3. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(0)=0, f(1)=1$, and $\left|f^{\prime}(x)\right| \leq 2$ for all real numbers $x$. If $a$ and $b$ are real numbers such that the set of possible values of $\int_{0}^{1} f(x) d x$ is the open interval $(a, b)$, determine $b-a$.
vertices $(0,0),(.75,1.5),(1,1),(.25,-.5)$. By the mean value theorem, no point of $(x, f(x))$ lies outside this parallelogram, but we can construct functions arbitrarily close to the top or the bottom of the parallelogram while satisfying the condition of the problem. So $(b-a)$ is the area of this parallelogram, which is $\frac{3}{4}$.