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Derivatives

A derivative is simply a rate of change. Whether you're modeling the movement of a particle or a supply/demand model, this is a key instrument of Calculus.

Instantaneous Rate of Change

         

Let \(f(x)=6 x^2+7x-5\). Then if the average rate of change of \(f(x)\) when \(x\) changes from \(0\) to \(18\) is the same as the rate of change of \(f(x)\) at \(x=a\), what is the value of \(a\)?

What is the rate of change of \( y = \frac{x(x-5)^2}{(x+3)^3} \) at \( x = 1 \)?

For a function \(f(x)= x^2+px+q\), if the average rate of change of \(f(x)\) when \(x\) changes from \(a\) to \(b\) is the same as the rate of change at \(x=c\), what is the value of \(c\)?

A girl \(180\) cm tall walks away from the base of a streetlight \(4\) m high. If she follows a straight line at a velocity of \(121\) m/min, what is the rate of change of the length of her shadow (in m/min)?

What is the rate of change of the function \( y = \ln ( 8 x) \) when \( x = \frac{ 1}{16} \)?

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