More Formulas for Derivative and their Applications

1.  How much of this table can you fill in as a result of the last class? Fill that in.

Function: / Derivative: / General Rules
f(x) = a / f’(x)=0 /
f(x) = x / f’(x)=1 /
f(x) =x2 / f’(x)=2 x /
f(x) = x3 / f’(x)=3 x2
f(x) = xn / f’(x)=n xn-1
f(x) = ln(x) /

Today we’ll do the formula for the derivative of a product:

2.  You already know the derivative of x2. What is it?

3.  Show that you get the same derivative for x2 as you did in #1 by writing x2 as a product, x2 = x∙x., and using the product rule.

Find the derivatives of the following:

4.  x2ex

5.  q∙ln(q) + q4

6.  e2t ln(t)

7. 

8.  (q2 +3q + 12) eq


Evaluate the following:

9.  if

10.  MC(10) if C(q) = 3217 + 254 qln(q).

11. 

Application 1

12.  What is the formula for R(q) in terms of D(q)?

13.  Using your answer to #12 find a formula for MR(q) in terms of MD(q).

14.  If D(2000) = 9000 and MD(2000) = –5, what is MR(2000)? (Use your answer to #13.)

15.  If you increase production from 2000 units, does the revenue increase or decrease?

16.  If D(1000) = 12,000 and MD(1000) = –3, what is MR(1000)? (Use your answer to #13.)

17.  If you increase production from 1000 units, does the revenue increase or decrease?

Application 2

18.  Write in words the meaning of the statements R(100) = 6,000,000 and MR(100) = 5,123.

19.  Write in words the meaning of the statements D(2000) = 9000 and MD(2000) = –5.

20.  Suppose 2500 units are demanded when the price is $300. Write this statement using one of the functions D,R,C,P.

21.  If the quantity demanded increases from 1512 items, the revenue drops by about $20 per item. Write this statements using one of the functions D,R,C,P.

22.  If the quantity demanded decreases from 750 items, the profit decreases by about $10 an item. Write this statements using one of the functions D,R,C,P.

23.  The increase in demand due to a dollar drop in price is 10 units. Write this statements using one of the functions D,R,C,P. (This one is hard!)

Definition

24.  Define the derivative. (Give a formula.)

25.  If f(3.001) = 12.9 and f(3) = 13.2, estimate f’(3).