The expression S(h)= (5+h)^ 2 -25 h

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

The expression S(h)=(5+h)2−25h,h≠0,S(h)=\frac{(5+h)^{2}-25}{h}, \quad h\neq0, models the average rate of change of s(x)=x2s(x)=x^{2} over the interval from x=5x=5 to x=5+hx=5+h. (a) Simplify S(h)S(h) completely for h≠0h\neq0. (b) Determine lim⁡h→0S(h)\displaystyle \lim_{h\to0}S(h), showing every algebraic step. (c) Interpret the value obtained in part (b) in the context of the instantaneous rate of change of s(x)s(x) at x=5x=5. Explain the meaning of your result without performing any additional differentiation rules.

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