Sketch the graphs of $y = 3e^x$ and $y = x + 3$ on the same diagram. Indicate clearly the coordinates of the points of intersection between the 2 graphs. Solve the inequality $3e^x > x + 3$. Hence find $\int_{-2}^{2} |3e^x - x - 3| \, \mathrm{d}x$, giving your answer in an exact form.

1. Sketch the graphs and find their points of intersection using a graphing calculator.

Using a graphing calculator, plot $y = 3e^x$ and $y = x + 3$.

• The graph of $y = 3e^x$ is an exponential curve with a horizontal asymptote at $y = 0$.
• The graph of $y = x + 3$ is a straight line with an $x$-intercept at $(-3, 0)$ and a $y$-intercept at $(0, 3)$.

Equating both functions $3e^x = x + 3$ on the calculator yields the points of intersection:

• First intersection: $(0, 3)$ exactly.
• Second intersection: $x \approx -2.8214, y \approx 0.1786 \implies (-2.82, 0.179)$ to 3 significant figures.

2. Determine the solution to the inequality $3e^x > x + 3$ using the sketched graphs.

The inequality $3e^x > x + 3$ is satisfied in the regions where the graph of $y = 3e^x$ lies strictly above the line $y = x + 3$.
From the sketch, the exponential curve is above the straight line for $x$-values less than the smaller root and greater than the larger root.
Thus, the solution is $x < -2.82$ or $x > 0$.

1. Split the definite integral by analyzing the sign of the expression inside the modulus.

\text{From part (a), the graphs of } y = 3e^x \text{ and } y = x + 3 \text{ intersect at } x = 0.
\text{For } x \in [-2, 0], 3e^x \leq x + 3 \implies 3e^x - x - 3 \leq 0.
\text{For } x \in [0, 2], 3e^x \geq x + 3 \implies 3e^x - x - 3 \geq 0.

\int_{-2}^{2} |3e^x - x - 3| , \mathrm{d}x
= \int_{-2}^{0} -(3e^x - x - 3) , \mathrm{d}x + \int_{0}^{2} (3e^x - x - 3) , \mathrm{d}x

2. Integrate and evaluate the limits to find the exact value.

\begin{aligned}
&\int_{-2}^{0} -(3e^x - x - 3) , \mathrm{d}x + \int_{0}^{2} (3e^x - x - 3) , \mathrm{d}x \
&= -\left[ 3e^x - \frac{1}{2}x^2 - 3x \right]{-2}^{0} + \left[ 3e^x - \frac{1}{2}x^2 - 3x \right]{0}^{2} \
&= -\left( (3e^0 - 0 - 0) - \left(3e^{-2} - \frac{1}{2}(-2)^2 - 3(-2)\right) \right) + \left( \left(3e^2 - \frac{1}{2}(2)^2 - 3(2)\right) - (3e^0 - 0 - 0) \right) \
&= -\left( 3 - (3e^{-2} - 2 + 6) \right) + \left( (3e^2 - 2 - 6) - 3 \right) \
&= -(3 - 3e^{-2} - 4) + (3e^2 - 11) \
&= -(-1 - 3e^{-2}) + 3e^2 - 11 \
&= 1 + 3e^{-2} + 3e^2 - 11 \
&= 3e^2 + \frac{3}{e^2} - 10
\end{aligned}