Exercise 1 (5 points) The plane is equipped with an orthogonal coordinate system $(\mathrm{O}, \mathrm{I}, \mathrm{J})$. We denote by $\Gamma$ the representative curve of the function $g$ defined on the interval $]0; 1]$ by $g(x) = \ln x$. Let $a$ be a real number in the interval $]0; 1]$. We denote by $M_a$ the point on the curve $\Gamma$ with abscissa $a$ and $d_a$ the tangent line to the curve $\Gamma$ at the point $M_a$. This line $d_a$ intersects the $x$-axis at point $N_a$ and the $y$-axis at point $P_a$. We are interested in the area of triangle $\mathrm{O}N_aP_a$ as the real number $a$ varies in the interval $]0; 1]$. In this question, we study the particular case where $a = 0.2$. a. Determine graphically an estimate of the area of triangle $\mathrm{O}N_{0.2}P_{0.2}$ in square units. b. Determine an equation of the tangent line $d_{0.2}$. c. Calculate the exact value of the area of triangle $\mathrm{O}N_{0.2}P_{0.2}$.
Exercise 1 (5 points)
The plane is equipped with an orthogonal coordinate system $(\mathrm{O}, \mathrm{I}, \mathrm{J})$.
We denote by $\Gamma$ the representative curve of the function $g$ defined on the interval $]0; 1]$ by $g(x) = \ln x$. Let $a$ be a real number in the interval $]0; 1]$. We denote by $M_a$ the point on the curve $\Gamma$ with abscissa $a$ and $d_a$ the tangent line to the curve $\Gamma$ at the point $M_a$. This line $d_a$ intersects the $x$-axis at point $N_a$ and the $y$-axis at point $P_a$. We are interested in the area of triangle $\mathrm{O}N_aP_a$ as the real number $a$ varies in the interval $]0; 1]$.
In this question, we study the particular case where $a = 0.2$.
a. Determine graphically an estimate of the area of triangle $\mathrm{O}N_{0.2}P_{0.2}$ in square units.
b. Determine an equation of the tangent line $d_{0.2}$.
c. Calculate the exact value of the area of triangle $\mathrm{O}N_{0.2}P_{0.2}$.