The notion of the derivative is often introduced through the idea of slope
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
The notion of the derivative is often introduced through the idea of slope. Answer the following questions to demonstrate your understanding of this connection.
(i) Define the derivative of a real-valued function f at a point x=a by means of a limit.
(ii) Explain, in words, how this limit represents the slope of the tangent to the curve y=f(x) at the point (a,f(a)).
(iii) For the specific function f(x)=x2, consider the secant line joining the points (1,1) and (1+h,(1+h)2). Describe what happens to the slope of this secant line as h→0 and relate your observation to the slope of the tangent at x=1 (no numerical calculation of the limit is required).