Let's analyze each of the statements one by one.
1. When you zoom in enough on any function, it can be approximated by a straight line.
- This statement refers to the concept of local linearity. Most continuous functions, when observed at a very small scale around any given point, can appear almost linear. This is the underlying principle behind differential calculus where the derivative at a point gives the slope of the tangent line to the function at that point, which serves as its local linear approximation.
- Answer: True (T)
2. At the point of tangency, the function's value and the local linear approximation of the function have the same value.
- This statement asserts that at the exact point where the tangent line touches the curve, both the function and the tangent line (which is the local linear approximation) intersect. Therefore, the function and its local linear approximation indeed share the same value and slope at the point of tangency.
- Answer: True (T)
3. If the tangent line to the graph [tex]\( f(x) = \sin x \)[/tex] at [tex]\((0,0)\)[/tex] is [tex]\( y = x \)[/tex], then [tex]\( y = x \)[/tex] is the best straight-line approximation for the graph at all [tex]\( x \)[/tex].
- This statement is examining a specific case: the function [tex]\( \sin x \)[/tex] and its tangent line at the origin. While it's true that the tangent line [tex]\( y = x \)[/tex] is the best approximation near [tex]\( x = 0 \)[/tex], this approximation does not remain accurate as [tex]\( x \)[/tex] moves further from zero. The sine function [tex]\( \sin x \)[/tex] eventually deviates significantly from the linear function [tex]\( y = x \)[/tex], especially as [tex]\( x \)[/tex] increases or decreases.
- Hence, [tex]\( y = x \)[/tex] is not the best straight-line approximation for [tex]\( \sin x \)[/tex] at all [tex]\( x \)[/tex]; it is only valid close to the point of tangency [tex]\( (0,0) \)[/tex].
- Answer: False (F)
Summarizing the answers:
1. True (T)
2. True (T)
3. False (F)
So, the final results would be: ['T', 'T', 'F'].
