Science:Math Exam Resources/Courses/MATH110/April 2017/Question 08 (c)

MATH110 April 2017

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Question 08 (c)
The function $f(x)$ is approximated near $a=0$ by the second degree Taylor polynomial $T_{2}(x)=3+x-8x^{2}$ (c) Find the equation of the linear approximation of $f(x)$ at $a=0$.

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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Recall the formulas for a first degree Taylor polynomial of $f(x)$ near $a$ and for the linear approximation of $f(x)$ near $a$. See if these formulas are identical or not. Then compare this with the formula for the second degree Taylor polynomial of $f(x)$ near $a$.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The linear approximation for $f(x)$ near $a=0$ is $f(a)+f'(a)(x-a)=f(0)+f'(0)x$ and that is the same as the first degree Taylor polynomial of $f(x)$ near $a=0$. Because the second degree Taylor polynomial of $f(x)$ near $a=0$ is $T_{2}(x)=f(a)+f'(a)(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}=f(0)+f'(0)x+{\frac {f''(0)}{2!}}x^{2}$ and because $T_{2}(x)=3+x-8x^{2}$ it follows that the linear approximation for $f(x)$ near $a=0$ is $\color {blue}3+x$