Given the function 𝑓(𝑥)=2𝑥2+3𝑥+1,Find the instantaneous rate of change when 𝑥=1 using the secant method.

The secant method involves finding the average rate of change over a small interval around the point of interest. This can be used to approximate the instantaneous rate of change at that point.

Here are the steps to find the instantaneous rate of change of the function f(x) = 2x^2 + 3x + 1 at x = 1 using the secant method:

1. Choose two points close to x = 1, say x = 0.99 and x = 1.01. These points are not too far from 1, so they should give a good approximation.

2. Calculate the function values at these points: f(0.99) = 2*(0.99)^2 + 30.99 + 1 = 2.9701 f(1.01) = 2(1.01)^2 + 3*1.01 + 1 = 3.0301

3. The secant line through the points (0.99, f(0.99)) and (1.01, f(1.01)) has slope given by the difference quotient: (f(1.01) - f(0.99)) / (1.01 - 0.99) = (3.0301 - 2.9701) / (1.01 - 0.99) = 0.06 / 0.02 = 3

So, the instantaneous rate of change of the function f at x = 1 is approximately 3, according to the secant method.