Calculus Tutorial 2.1.007

 
Problem Statement

eval rand(5,17,2)
eval round(1 + 7.91/100, 2)
eval round( 100*( pow(1.08,.5) - pow(1.08,0) ) / .5, 2);
eval round( 100*( pow(1.08,1) - pow(1.08,0) ) / 1, 2);
eval round( 100*( pow(1.08,.5) - pow(1.08,0) ) / .5, 2);
eval round( 100*( pow(1.08,.51) - pow(1.08,.5) ) / .01, 4);
eval round( 100*( pow(1.08,.501) - pow(1.08,.5) ) / .001, 4);
eval round( 100*( pow(1.08,.5001) - pow(1.08,.5) ) / .0001, 4);
eval round( 100*( pow(1.08,.500001) - pow(1.08,.5) ) / .000001, 2);
eval round( 100*( pow(1.08,.5) - pow(1.08,.49) ) / .01, 4);
eval round( 100*( pow(1.08,.5) - pow(1.08,.499) ) / .001, 4);
eval round( 100*( pow(1.08,.5) - pow(1.08,.4999) ) / .0001, 4);
eval round( 100*( pow(1.08,.5) - pow(1.08,.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.

 
Step 1

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

Question Sequence

Question 1

The account balance f(t) is measured in dollars and t is measured in years.

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

Question 2

State the units of the rate of change of f(t).

A.
B.
C.
D.
E.
2
Correct.
Recall, the units for rato of change are units of output per units of input.
Incorrect.

 
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 3

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

2
Correct.
Try again.
Incorrect.

Question 4

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

2
Correct.
Try again.
Incorrect.

Question 5

Thus the average rate of change is dollars per year.

2
Correct.
Try again.
Incorrect.

 
Step 3

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

Question Sequence

Question 6

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

In this case, f(t1) = 100*()1 and f(t0) = 100*()0

2
Correct.
Try again.
Incorrect.

Question 7

Thus the average rate of change is dollars per year.

2
Correct.
Try again.
Incorrect.

 
Step 4

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

Question Sequence

Question 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 of t =

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

Question 9

First 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

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

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

Question 10

Now 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

This table suggests the limit of the average rates of change as t approaches 0.5 from the right is approximately 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 11

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

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