Mathematics

Mathematics

Problem

The following items concern the idea of rate of change for a differentiable function. a. (i) State the definition of the derivative of a real-valued function y=f(x)y=f(x) at a point x=ax=a. b. (ii) Distinguish between the average rate of change of ff over an interval [a,a+h][a, a+h] and the instantaneous rate of change at x=ax=a. c. (iii) For the displacement function s(t)=3t2−2t+1s(t)=3t^{2}-2t+1, write the algebraic limit expression that represents the instantaneous velocity at a general time tt; do not evaluate the limit. d. (iv) Explain what information about the motion of the particle can be inferred when the derivative obtained in part (iii) is positive for t>0t>0.
Intuitive explanation

The core concept here is that the derivative measures how fast a function is changing at any instant—it's the slope of the tangent line, not the secant line. The key insight is recognizing that rate of change isn't about distance traveled over a whole interval, but about the limiting behavior as that interval shrinks to zero. This limit transforms an average rate (rise over run) into an instantaneous one, which is what makes calculus powerful for modeling real-world motion and growth.

Intuitive explanation

The core concept here is that the derivative measures how fast a function is changing at any instant—it's the slope of the tangent line, not the secant line. The key insight is recognizing that rate of change isn't about distance traveled over a whole interval, but about the limiting behavior as that interval shrinks to zero. This limit transforms an average rate (rise over run) into an instantaneous one, which is what makes calculus powerful for modeling real-world motion and growth.

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