Science:Math Exam Resources/Courses/MATH110/April 2013/Question 09 (a)

MATH110 April 2013

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Question 09 (a)
Use an appropriate line approximation to estimate e0.1.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

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
The linear approximation method requires choosing a value that is easy to calculate (usually denoted by the variable a), near the value being approximated. What would be a good choice for this question?

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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 formula is $f(x)\approx L(x)=f'(a)(x-a)+f(a)$ For some value a that is near the value to be approximated. In our case, $f(x)=e^{x}$ and a sensible choice for a would be 0 as it is easy to calculate $\displaystyle f(0)=e^{0}$ and $\displaystyle f'(0)=e^{0}$. Plugging this into our formula for L(x) we have: $\displaystyle L(x)=e^{0}(x-0)+e^{0}$ $\displaystyle L(x)=x+1$ To approximate $e^{0.1}$, we simply plug in $x=0.1$ to get $\displaystyle L(0.1)=0.1+1=1.1$

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