Solution — 2019 Question 2 Exponential & Logarithmic Functions| Graphs| Differentiation & Applications

(a)

Using a graphing calculator to solve $y = 0$ :

$x = - 0.677$

So $A = (- 0.677, 0) ∎$

(b)

\begin{matrix} 3 x + 4 & = 0 \\ 3 x & = - 4 \\ x & = - \frac{4}{3} ∎ \end{matrix}

(c)

\begin{matrix} y & = x + \ln (3 x + 4) \\ \frac{d y}{d x} & = 1 + \frac{3}{3 x + 4} \end{matrix}

At point $B,$ $x = 0,$

\begin{matrix} y & = x + \ln (3 x + 4) \\ = 0 + \ln (3 (0) + 4) \\ = \ln 4 \end{matrix}

\begin{matrix} \frac{d y}{d x} & = 1 + \frac{3}{3 (0) + 4} \\ = \frac{7}{4} \end{matrix}

Equation of tangent:

\begin{matrix} y - \ln 4 & = \frac{7}{4} (x - 0) \\ = \frac{7}{4} x \\ y & = \frac{7}{4} x + \ln 4 ∎ \end{matrix}