Chapter 2.2, Problem 28E

To determine

To find:

The graph of $f$ has a vertical tangent or a vertical cusp at c

Answer to Problem 28E

  $f\text{'}\left(1\right)=\infty$

Explanation of Solution

Given:

The function $f\left(x\right)=2^x;\left(0,1\right)$

Concept used:

The definition of a vertical cusp is that the one sided limits of the derivative approach opposite $\pm \infty$ : positive infinity on one side and negative infinity on the other side . A vertical tangent has the one sided limits of the derivative equal to the same sign of infinity

The derivative at the relevant point is undefined in both the cusp and the vertical tangent Calculation:

The function is $f\left(x\right)=2^x;\left(0,1\right)\mathrm{.........}\left(1\right)$

The first derivative and chain rule

  $f\text{'}\left(x\right)=\frac{2^x-1}{x}$

To draw a table

The vertical tangent means that the derivative at that point approaches infinity

Since the slope is infinitely large

Test one point in each of the region formed by the graph

If the point satisfies the function then shade the entire region to denote that every point in the region satisfies the function .

$x-axis$	$0$	$-1$	$-4$	$1$
$y-axis$	$1$	$0.5$	$0.05$	$2$

To draw a graph