if f(t) were continious on the interval [-2,1] but youo were only given the function for t=-2,-3/2,-1,-1/2 ..,3/2 and 2, explain how would you apporximate f'(1)
I'd take f(1.5)-f(0.5)
the slope of the line passing through points near f(1) is probably pretty close to f'(1). As the two surrounding points get closer together, the slope more closely approximates f'(1).
To approximate f'(1), we can use the concept of the derivative. The derivative of a function measures the rate at which it is changing at a specific point. In this case, we want to find the approximate value of f'(1) given the available function values at discrete points from -2 to 2.
To calculate an approximation for f'(1), we can use the finite difference approximation. This method approximates the derivative by computing the average rate of change between two neighboring points.
Here's how you can approach this:
1. Find the values of f(t) at the closest points to t = 1. From the given function values, identify the points that are closest to t = 1. In this case, it would be t = 1 and t = 3/2. Let's denote these points as (x1, y1) and (x2, y2).
2. Calculate the average rate of change between these two points. The average rate of change can be found using the formula:
average rate of change = (y2 - y1) / (x2 - x1)
3. Use the average rate of change as an approximation for f'(1). Since the derivative of a function represents the instantaneous rate of change, this approximation gives us an estimate of the derivative at t = 1.
However, it's important to note that this approximation is only an estimate and may not be completely accurate. As the interval between the given points becomes smaller, the approximation will be more precise.
In summary, to approximate f'(1) given the function values at discrete points, you can use the finite difference approximation by finding the closest points to t = 1 and calculating the average rate of change between them.