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 person $$180$$ 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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