Define the function f(x)= cases 1- (ax) x^2 , & x 0

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

Problem

Define the function f(x)={1−cos⁡(ax)x2,x≠0,c,x=0,f(x)=\begin{cases}\dfrac{1-\cos(ax)}{x^2}, & x\neq0,\\ c, & x=0,\end{cases} where aa and cc are real constants. (1) Part 1: Determine the value of cc in terms of aa that makes ff continuous at x=0x=0. Show all work. (2) Part 2: For the value of cc found in part (a), compute f′(0)=lim⁡x→0f(x)−f(0)xf'(0)=\lim_{x\to0}\frac{f(x)-f(0)}{x} in terms of aa. Explain each algebraic step in your computation. (3) Part 3: Find every real value of aa for which the derivative f′(0)f'(0) exists and is finite. Justify your conclusion fully.

Related theory

Browse the problem library