1. Differentiation Techniques: The Product and Quotient Rules
4.2
OBJECTIVES
Differentiate using the Product and the Quotient Rules.
Use the Quotient Rule to differentiate the average cost, revenue, and profit functions.
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2. 4.2 Differentiation Techniques: The Product and Quotient Rules
THEOREM 5: The Product Rule
Let
Then, OR
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3. 4.2 Differentiation Techniques: The Product and Quotient Rules
Example 1: Find
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5. If find y’. Example 2: Answer:
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6. Differentiation Techniques: The Product and Quotient Rules
THEOREM 6: The Quotient Rule
If then,
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7. Differentiation Techniques: The Product and Quotient Rules
Example 3: Differentiate
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9. Find the derivative of Answer: Example 4:
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10. Differentiation Techniques: The Product and Quotient Rules
DEFINITION:
If C(x) is the cost of producing x items, then the
average cost of producing x items is
If R(x) is the revenue from the sale of x items, then the
average revenue from selling x items is
If P(x) is the profit from the sale of x items, then the
average profit from selling x items is
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11. Differentiation Techniques: The Product and Quotient Rules
Example 6: Paulsen’s Greenhouse finds that
the cost, in dollars, of growing x hundred geraniums is
given by If the revenue from the sale of x hundred geraniums is given by
find each of the following.
a) The average cost, the average revenue, and the average profit when x hundred geraniums are grown and sold.
b) The rate at which average profit is changing when 300 geraniums are being grown.
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12. Example 6 (continued): a) We let , , and represent average cost,
average revenue, and average profit.
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13. Differentiation Techniques: The Product and Quotient Rules
Example 6 (continued):
b) First we must find Then we can substitute 3
(hundred) into
NOTE: In the textbook, on p. 164, they substitute in 300 (instead of 3).
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14. Example 6 (concluded):
Thus, at 300 geraniums, Paulsen’s average profit is
increasing by about $11.20 per plant.
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15. Do p. 258 ex. 5 Do p.260 #40 (a & b) Example 7
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16. Example 8: Average Profit
The total profit (in tens of dollars) from selling x
self-help books is
Find the average profit from each sales level.
8 books B) 15 books
Answers: A) $0.224 tens or $ B) $1.39
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