10. For f(x)=lnx, construct tables, rounded to four decimals, near x=1, x=2, x=5, and x=10. Use tables to estimate f'(1), f'(2), f'(5) and f'(10). Then guess a general formula for f'(x).
I don't understand how to do number 11
11. Estimate f'(2) using the values of f in the table.
ln (1) = 0
ln(1.1) = .09531
dy/dx = .09531/.1 = .9531
ln 2 = .69315
ln 2.1 = .74194
dy/dx = .04879/.1 = .4879
ln 5 = etc
ln 10 = 2.30258
ln 10.1 = 2.31253
dy/dx = 0.09950 which is about 1/10
looks like 1/x in the end
To estimate f'(2) using the values of f in the table, you can use the formula for the derivative of f(x) = lnx, which is f'(x) = 1/x.
Let's construct the table first:
For x = 1:
f(1) = ln(1) = 0
For x = 2:
f(2) = ln(2) ≈ 0.6931
For x = 5:
f(5) = ln(5) ≈ 1.6094
For x = 10:
f(10) = ln(10) ≈ 2.3026
Now, let's estimate f'(2) using the values from the table.
The formula f'(x) = 1/x implies that f'(2) = 1/2, which is approximately 0.5.
Therefore, the estimated value of f'(2) using the table is approximately 0.5.
To estimate f'(2) using the values in the table, we need to use the concept of the derivative. The derivative of a function f(x) represents the rate at which the function is changing at a particular point. In other words, it measures the slope of the function at that point.
To estimate f'(2), we can use the concept of the average rate of change. The average rate of change of a function f(x) between two points (x1, f(x1)) and (x2, f(x2)) is given by:
average rate of change = (f(x2) - f(x1)) / (x2 - x1)
We can use this formula to estimate the derivative f'(2) by taking two surrounding points from the table and plugging them into the formula.
Let's say we have the following values from the table for the function f(x):
x | f(x)
---------
1 | 0
2 | 0.6931
5 | 1.6094
10 | 2.3026
To estimate f'(2), we can choose the points (1, 0) and (5, 1.6094) from the table:
average rate of change = (f(5) - f(1)) / (5 - 1)
= (1.6094 - 0) / 4
= 0.40235
This estimate of the average rate of change gives us an estimate for f'(2).
Therefore, the estimated value of f'(2) using the values from the table is approximately 0.4024.
Note: The above method provides an estimate for f'(2) based on the data provided in the table. To find a more accurate value for f'(2), we would need to use calculus techniques such as taking the limit as the interval approaches 0 or finding the derivative function algebraically.