Given a continuous function $f$, define the following subsets of the set $\mathbb{R}$ of real numbers. $T =$ set of slopes of all possible tangents to the graph of $f$. $S =$ set of slopes of all possible secants, i.e. lines joining two points on the graph of $f$. For each statement below, state if it is true or false. (i) If $f$ is differentiable, then $S \subset T$. (ii) If $f$ is differentiable, then $T \subset S$. (iii) If $T = S = \mathbb{R}$, then $f$ must be differentiable everywhere. (iv) Suppose 0 and 1 are in $S$. Then every number between 0 and 1 must also be in $S$.
Given a continuous function $f$, define the following subsets of the set $\mathbb{R}$ of real numbers.
$T =$ set of slopes of all possible tangents to the graph of $f$.
$S =$ set of slopes of all possible secants, i.e. lines joining two points on the graph of $f$.
For each statement below, state if it is true or false.
(i) If $f$ is differentiable, then $S \subset T$.
(ii) If $f$ is differentiable, then $T \subset S$.
(iii) If $T = S = \mathbb{R}$, then $f$ must be differentiable everywhere.
(iv) Suppose 0 and 1 are in $S$. Then every number between 0 and 1 must also be in $S$.