Let f(x)=2.2 3x. Estimate the derivative f'(2.3) to within two decimal places by using a small enough interval.
To estimate the derivative f'(2.3) using a small enough interval, we can use the concept of the difference quotient.
The difference quotient is given by the formula:
f'(x) ≈ [f(x + h) - f(x)] / h
where h is a small interval.
Let's choose h to be 0.01. Now we can calculate the estimate for f'(2.3) using the difference quotient:
f'(2.3) ≈ [f(2.3 + 0.01) - f(2.3)] / 0.01
Substituting the function f(x)=2.2 * 3x:
f'(2.3) ≈ [2.2 * 3(2.3 + 0.01) - 2.2 * 3(2.3)] / 0.01
Simplifying this expression will give us the estimate for f'(2.3) to within two decimal places.
To estimate the derivative of f(x) at x = 2.3, we can use the concept of the average rate of change over a small interval.
The derivative of a function at a specific point represents the instantaneous rate of change of the function at that point. However, we can estimate this instantaneous rate of change by calculating the average rate of change over a small interval around that point.
Let's choose a small interval centered around x = 2.3. We can use a small value like 0.1 to ensure that the interval is small enough.
First, we calculate the value of f(2.3), which is given by:
f(2.3) = 2.2 * 3^(2.3)
Next, we calculate the value of f(2.3 + 0.1), which is given by:
f(2.3 + 0.1) = 2.2 * 3^(2.3 + 0.1)
Now, we can calculate the average rate of change over the interval [2.3, 2.3 + 0.1] as the difference in function values divided by the difference in x-values:
Average rate of change = (f(2.3 + 0.1) - f(2.3)) / (2.3 + 0.1 - 2.3)
Finally, we can approximate the derivative f'(2.3) by taking the calculated average rate of change as our estimate.
Note that the more narrow the interval, the better our estimate will be. If you want a more accurate estimate, you can choose an even smaller interval size, like 0.01 or 0.001, and repeat the calculation.
By following this process and performing the calculations, you can estimate the derivative f'(2.3) to within two decimal places.