(a) Make a table of values rounded to two decimal places for the function
f(x) = ex for x = 4, 4.5, 5, 5.5, and 6. Then use the table to answer parts (b) and (c).
(b) Find the average rate of change of
f(x) between x = 4 and x = 6. (Round your answer to two decimal places.)
I am sure you meant:
f(x) = e^x
b)
f(6) = e^6
f(4) = e^4
avg rate of change from x=4 to x=6
= (e^6 - e^4)/(6-4)
= ... , use your calculator
To create a table of values for the function f(x) = ex for the given x-values, we can use a calculator or a mathematical software tool. Since we are asked to round the values to two decimal places, we will do so.
First, let's calculate the values for x = 4, 4.5, 5, 5.5, and 6 using the exponential function f(x) = ex:
For x = 4:
f(4) = e^4 ≈ 54.60
For x = 4.5:
f(4.5) = e^4.5 ≈ 90.02
For x = 5:
f(5) = e^5 ≈ 148.41
For x = 5.5:
f(5.5) = e^5.5 ≈ 244.69
For x = 6:
f(6) = e^6 ≈ 403.43
Now, let's create the table:
| x | f(x) |
|:-------:|:-------:|
| 4.00 | 54.60 |
| 4.50 | 90.02 |
| 5.00 | 148.41 |
| 5.50 | 244.69 |
| 6.00 | 403.43 |
Now, let's move on to part (b), finding the average rate of change of f(x) between x = 4 and x = 6.
To find the average rate of change of a function, we use the formula:
Average rate of change = (f(b) - f(a))/(b - a),
where a and b are the initial and final values of x that we are considering. In this case, a = 4 and b = 6.
Using the table, we can substitute the values:
Average rate of change = (f(6) - f(4))/(6 - 4)
Substituting the corresponding values from the table:
Average rate of change = (403.43 - 54.60)/(6 - 4)
= 348.83/2
= 174.41
Therefore, the average rate of change of f(x) between x = 4 and x = 6 is approximately 174.41.