Slope Unity in Mean Value Theorem: Average Meets Instant

1. finding the average rate of change for the interval \( [a,b] \) using the formula \( f'(c) = \frac{f(b) – f(a)}{b – a} \)
2. finding the derivative of the function using rules of differentiation
3. setting the derivative equal to average rate and solving for \( x \), which will be the coordinate of the \( x \) point that we are looking for, and we call it \(c\)

• secant line passes through the first and the last point of the function, so by finding one of these points using \( (x,f(x)) \) for our function, and using point-slope form, (with the slope being average rate of change of the function in the interval), we get the equation for the secant line.

• to find the equation for the tangent line, we put the point that we found in step 3 above using the mean value theorem, into the original function to find the corresponding \( y \) value of that \( x \). using point-slope form for this \( (x,y) \) point with the slope being average rate of change of the function in the interval, we get the equation for the tangent line.

And \( y=7x-7.1 \) is the tangent line, touching the curve \( y=x^3 \) at the point \( \sqrt{\frac{7}{3}} \). this point is the where the average rate of change of the function in the interval \( [1,2] \) is exactly the same as the instantaneous rate of change of \( y=x^3 \).

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