Calculus I - Interpretation of the Derivative

Section 3.2 : Interpretation of the Derivative

2. Use the graph of the function, $f\left( x \right)$, estimate the value of $f'\left( a \right)$ for

$a = 1$
$a = 4$

$The graph of $f\left( x \right)$ on $0 \le x \le 5$. The graph starts at approximately $\left( {0, - 5} \right)$ and increases the whole way, slowly at first and then much more steeply. It crosses the x-axis at about $x = 3.3$ and reaches approximately $\left( {5,7} \right)$ at the right end.$

Show All Solutions Hide All Solutions

a $a = 1$ Show All Steps Hide All Steps

Start Solution

Given that one of the interpretations of the derivative is that it is the slope of the tangent line to the function at a particular point let’s first sketch in a tangent line at the point on the graph.

$The graph of $f\left( x \right)$ with a tangent line sketched in at $x = 1$. The point on the graph at $x = 1$ is marked with a dot and the tangent line drawn through it rises gently from the lower left to the upper right, showing that the function is increasing and the derivative is positive there.$

Show Step 2

The function is clearly increasing here and so we know that the derivative at this point will be positive. Now, from this sketch of the tangent line it looks like if we run over 1 we go up 1 and so we can estimate that,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{f'\left( 1 \right) = 1}}\]

b $a = 4$ Show All Steps Hide All Steps

Start Solution

Given that one of the interpretations of the derivative is that it is the slope of the tangent line to the function at a particular point. Let’s first sketch in a tangent line at the point.

$The graph of $f\left( x \right)$ with a tangent line sketched in at $x = 4$. The point on the graph at $x = 4$ is marked with a dot and the tangent line drawn through it rises steeply from the lower left to the upper right, showing that the function is increasing and the derivative is positive there.$

Show Step 2

\[\require{bbox} \bbox[2pt,border:1px solid black]{{f'\left( 4 \right) = 5}}\]