickTime MOV format. This format is also used for desktop videos. QuickTime is the name of the program that is used to view and edit these files. You need QuickTime to play a video file. To install QuickTime, visit the Apple website at www.apple.com/quicktime/. The download features a link to a version of QuickTime Player that you can run right away. You can download a

Q: Approximating the error $\Vert \int_{0}^{1} f(x) \,\mathrm{d}x – \int_{0}^{1} f_n(x) \,\mathrm{d}x\Vert$ for $f_n(x)=\sum_{i=1}^{n}2^{i-1}\cos(2^ix)$ Let $f_n(x)=\sum_{i=1}^{n}2^{i-1}\cos(2^ix)$. Find the error of the approximation $\int_{0}^{1} f(x) \,\mathrm{d}x$ using the same technique, and approximate the error with $n=10$. For the first question, we have $$\Vert \int_{0}^{1} f(x) \,\mathrm{d}x – \int_{0}^{1} f_n(x) \,\mathrm{d}x\Vert=\int_{0}^{1} \Vert f(x)-f_n(x)\Vert \,\mathrm{d}x=\int_{0}^{1} 2 \sin (2^n x) \,\mathrm{d}x$$ Is the answer for the first question $\frac{0.4142}{2^n}$? For the second part, we have $$\Vert \int_{0}^{1} f(x) \,\mathrm{d}x\Vert=\int_{0}^{1} \Vert f(x)\Vert \,\mathrm{d}x=\int_{0}^{1} \sqrt{2} \cosh (2^n x) \,\mathrm{d}x$$ Approximating the integral by the left Riemann sum and using the integral limit and the desired value as $n \rightarrow \infty$, the answer is $\frac{0.6737 \sqrt{2}}{2^{n/2}}$, but I’m not sure if it is correct. A: For the first question, you can apply Cauchy’s inequality,  \left| \int_0^1 f(x)\,dx\right| \leq \int_