Answer the following questions concerning the notion of derivative.
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Answer the following questions concerning the notion of derivative.
(i) State the definition of the derivative of a real-valued function f at the point x=a.
(ii) Describe geometrically why the derivative of f at a is interpreted as the slope of the tangent to the graph of y=f(x) at the point (a,f(a)).
(iii) For the function f(x)=∣x∣, test whether the derivative exists at x=0. Show every step of your working and give a clear reason for your conclusion.