Calculus I - Interpretation of the Derivative

Section 3.2 : Interpretation of the Derivative

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

$a = - 2$
$a = 3$

$The graph of $f\left( x \right)$ on $ - 3 \le x \le 4$. It is a parabola shaped curve that comes down from approximately $\left( { - 3,6} \right)$, crosses the x-axis at about $x = - 1.8$, reaches its lowest point at approximately $\left( { - 0.2, - 2.5} \right)$ and then increases, crossing the x-axis at $x = 2$ and reaching approximately $\left( {4,3.5} \right)$ at the right end.$

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a $a = - 2$ Show All Steps Hide All Steps

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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 $x = - 2$. The point on the graph at $x = - 2$ is marked with a dot and the tangent line drawn through it falls steeply from the upper left to the lower right, showing that the function is decreasing and the derivative is negative there.$

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The function is clearly decreasing here and so we know that the derivative at this point will be negative. Now, from this sketch of the tangent line it looks like if we run over 1 we go down 4 and so we can estimate that,

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

b $a = 3$ 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 $x = 3$. The point on the graph at $x = 3$ is marked with a dot and the tangent line drawn through it rises 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 2 and so we can estimate that,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{f'\left( 3 \right) = 2}}\]