Taylor series Quiz

1. What is the coefficient of $(x-a)^3$ in the Taylor series of $f(x)$ about $x=a$?

• A. $\frac{f'''(a)}{3}$
• B. $\frac{f'''(a)}{3!}$
• C. $3!\,f'''(a)$
• D. $\frac{f'''(0)}{3!}$

2. What is the coefficient of $(x-1)^2$ in the Taylor series of $\mathrm{e}^x$ about $x=1$?

• A. $\frac12$
• B. $\mathrm{e}$
• C. $2\mathrm{e}$
• D. $\frac{\mathrm{e}}{2}$

3. What is the Taylor series of $\cos x$ about $\frac{\pi}{2}$ up to $\left(x-\frac{\pi}{2}\right)^3$?

• A. $-\left(x-\frac{\pi}{2}\right)+\frac16\left(x-\frac{\pi}{2}\right)^3$
• B. $\left(x-\frac{\pi}{2}\right)-\frac16\left(x-\frac{\pi}{2}\right)^3$
• C. $1-\frac12\left(x-\frac{\pi}{2}\right)^2$
• D. $-\left(x-\frac{\pi}{2}\right)-\frac16\left(x-\frac{\pi}{2}\right)^3$

4. What is the coefficient of $(x-1)^2$ in the Taylor series of $\ln x$ about $x=1$?

• A. $\frac12$
• B. $-1$
• C. $-\frac12$
• D. $1$

5. To estimate $\sqrt{10}$ with a Taylor series of $\sqrt{x}$, which centre is most suitable?

• A. $a=0$
• B. $a=9$
• C. $a=16$
• D. $a=1$