5128_Ch03_pp098-184

Section 3.3 Rules for Differentiation 123
EXAMPLE 8
Finding Higher Order Derivatives
Find the first four derivatives of y x 3 5x 2 2.
SOLUTION
The first four derivatives are:
First derivative: y 3x 2 10x;
Second derivative: y 6x 10;
Third derivative: y 6;
Fourth derivative: y 4 0.
This function has derivatives of all orders, the fourth and higher order derivatives all
being zero. Now try Exercise 33.
EXAMPLE 9
Finding Instantaneous Rate of Change
An orange farmer currently has 200 trees yielding an average of 15 bushels of oranges
per tree. She is expanding her farm at the rate of 15 trees per year, while improved husbandry
is improving her average annual yield by 1.2 bushels per tree. What is the current
(instantaneous) rate of increase of her total annual production of oranges?
SOLUTION
Let the functions t and y be defined as follows.
t(x) the number of trees x years from now.
y(x) yield per tree x years from now.
Then p(x) t(x)y(x) is the total production of oranges in year x. We know the following
values.
t(0) 200, y(0) 15
t(0) 15, y(0) 1.2
We need to find p(0), where p ty.
p(0) t(0)y(0) y(0)t(0)
(200)(1.2) (15)(15)
465
The rate we seek is 465 bushels per year. Now try Exercise 51.
Quick Review 3.3 (For help, go to Sections 1.2 and 3.1.)
In Exercises 1–6, write the expression as a sum of powers of x.
5. x
1
1. x 2 2x 1 1 2. ( x
x 2 x x
)
1 2x 2 1 6. x1
x3
x 2 x
x 3 x 1 2x 2 2
1
1
7. Find the positive roots of the equation
x x 2 – 2x 1 – 2
2x 3 5x 2 2x 6 0
3. 3x 2 2 x x
52 4. 2x
3 4
and evaluate the function y 500x 3x4
6 at each root. Round your
2x2
3 2 x2 x 2x 2 answers to the nearest integer, but only in the final step.
3x 2 – 2x 1 5x 2 Root: x 1.173, 500x 6 1305
Root: x 2.394, 500x 6 94, 212
x 2