# Math Insight

### Derivative intuition

Math 1241, Fall 2020
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Due date: Oct. 9, 2020, 11:59 p.m.
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Total points: 3
1. The graph of a piecewise linear function $f(x)$ is given below. It is composed of three linear pieces.
Feedback from applet
derivative sketch:
1. For each of the three linear pieces, pick two points on the graph, estimate their coordinates, and from those coordinates estimate the slope of that line.

Slope of left segment:

Slope of middle segment:

Slope of right segment:

(Round answer to nearest 10th or less.)

Should your estimate of the slope depend on your choice of points?

2. Imagine that, for each of the three linear pieces, you picked points that were closer and closer together. The slope that you calculated would be the derivative. For each linear piece, what is the derivative of $f(x)$?

$f'(x)$ for left segment:

$f'(x)$ for middle segment:

$f'(x)$ for right segment:

(Round answer to nearest 10th or less.)

3. Sketch the derivative of $f$ on the above graph.

2. The graph of a function $g(x)$ is shown below.
Feedback from applet
tangents:
Use this second plot to sketch the graph of the derivative $\diff{g}{x}$, as outlined below.
Feedback from applet
points:

Online, see the “Need Help” sections for instructions on how to use the applets to answer the questions. The applets are particular about which points you move to which positions.

1. First, find the point where the graph of $g$ has a horizontal tangent line (there is only one). This occurs for $x=$
.
Sketch the horizontal tangent line on the graph. What is the derivative at this point?

$g'($
$) =$

On the second graph, as a first step of sketching the derivative of $g$, draw the single point on the graph of the derivative that we've calculated so far. Since we've calculate the derivative at $x=$
, the point to draw is

$(x,g'(x)) = ($
,
$)$.

2. There are two other places where the tangent line is nearly horizontal. Sketch those tangent lines on the first graph. The derivative at those points is very similar to the derivative from the previous part. Sketch corresponding points $(x,g'(x))$ for the derivative on the second graph. (Online, the order matters; see hint.)
3. There's one point where the tangent line has a maximally positive slope and another point where it has a maximally negative slope. Sketch those tangent lines on the first graph. Estimate the slope of those tangent lines, i.e., the derivative $g'(x)$ at those points. Then, add the points $(x,g'(x))$ for those two values of $x$ to the graph of the derivative in the second plot.
4. Finish the sketch of the derivative $\diff{g}{x}$ on the second graph by drawing a smooth curve connecting the points you sketched. Make sure that your sketch has the maximum and minimum values where you indicated and that it is zero only where the tangent line is horizontal.

3. Follow a similar procedure to sketch the derivative of the below curve. Be sure to identify where the derivative is zero, positive or negative.
Feedback from applet
derivative:
min or max location:
min or max value:
zero points:
(Online, you don't have a way to identify where derivative is zero, positive, or negative, but on paper you can make this identification as you would on an exam.)

4. In each plot, identify which of the curves is the graph of the derivative of the other.