Science:Math Exam Resources/Courses/MATH110/April 2018/Question 08 (c)/Solution 2

This answer is outside the curriculum of MATH110. But this can be a valid answer for other 100 level courses of which curriculum contains the error formula.

In addition to the explanation from part (a), the accuracy of the linear approximation will also depend on how far the function is from being linear: if a function is truly linear then the linear approximation would give an accurate estimate even if the two ${\textstyle x}$-values used are very far apart, whereas an exponential function diverges very quickly from it’s tangent line. This is measured by the second derivative of the function. The second derivative of the function used in (b) at ${\textstyle x=1}$ is significantly smaller in absolute value than the second derivative of the function used in (a) at ${\textstyle x=0}$, so this would suggest that (b) should give a more accurate approximation.

Answer: One would expect the linear approximation from ${\textstyle {\color {blue}(b)}}$ to be more accurate.