Chapter 1. Calculus Tutorial 2.1.007

1.1 Problem Statement

eval rand(5,17,2)
eval round(1 + $avg/100, 2)
eval round( 100*( pow($b,.5) - pow($b,0) ) / .5, 2);
eval round( 100*( pow($b,1) - pow($b,0) ) / 1, 2);
eval round( 100*( pow($b,.5) - pow($b,0) ) / .5, 2);
eval round( 100*( pow($b,.51) - pow($b,.5) ) / .01, 4);
eval round( 100*( pow($b,.501) - pow($b,.5) ) / .001, 4);
eval round( 100*( pow($b,.5001) - pow($b,.5) ) / .0001, 4);
eval round( 100*( pow($b,.500001) - pow($b,.5) ) / .000001, 2);
eval round( 100*( pow($b,.5) - pow($b,.49) ) / .01, 4);
eval round( 100*( pow($b,.5) - pow($b,.499) ) / .001, 4);
eval round( 100*( pow($b,.5) - pow($b,.4999) ) / .0001, 4);
eval round( 100*( pow($b,.5) - pow($b,.499999) ) / .000001, 2);

With an initial deposit of $100, the balance in a bank account after t years is dollars. Find the average rate of change over [0, 0.5] and [0, 1], then estimate the instantaneous rate of change at t = 0.5.

1.2 Step 1

The rate of change of f(t) is the ratio of the change in f(t) divided by the unit change in t.

Question Sequence

Question 1.1

The account balance f(t) is measured in dollarsJvMTWtHHPxcVgRd4iHmpHpAOkxQe6GvTNzwj4lqqJTw= and t is measured in yearsEFBAAtjIr6f0KWQKxGhjyzhRKP+ZNnARzqfy/+h/RGY=.

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

Question 1.2

Giz6sfD2ivlgA1C+zA3suJnRJfgXYhiZ2COcvr9azBn3EO2LeJEH2Wb7PlR6rZWj81cCCzNfzGvME32GdEXwAdktKpKch0TwR0nbeyljDrUDVfgSDxluUAexCUgUH0nFZry+Hf1rrO+KV/WGt8J8j2RIUir5rVL8PlcPTdgEX1LWNrHOBKSw/TEH4UasuhEMoKI2kMydLiAGITq+Pe2vmkG+oIgDGOVL9TM5dQ==
2
Correct.
That's not right. Check your work.
Incorrect.

1.3 Step 2

The average rate of change of y = f(t) over the interval [t0, t1] is given by the following.

Average rate of change =

Find the average rate of change over the interval [0, 0.5]. (Round your answers to two decimal places.)

Question Sequence

Question 1.3

For the interval [0.0, 0.5], let t0= 0 and t1 = ot+qaZUmHUo=.

2
Correct.
Try again.
Incorrect.

Question 1.4

In this case, f(t1) = 100*(iSba6t70dtA=)0.5 and f(t0) = 100*(iSba6t70dtA=)0

2
Correct.
Try again.
Incorrect.

Question 1.5

Thus the average rate of change is ogclm5qspRNIZA6hUagrKN0l3ew=.

2
Correct.
Try again.
Incorrect.

1.4 Step 3

Find the average rate of change over the interval [0, 1]. (Round your answers to two decimal places.)

Question Sequence

Question 1.6

For the interval [0, 1], let t0= 0 and t1 = 0VV1JcqyBrI=.

In this case, f(t1) = 100*(FB:*$b)1 and f(t0) = 100*(FB:*$b)0

2
Correct.
Try again.
Incorrect.

Question 1.7

Thus the average rate of change is FB:*$avgroc00to10.

2
Correct.
Try again.
Incorrect.

1.5 Step 4

Recall that the instantaneous rate of change at t = t0 is the limit of the average rates of change.

Question Sequence

Question 1.8

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 t = ot+qaZUmHUo=

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

Question 1.9

First calculate the average rate of change over three intervals to the right of t = 0.5. (Round your answers to four decimal places.)

Interval! [0.5, 0.51] [0.5, 0.501] [0.5, 0.5001]
Average rate of change KM4Y0fTBDKXkN24kA9uYBg== tOpUvmG3/QKqNS+t06QXwA== BM6dDvGjjeFBDTN8BeDEVg==
Table

This table suggests the limit of the average rates of change as t approaches 0.5 from the left is approximately XFnw9W4YV2DY62VqrYm0C8FUhMM= dollars per year (rounded to two decimal places).

2
_feedback_correct: Correct.
_feedback_hint: That's not right. Check your work.
_feedback_incorrect: Incorrect.

Question 1.10

Now calculate the average rate of change over three intervals to the left of t = 0.5. (Round your answers to four decimal places.)

Interval [0.49, 0.5] [0.499, 0.500] [0.4999, 0.5000]
Average rate of change bECI2HMu3zb8FISFiyvzAg== AqHQgaV4SHbsguD1jWG/hg== HZJ3TiwmXftPFf5AM9MwDw==

This table suggests the limit of the average rates of change as t approaches 0.5 from the left is approximately FB:*$avg499999toH dollars per year (rounded to two decimal places).

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

Question 1.11

Based on the previous questions, the instantaneous rate of change at t = 0.5 is approximately L1roKTQ32O/N3Yl8O0Zde3nASJA= dollars per year (rounded to two decimal places).

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