Math Natural exponential function Curve sketching

# Curve sketching of the natural exponential function

Even with natural exponential functions a curve sketching can be done

!

### Remember

When setting to zero and calculating an equation with $e$, keep in mind that $e$ to the power of anything never becomes zero.

$e^{x}>0$ with $x\in\mathbb{R}$

### Example

Examine $f(x)=x\cdot e^x$ on the following properties:

• Zeros
• Extrema
• Inflection point
1. #### Calculate derivative

To compute the drivative, use the product rule.

$f(x)=x\cdot e^x$

$f'(x)=x\cdot e^x+e^x$ $=e^x(x+1)$

$f''(x)=x\cdot e^x+e^x+e^x$ $=e^x(x+2)$

$f'''(x)=x\cdot e^x+e^x+e^x+e^x$ $=e^x(x+3)$

2. #### Zeros

Calculating Zeros: Set function equal to zero

$f(x)=0$
$x\cdot e^x=0$

Product of zero: A product becomes zero if one of the factors becomes zero.

$e^x>0$ (can never be zero!) and
$x_N=0$
3. #### Extrema

Calculate extrema: Set first derivative equal to zero
$f'(x)=0$
$e^x(x+1)=0$

$e^x>0$ (can never be zero!) and
$x+1=0\quad|-1$
$x_E=-1$

use suspicious points for extrema in the second derivative test:

$f''(-1)=e^{-1}>0$ => minimum

Calculate the y-coordinate and specify the minimum:

$f(-1)$ $=-1\cdot e^{-1}$ $=-e^{-1}$ $\approx-0.37$

$T(-1|-0.37)$
4. #### Inflection point

Calculate inflection point: Set second derivative equal to zero
$f''(x)=0$
$e^x(x+2)=0$

$e^x>0$ (can never be zero!) and
$x+2=0\quad|-2$
$x_W=-2$

Insert point suspected of having an inflection point in the third derivative:

$f'''(-2)=e^{-2}\neq0$ => Inflection point

Calculate y-coordinate and specify inflection point:

$f(-2)$ $=-2e^{-2}$ $\approx-0.27$

$W(-2|-0.27)$