Calculus I - Interpretation of the Derivative

Section 3.2 : Interpretation of the Derivative

4. Sketch the graph of a function that satisfies$f\left( { - 3} \right) = 5$, $f'\left( { - 3} \right) = - 2$, $f\left( 1 \right) = 2$, $f'\left( 1 \right) = 0$, $f\left( 4 \right) = - 2$, $f'\left( 4 \right) = - 3$.

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First, recall 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. So, let’s start off with a graph that has the given points on it and a sketch of a tangent line at the points whose slope is the value of the derivative at the points.

$A sketch showing just the three given points and the tangent lines at those points. The point $\left( { - 3,5} \right)$ has a short line through it falling to the right with a slope of -2, the point $\left( {1,2} \right)$ has a short horizontal line through it since the slope there is 0, and the point $\left( {4, - 2} \right)$ has a short line through it falling steeply to the right with a slope of -3.$

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Now, all that we need to do is sketch in a graph that goes through the indicated points and at the same time it must be parallel to the tangents that we sketched. There are many possible sketches that we can make here and so don’t worry if your sketch is not the same as the one here. This is just one possible sketch that meets the given conditions.

$A sketch of a function that satisfies the given conditions, drawn in with the three points and their tangent lines. The curve comes down from the upper left through $\left( { - 3,5} \right)$ parallel to the falling tangent line there, flattens out and just touches the horizontal tangent line as it passes through $\left( {1,2} \right)$, and then falls away steeply through $\left( {4, - 2} \right)$ parallel to the tangent line there.$

While, it’s not really needed here is a sketch of the function without all the extra bits that we put in to help with the sketch.

$A sketch of a function that satisfies the given conditions with the points and tangent lines removed. The curve comes down from the upper left, flattens out into a nearly level stretch at about $y = 2$ between $x = 0$ and $x = 2$, and then falls away steeply, crossing the x-axis near $x = 3$ and continuing down to the lower right.$