Calculus Tutorial 2.1.013

Problem Statement

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eval rand(2,6);

eval 3 + 0.01;

eval 3 + 0.001;

eval 3 + 0.0001;

eval 3 + 0.00001;

eval 3 - 0.01;

eval 3 - 0.001;

eval 3 - 0.0001;

eval 3 - 0.00001;

eval round( ( 1/(3.01 + 2) - 1/(3 + 2) )/ 0.01 , 6);

eval round( ( 1/(3.001 + 2) - 1/(3 + 2) )/ 0.001 , 6);

eval round( ( 1/(3.0001 + 2) - 1/(3 + 2) )/ 0.0001 , 6);

eval round( ( 1/(3.00001 + 2) - 1/(3 + 2) )/ 0.00001 , 6);

eval round( ( 1/(3 + 2) - 1/(2.99 + 2) )/ 0.01 , 6);

eval round( ( 1/(3 + 2) - 1/(2.999 + 2) )/ 0.001 , 6);

eval round( ( 1/(3 + 2) - 1/(2.9999 + 2) )/ 0.0001 , 6);

eval round( ( 1/(3 + 2) - 1/(2.99999 + 2) )/ 0.00001 , 6);

eval round( (-0.040000 + -0.040000)/2 , 4);

Estimate the instantaneous rate of change at the point indicated.

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Step 1

Recall that the instantaneous rate of change at x = x₀ is the limit of the average rates of change.

Question 1

For x = x₀, the average rate of change of y = f(x) over [x₀, x₁] has the following formula.

Average rate of change =

where and .

To estimate the instantaneous rate of change of the given problem, we calculate the average rate of change over smaller and smaller intervals to the of x = . That is, we find the limit of the average rates of change as x approaches 3.

Correct.

That's not right. Check your work.

Incorrect.

Step 3

Question 2

First calculate the average rate of change over four intervals to the left of x = 3. (Round your answers to six decimal places.)

Interval	[2.99,3]	[2.999,3]	[2.9999,3]	[2.99999,3]
Average rate of change

This table suggests the limit of the average rates of change as x approaches 3 from the left is approximately (rounded to four decimal places).

Correct.

Determine the rate of changes of smaller and smaller intervals by dividing thechange of output by the change of input.

Incorrect.

Step 2

Question 3

Now calculate the average rate of change over four intervals to the right of x = 3. (Round your answers to six decimal places.)

Interval	[3,3.01]	[3,3.001]	[3,3.0001]	[3,3.00001]
Average rate of change

This table suggests the limit of the average rates of change as x approaches 3 from the right is approximately (rounded to four decimal places).

Correct.

Determine the rate of changes of smaller and smaller intervals by dividing thechange of output by the change of input.

Incorrect.

Step 4

Question 4

Thus, from steps 2 and 3, the limits of the average rates of change from the left and from the right as x approaches 3 is -0.0400.

Thus, the estimate of the instantaneous rate of change is (rounded to four decimal places).

Correct.

That's not right. Check your work.

Incorrect.