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In this lesson, we will learn how to move from average rate of change to instantiations rate of change through the use of limits.

Q1:

Determine the average rate of change function π΄ ( β ) for π ( π₯ ) = π₯ + 2 8 π₯ 2 when π₯ changes from π₯ 1 to π₯ + β 1 .

Q2:

If the function π ( π₯ ) = β 3 π₯ β 5 9 , find l i m β β 0 π ( π₯ + β ) β π ( π₯ ) β .

Q3:

If the function π ( π₯ ) = 3 π₯ + 7 3 , find l i m β β 0 π ( π₯ + β ) β π ( π₯ ) β .

Q4:

If the function π ( π₯ ) = β π₯ + 3 9 , find l i m β β 0 π ( π₯ + β ) β π ( π₯ ) β .

Q5:

If the function π ( π₯ ) = 2 π₯ β 2 6 , find l i m β β 0 π ( π₯ + β ) β π ( π₯ ) β .

Q6:

If the function π ( π₯ ) = 4 π₯ + 1 0 7 , find l i m β β 0 π ( π₯ + β ) β π ( π₯ ) β .

Q7:

If the function π ( π₯ ) = π₯ + 1 6 4 , find l i m β β 0 π ( π₯ + β ) β π ( π₯ ) β .

Q8:

For a function π ( π₯ ) , the average rate of change between a fixed point π₯ and another point π₯ + β is π΄ ( β ) = π ( π₯ + β ) β π ( π₯ ) β . Given that π ( π₯ ) = π₯ β 6 π₯ + 5 2 , find π΄ ( 0 . 5 ) when π₯ = 4 .

Q9:

Find the average rate of change of π ( π₯ ) = 4 π₯ β 6 π₯ + 2 2 when π₯ varies from 1.3 to 2.

Q10:

Consider the average rate of change of the function π ( π₯ ) = 1 π₯ over the interval [ 3 , 3 + β ] with small values of β .

Simplify the expression π ( 3 + β ) β π ( 3 ) ( 3 + β ) β 3 .

The average rate of change gets closer and closer to β 1 9 as β becomes smaller and smaller. Simplify the expression for the difference πΏ ( β ) between π ( 3 + β ) β π ( 3 ) ( 3 + β ) β 3 and β 1 9 .

For what values of β is the difference πΏ ( β ) exactly 1 1 0 , 1 1 0 0 , 1 1 0 4 . Give your answer as a fraction.

Q11:

What is the average rate of change function π΄ ( β ) for the function π ( π₯ ) = 7 5 π₯ ?

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