Question

Write the Taylor polynomial T5(x) for the function f(x)=cos(x) centered at x=0.
Answer: T5(x)=

Answer 1

Write the Taylor polynomial $T_{5}(x)$ for the function $f(x) = \cos x$ centered at $x = 0$ .

**Solution:**

$$T _ {5} (x) = f (0) + \frac {x}{1 !} f ^ {\prime} (0) + \frac {x ^ {2}}{2 !} f ^ {\prime \prime} (0) + \frac {x ^ {3}}{3 !} f ^ {\prime \prime \prime} (0) + \frac {x ^ {4}}{4 !} f ^ {(4)} (0) + \frac {x ^ {5}}{5 !} f ^ {(5)} (0)$$ $$f (x) = \cos x \quad f (0) = 1$$ $$f ^ {\prime} (x) = - \sin x \quad f ^ {\prime} (0) = 0$$ $$f ^ {\prime \prime} (x) = - \cos x \quad f ^ {\prime \prime} (0) = - 1$$ $$f ^ {\prime \prime \prime} (x) = \sin x \quad f ^ {\prime \prime \prime} (0) = 0$$ $$f ^ {(4)} (x) = \cos x \quad f ^ {(4)} (0) = 1$$ $$f ^ {(5)} (x) = - \sin x \quad f ^ {(5)} (0) = 0$$ $$T _ {5} (x) = 1 - \frac {x ^ {2}}{2} + \frac {x ^ {4}}{24}$$

**Answer:** $T_{5}(x) = 1 - \frac{x^{2}}{2} + \frac{x^{4}}{24}$