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A runner runs along a straight east-west road. The function f ( t ) gives how many feet eastward of her starting point she is after t seconds. Interpret each of the following as it relates to the runner.

  1. f ( 0 ) = 0
  2. f ( 10 ) = 150
  3. f ( 10 ) = 15
  4. f ( 20 ) = 10
  5. f ( 40 ) = −100

  1. After zero seconds, she has traveled 0 feet.
  2. After 10 seconds, she has traveled 150 feet east.
  3. After 10 seconds, she is moving eastward at a rate of 15 ft/sec.
  4. After 20 seconds, she is moving westward at a rate of 10 ft/sec.
  5. After 40 seconds, she is 100 feet westward of her starting point.
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Finding points where a function’s derivative does not exist

To understand where a function’s derivative does not exist, we need to recall what normally happens when a function f ( x ) has a derivative at x = a . Suppose we use a graphing utility to zoom in on x = a . If the function f ( x ) is differentiable    , that is, if it is a function that can be differentiated, then the closer one zooms in, the more closely the graph approaches a straight line. This characteristic is called linearity .

Look at the graph in [link] . The closer we zoom in on the point, the more linear the curve appears.

Graph of a negative parabola that is zoomed in on a point to show that the curve becomes linear the closer it is zoomed in.

We might presume the same thing would happen with any continuous function, but that is not so. The function f ( x ) = | x | , for example, is continuous at x = 0 , but not differentiable at x = 0. As we zoom in close to 0 in [link] , the graph does not approach a straight line. No matter how close we zoom in, the graph maintains its sharp corner.

Graph of an absolute function.
Graph of the function f ( x ) = | x | , with x -axis from –0.1 to 0.1 and y -axis from –0.1 to 0.1.

We zoom in closer by narrowing the range to produce [link] and continue to observe the same shape. This graph does not appear linear at x = 0.

Graph of an absolute function.
Graph of the function f ( x ) = | x | , with x -axis from –0.001 to 0.001 and y -axis from—0.001 to 0.001.

What are the characteristics of a graph that is not differentiable at a point? Here are some examples in which function f ( x ) is not differentiable at x = a .

In [link] , we see the graph of

f ( x ) = { x 2 , x 2 8 x , x > 2 .

Notice that, as x approaches 2 from the left, the left-hand limit may be observed to be 4, while as x approaches 2 from the right, the right-hand limit may be observed to be 6. We see that it has a discontinuity at x = 2.

Graph of a piecewise function where from negative infinity to (2, 4) is a positive parabola and from (2, 6) to positive infinity is a linear line.
The graph of f ( x ) has a discontinuity at x = 2.

In [link] , we see the graph of f ( x ) = | x | . We see that the graph has a corner point at x = 0.

Graph of an absolute function.
The graph of f ( x ) = | x | has a corner point at x = 0 .

In [link] , we see that the graph of f ( x ) = x 2 3 has a cusp at x = 0. A cusp has a unique feature. Moving away from the cusp, both the left-hand and right-hand limits approach either infinity or negative infinity. Notice the tangent lines as x approaches 0 from both the left and the right appear to get increasingly steeper, but one has a negative slope, the other has a positive slope.

Graph of f(x) = x^(2/3) with a viewing window of [-3, 3] by [-2, 3].
The graph of f ( x ) = x 2 3 has a cusp at x = 0.

In [link] , we see that the graph of f ( x ) = x 1 3 has a vertical tangent at x = 0. Recall that vertical tangents are vertical lines, so where a vertical tangent exists, the slope of the line is undefined. This is why the derivative, which measures the slope, does not exist there.

Practice Key Terms 7

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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