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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

  1. \(a = 1\)
  2. \(a = 4\)
Hint : Remember that one of the interpretations of the derivative is the slope of the tangent line to the function.

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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.

Show Step 2

The function is clearly decreasing 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.

Show Step 2

The function is clearly decreasing 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 5 and so we can estimate that,

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