5128_Ch03_pp098-184

118 Chapter 3 Derivatives
EXAMPLE 1 Differentiating a Polynomial
Find d p
if p t
dt
3 6t 2 5 3 t 16.
SOLUTION
By Rule 4 we can differentiate the polynomial term-by-term, applying Rules 1 through 3
as we go.
d p d
t dt
d t
3 d
6t d t
2 d
d
( t
5 3 ) t d
16 Sum and Difference Rule
dt
3t 2 6 • 2t 5 3 0
Constant and Power Rules
3t 2 12t 5 3 Now try Exercise 5.
EXAMPLE 2
Finding Horizontal Tangents
Does the curve y x 4 2x 2 2 have any horizontal tangents? If so, where?
SOLUTION
The horizontal tangents, if any, occur where the slope dydx is zero. To find these
points, we
(a) calculate dydx:
d y d
x dx
d x
4 2x 2 2 4x 3 4x.
(b) solve the equation dydx 0 for x:
4x 3 4x 0
4xx 2 1 0
x 0, 1, 1.
The curve has horizontal tangents at x 0, 1, and 1. The corresponding points on the
curve (found from the equation y x 4 2x 2 2) are 0, 2, 1, 1, and 1, 1. You
might wish to graph the curve to see where the horizontal tangents go.
Now try Exercise 7.
The derivative in Example 2 was easily factored, making an algebraic solution of the
equation dydx 0 correspondingly simple. When a simple algebraic solution is not possible,
the solutions to dydx 0 can still be found to a high degree of accuracy by using
the SOLVE capability of your calculator.
[–10, 10] by [–10, 10]
Figure 3.18 The graph of
y 0.2x 4 0.7x 3 2x 2 5x 4
has three horizontal tangents. (Example 3)
EXAMPLE 3
Using Calculus and Calculator
As can be seen in the viewing window 10, 10 by 10, 10, the graph of
y 0.2x 4 0.7x 3 2x 2 5x 4 has three horizontal tangents (Figure 3.18).
At what points do these horizontal tangents occur?
continued