Chapter 1. Calculus Tutorial 2.1.013

1.1 Problem Statement

true
eval rand(2,6);
eval $a + 0.01;
eval $a + 0.001;
eval $a + 0.0001;
eval $a + 0.00001;
eval $a - 0.01;
eval $a - 0.001;
eval $a - 0.0001;
eval $a - 0.00001;
eval round( ( 1/($ap01 + 2) - 1/($a + 2) )/ 0.01 , 6);
eval round( ( 1/($ap001 + 2) - 1/($a + 2) )/ 0.001 , 6);
eval round( ( 1/($ap0001 + 2) - 1/($a + 2) )/ 0.0001 , 6);
eval round( ( 1/($ap00001 + 2) - 1/($a + 2) )/ 0.00001 , 6);
eval round( ( 1/($a + 2) - 1/($am01 + 2) )/ 0.01 , 6);
eval round( ( 1/($a + 2) - 1/($am001 + 2) )/ 0.001 , 6);
eval round( ( 1/($a + 2) - 1/($am0001 + 2) )/ 0.0001 , 6);
eval round( ( 1/($a + 2) - 1/($am00001 + 2) )/ 0.00001 , 6);
eval round( ($avgap00001 + $avgam00001)/2 , 4);

Estimate the instantaneous rate of change at the point indicated.

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1.2 Step 1

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

Question 1.1

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 /gyv4StTnixZTcnynbgrMi0adcL37d1XJwOewtY/zU4we9prEKHQkff9E5b6rMXcANV3O0wRGR7OyJpSvzSK6HjPIihasfwJzK6eSV9Xq0+3P7+1tLvcieviMG5MiIgW.

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 nO/+DAdRVq+2oOvmzNgeN0ocEUKz3yEuOTAqpssC58Q= of x = nc1ItEz0kR4=. That is, we find the limit of the average rates of change as x approaches $a.

2
Correct.
That's not right. Check your work.
Incorrect.

1.3 Step 3

Question 1.2

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

Interval [$am01,$a] [$am001,$a] [$am0001,$a] [$am00001,$a]
Average rate of change sAF6ALj0Z4+paJAF OSbqcGxs4qnq/dTMRDVB2g== PgSsY9hDTloSRysei0Vhtg== R78Nu8s7sf9aKF8Uf2d7mQ==

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

2
Correct.
Determine the rate of changes of smaller and smaller intervals by dividing thechange of output by the change of input.
Incorrect.

1.4 Step 2

Question 1.3

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

Interval [$a,$ap01] [$a,$ap001] [$a,$ap0001] [$a,$ap00001]
Average rate of change RhjwTClEuoy99C6T CWO/+PXAHUZw9M4bw+br6Q== BW3epklUTIVOoCgS43LxhQ== F8CIPolyEJQLPTb6wPCEFw==

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

2
Correct.
Determine the rate of changes of smaller and smaller intervals by dividing thechange of output by the change of input.
Incorrect.

1.5 Step 4

Question 1.4

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

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

2
Correct.
That's not right. Check your work.
Incorrect.