Transcript 2.3 Some Differentiation Formulas

Section 2.3
1. Find the derivative of f (x) = x 4 .
Use the power rule: If y = x n then y’ = nx n – 1
2. Find the derivative of f (x) = x 1/2 .
Use the power rule: If y = x n then y’ = nx n – 1
3. Find the derivative of

g(w)  63 w
Use the power rule: If y = x n then y’ = nx n – 1
4. Find the derivative of f (x) = 4x 2 - 3x + 2.
Use the power rule: If y = x n then y’ = nx n – 1
5. Find the derivative of
f (x) 
6
3
x
Use the power rule: If y = x n then y’ = nx n – 1
6. Find the derivative of
f (x) 
1 3 1 2
x  x  x1
6
2
Use the power rule: If y = x n then y’ = nx n – 1
7. Find the derivative of
h( x)  6
3
x
Use the
power rule: If y = x n then y’ = nx n – 1
2
12
 3
x
8. Find the derivative of the following function at x = - 2.
f (x)  x 5

9. Find the derivative of the following function at x = - 3.
f (x)  x 3

f ‘ (x) = 3x 2 so f ‘ (- 3) = 3 ( - 3) 2 = 27
10. a. Find the equation of the tangent line to f (x) = x 2 + 2 at x = 3.
b. Graph the function and the tangent line on the window [-1,6] by [-10,20].
Use your calculator and the draw – tangent button.
OR
10. a. Find the equation of the tangent line to f (x)  x 2  2 at x =3.
b. Graph the function and the tangent line on the window [-1,6] by [-10,20].

11. a. Find the equation of the tangent line to f (x) = x 3 - 3x 2 + 2x - 2 at x = 2.
b. Graph the function and the tangent line on the window [-1,4] by [-7,5].
Use your calculator and the draw – tangent button.
OR
11. a. Find the equation of the tangent line to f (x)  x 3 2x  2 at x =2.
b. Graph the function and the tangent line on the window [1,4] by [-7,5]

12. Business: Software Costs Businesses can buy multiple licenses for PowerZip
data compression software at a total cost of approximately:
C (x) = 24x 2/3 dollars for x licenses. Find the derivative of this cost function at:
a. x = 8 and interpret your answer.
b. x = 64 and interpret your answer.
13. Business: Marginal Cost (12 continued) Use a calculator to find the actual cost
of the 64th license by evaluating C(64)-C(63) for the cost function in 12. Is your answer
close to the $4 that you found for part (b) of that exercise?
14. Business: Marketing to Young Adults Companies selling products to young adults
often try to predict the size of that population in the future years. According to the
predictions by the Census Bureau, the 18-24-year old population in the United States
will follow the function
1
P (x)   x 3  25x 2  300x  31, 000 (in thousands),
3
where x is the number of years after 2010. Find the rate of change of this population:
a. In the year 2030 and interpret your answer.
b. In the year 2010 and interpret your answer.
15. General: Internet Access The percentage of U.S. households with broadband
Internet access is approximated by
f (x) 
, where x is the number
1 2
x  5x  6
4
of years after the year 2000. Find the rate of change of this percentage in the year
2010 and interpret your answer.

16. Psychology: Learning Rates A language school has found that it’s students
can memorize P(t) = 24 t , phrases in t hours of class (for 1  t  10). Find the
instantaneous rate of change of this quality after 4 hours of class and interpret your
answer.
17. Economics: Marginal Utility Generally, the more you have of something, the less
valuable each additional unit becomes. For example, a dollar is less valuable to a
millionaire than to a beggar. Economists define a person’s “utility function” U(x)
for a product as the “perceived value” of having x units of that product. The derivative
of U(x) is called marginal utility function, MU(x)=U’(x). Suppose that a person’s
utility function for money is given by the function below. That is, U(x) is the utility
(perceived value) of x dollars.
a. Find the marginal utility function. MU(x).
b. Find MU(1), the marginal utility of the first dollar.
c. Find MU(1,000,000), the marginal utility of the millionth dollar.
U (x) = 100 x
18. General: Smoking and Education According to a study, the probability that a smoker
will quit smoking increases with the smoker’s educational level. The probability
(expressed as a percent) that a smoker with x years of education will quit is
approximated by the equation f (x) = 0.831 x 2 – 18.1 x + 137.3 (for 10  x  16)
a.Find f(12) and f’(12) and interpret these numbers. [Hint: x = 12 corresponds
to a high school graduate.]
a.b. Find f(16) and f’(16) and interpret these numbers. [Hint: x = 16 corresponds
to a college graduate.]