The problem presented is as follows:

Given functions: $f(x) = e^x$ and $g(x) = e^{-x}$

Tasks:

1. (a) Draw the graphs of $f(x) = e^x$ and $g(x) = e^{-x}$ in the interval $-2 \leq x \leq 2$.
2. (b) Draw the tangents to the graphs of $f$ and $g$ at the point $P(0,1)$ and provide the equations for these tangents.

Solution Outline:

Part (a): Drawing the Graphs of $f(x) = e^x$ and $g(x) = e^{-x}$

The graphs of $f(x) = e^x$ and $g(x) = e^{-x}$ are exponential functions:

• $f(x) = e^x$: an increasing exponential function.
• $g(x) = e^{-x}$: a decreasing exponential function.

For the interval $-2 \leq x \leq 2$, both functions exhibit typical exponential behavior:

• $f(x)$ grows as $x$ increases.
• $g(x)$ decreases as $x$ increases.

Part (b): Tangent Lines at $P(0,1)$

1. Finding the Derivative for $f(x)$: $f'(x) = e^x$ At $x = 0$, $f'(0) = e^0 = 1$.

   So, the tangent line to $f(x)$ at $P(0,1)$ has a slope of $1$. The equation of this tangent is: $y - 1 = 1(x - 0) \implies y = x + 1$

2. Finding the Derivative for $g(x)$: $g'(x) = -e^{-x}$ At $x = 0$, $g'(0) = -e^0 = -1$.

   So, the tangent line to $g(x)$ at $P(0,1)$ has a slope of $-1$. The equation of this tangent is: $y - 1 = -1(x - 0) \implies y = -x + 1$

Summary:

• Graphs of $f(x) = e^x$ and $g(x) = e^{-x}$ should be plotted in the interval $-2 \leq x \leq 2$.
• Tangent equations:
  • For $f(x)$: $y = x + 1$
  • For $g(x)$: $y = -x + 1$

Would you like assistance with plotting these graphs or further explanation?

Here are 5 related questions to deepen your understanding:

1. How does the rate of change of an exponential function compare to that of a linear function?
2. Why does $g(x) = e^{-x}$ have a negative slope at $x = 0$?
3. What effect does changing the base of the exponential function (from $e$ to another base) have on the graph?
4. How would the tangent lines change if the point were not $P(0,1)$ but a different point on the graph?
5. How does the symmetry of $f(x) = e^x$ and $g(x) = e^{-x}$ reflect in their respective tangent lines?

Tip: Remember, the slope of the tangent line at a point is given by the derivative evaluated at that point.