Page 137 - Calculus Workbook For Dummies

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Chapter 7: Analyzing Those Shapely Curves with the Derivative

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3
q For g x = x + x - , find all the values c in the interval - , 2 1i that satisfy the Mean Value
x
^ h
_
- 1 - 7 - 1 + 7
Theorem. The values of c are and .
3 3
1. Find the first derivative.
2
3
g x = x + x - x
^ h
2
g x = 3 x + 2 x - 1
l ^ h
2. Figure the slope between the endpoints of the interval.
2 -
^
3 2 g - h g 1 ^ h
2 - -
^
^
g - h ^ 2 + - h ^ 2h m =
2 = - h
- 2 - 1
= - 2 - 2 - 1
=
g 1 = 1 - 2 - 1
^ h
= 1
3. Set the derivative equal to this slope and solve.
^
2
1
3 x + 2 x - = 1 x = - 2 ! 4 - - 24h
2
2
3 x + 2 x - = 0 6
- 2 ! 2 7
=
6
- 1 - 7 - 1 + 7
= or
3 3
Both are inside the given interval, so you’ve got two answers.
r For s t = t / 4 3 - t 3 / 1 3 , find all the values c in the interval (0, 3) that satisfy the Mean Value
^ h
Theorem. The value of c is ⁄4.
3
1. Find the first derivative.
s t = t / 4 3 - t 3 / 1 3
^ h
4 / 1 3 - / 2 3
s t = t - t
l ^ h
3
2. Figure the slope between the endpoints of the interval.
^
s 3 - ^h s 0h
s 0 = 0 m =
^ h
3 - 0
/ 4 3
3 3
s 3 = 3 - $ / 1 3 0 - 0
^ h
=
= 0 3
= 0
3. Set the derivative equal to the result from Step 2 and solve.
4 t - t - / 2 3 = 0
/ 1 3
3
4
t - / 2 3 c t - m 0
1
1 =
3
1
t - / 2 3 = 0 or 4 t - = 0
1
3
3
Q or t =
4
3
Graph s and check that its slope at t = is zero.
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