Answer
The estimated slope of the curve at $P_1$ is $2.5$.
The estimated slope of the curve at $P_2$ is $-1$.
Work Step by Step
To estimate the slope $m$ of the curve at point $P(a,b)$ without the equation of the function, we find another point $Q(c, d)$ as near to $P$ as possible in the graph, then the slope of the curve will approximately be equal with the slope of $PQ$: $$m=\frac{d-b}{c-a}$$
1) Find the slope $m_1$ of the curve at $P_1(0.4,1.1)$.
In the graph, we see that very near to $P_1$, there is a point $Q(0.2,0.6)$.
By calculating the slope of $P_1Q$, we will have the estimated slope $m_1$ of the curve at $P_1$: $$m_1=\frac{1.1-0.6}{0.4-0.2}=\frac{0.5}{0.2}=2.5$$
2) Find the slope $m_2$ of the curve at $P_2(1.9,1.3)$:
In the graph, we see that very near to $P_2$, there is this point $R(1.7,1.5)$.
By calculating the slope of $P_2R$, we will have the estimated slope $m_2$ of the curve at $P_2$: $$m_2=\frac{1.5-1.3}{1.7-1.9}=\frac{0.2}{-0.2}=-1$$
