2. Let A(x) = [[|t|dt. (a) Write down a piecewise expression for A(x). (HINT: draw the...

Transcribed Image Text:

2. Let A(x) = [[|t|dt. (a) Write down a piecewise expression for A(x). (HINT: draw the graph of |t| and calculate the area A(x) geometrically.) (b) Sketch a graph of the function from part (a) on the interval [-3,3]. (c) Differentiate the answer from (a) to calculate A'(x). NOTE : To give a correct argument for the derivative of the function, it is best to split the calculation of the derivative A'(xo) into three different cases. The case where xo > 0, the case where x < 0, and the case where xo 0. In the last case you will need to use the limit definition of the derivative, and consider lim and lim separately. = h0+ h0- (d) What does the Fundamental Theorem of Calculus say that A'(x) will be? Does this agree with your answer from (c)? 2. Let A(x) = [[|t|dt. (a) Write down a piecewise expression for A(x). (HINT: draw the graph of |t| and calculate the area A(x) geometrically.) (b) Sketch a graph of the function from part (a) on the interval [-3,3]. (c) Differentiate the answer from (a) to calculate A'(x). NOTE : To give a correct argument for the derivative of the function, it is best to split the calculation of the derivative A'(xo) into three different cases. The case where xo > 0, the case where x < 0, and the case where xo 0. In the last case you will need to use the limit definition of the derivative, and consider lim and lim separately. = h0+ h0- (d) What does the Fundamental Theorem of Calculus say that A'(x) will be? Does this agree with your answer from (c)?