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Chapter 3 | Derivatives
Figure 3.17 For the car to move smoothly along the track, the function must be both continuous and differentiable.
Solution For the function to be continuous at x =−10, lim x →10 −
f ( x ) = f (−10). Thus, since
lim x →−10 − 2 −10 b + c =10−10 b + c and f (−10) =5, we must have 10−10 b + c =5. Equivalently, we have c =10 b −5. For the function to be differentiable at −10, f ′(10) = lim x →−10 f ( x )− f (−10) x +10 must exist. Since f ( x ) is defined using different rules on the right and the left, we must evaluate this limit from the right and the left and then set them equal to each other: f ( x ) = 1 10 (−10)
x 2 + bx + c −5 x +10
x +10 = lim x →−10 − 1 10
f ( x )− f (−10)
lim x →−10 −
1 10 x 2 + bx +(10 b −5)−5 x +10 x 2 −100+10 bx +100 b 10( x +10) ( x +10)( x −10+10 b ) 10( x +10)
= lim
Substitute
c =10 b −5.
x →−10 −
= lim
x →−10 −
= lim = b −2.
Factor by grouping.
x →−10 −
We also have
x + 5 x +10
− 1 4
2 −5
f ( x )− f (−10)
lim x →−10 +
x +10 = lim x →−10 +
−( x +10) 4( x +10)
= lim
x →−10 +
= − 1 4 .
and c =10 ⎛
⎞ ⎠ −5= 25 2 .
⎝ 7 4
This gives us b −2= − 1 4 .
Thus b = 7 4
Find values of a and b that make f ( x ) = ⎧ ⎩
⎨ ax + b if x <3 x 2 if x ≥3
3.8
both continuous and differentiable at 3.
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