Chapter 3 | Derivatives
239
Figure 3.16 The function f ( x ) = ⎧ ⎩
⎛ ⎝ 1 x
⎞ ⎠ if x ≠0
⎨ x sin
is not
0 if x =0
differentiable at 0.
In summary: 1. We observe that if a function is not continuous, it cannot be differentiable, since every differentiable function must be continuous. However, if a function is continuous, it may still fail to be differentiable. 2. We saw that f ( x ) = | x | failed to be differentiable at 0 because the limit of the slopes of the tangent lines on the left and right were not the same. Visually, this resulted in a sharp corner on the graph of the function at 0. From this we conclude that in order to be differentiable at a point, a function must be “smooth” at that point. 3. As we saw in the example of f ( x ) = x 3 , a function fails to be differentiable at a point where there is a vertical tangent line.
4. As we saw with f ( x ) = ⎧ ⎩
⎛ ⎝ 1 x
⎞ ⎠ if x ≠0
⎨ x sin
a function may fail to be differentiable at a point in more complicated
0 if x =0
ways as well.
Example 3.14 A Piecewise Function that is Continuous and Differentiable
A toy company wants to design a track for a toy car that starts out along a parabolic curve and then converts to a straight line ( Figure 3.17 ). The function that describes the track is to have the form
⎧ ⎩ ⎨ 1
x 2 + bx + c if x <−10 − 1 4 x + 5 2 if x ≥−10
10
f ( x ) =
where x and f ( x ) are in inches. For the car to move smoothly along the
track, the function f ( x ) must be both continuous and differentiable at −10. Find values of b and c thatmake f ( x ) both continuous and differentiable.
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