Chapter 3 | Derivatives
259
3.19 Find d dx ⎛
⎝ 3 f ( x )−2 g ( x ) ⎞ ⎠ .
Example 3.31 Determining Where a Function Has a Horizontal Tangent
Determine the values of x for which f ( x ) = x 3 −7 x 2 +8 x +1 has a horizontal tangent line.
Solution To find the values of x for which f ( x ) has a horizontal tangent line, we must solve f ′( x ) =0. Since f ′( x ) =3 x 2 −14 x +8= (3 x −2)( x −4), we must solve (3 x −2)( x −4) =0. Thus we see that the function has horizontal tangent lines at x = 2 3 and x =4 as shown in the following graph.
Figure 3.19 This function has horizontal tangent lines at x = 2/3 and x = 4.
Example 3.32 Finding a Velocity The position of an object on a coordinate axis at time t is given by s ( t ) = t t 2 +1
. What is the initial velocity of
the object?
Solution Since the initial velocity is v (0) = s ′(0), begin by finding s ′( t ) by applying the quotient rule:
⎛ ⎝ t 2 +1
⎞ ⎠ −2 t ( t )
1
t 2
s ′( t ) =
= 1−
2 .
2
⎛ ⎝ t 2 +1
⎞ ⎠
⎛ ⎝ t 2 +1
⎞ ⎠
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