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Differentiability

Those friendly functions that don't contain breaks, bends or cusps are "differentiable". Take their derivative, or just infer some facts about them from the Mean Value Theorem.

Approximate Rate of Change

         

What is the average rate of change of the function \(f(x)=6x^2+px+q\) on the interval \([a,b] ?\)

Given the symmetric difference quotient, \[f'(a) \approx \frac{f(a+h)-f(a-h)}{2h}, \] approximate \(f'(2)\) using the symmetric difference quotient for the function \(f(x)=x^4\) with \(h=0.04.\)

What is the average rate of change of the function \(y=2x^2+7\) when the value of \(x\) changes from \(5\) to \(7?\)

What is the slope of the secant line that passes through the two points \((x,f(x))\) on the graph of the function \(f(x)=x^2,\) for \(x=2\) and \(x=2.004?\)

If \(f(x)=-7x+3,\) what is the average rate of change of \(f(x)\) when \(x\) changes from \(4\) to \(7?\)

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