Let f(x)=x^3+3. Estimate the following to within two decimal places by using small enough intervals.
Estimate what?
To estimate the values of the function f(x) within two decimal places, we can use the concept of limits. By selecting small enough intervals and evaluating the function at the endpoints of these intervals, we can get an approximation of the function's values.
Let's estimate the value of f(x) for x = 1.
Step 1: Choose an interval around x = 1. Let's select the interval [0.9, 1.1].
Step 2: Evaluate the function at the endpoints of the interval:
- f(0.9) = (0.9)^3 + 3 ≈ 4.03
- f(1.1) = (1.1)^3 + 3 ≈ 4.43
Hence, the estimated value of f(1) within two decimal places is between 4.03 and 4.43.
You can repeat this process for other values you want to estimate, selecting suitable intervals and evaluating the function at their endpoints.