Problem 11E

Give a recursive definition of Pm(n), the product of the integer m and the nonnegative integer n.

Product Rule If F(x)=f(x)g(x), Then F’(x)=f(x)g’(x) + f’(x)g(x) Ex. d/dx [ (4x^3 - x^2 -1)(x^3 -2x^2 + 3x +1)] = (4x^3 - X^2 - 1) (3x^2- 4x + 3) = (12x^2 -2x)(x^3 - 2x^2 + 3x + 1) However the product rule is more useful for the product of different types of functions Ex. f(x)= x^2cos(x) g(x) = e^x arctan(x) Proof of the Product Rule Let F(x) = f(x)g(x) F’(x) = lim ((F(x + h ) - F(x))/h) h 0 =lim ((f(x + h )g( x + h) - f(x)g(x))/h) h 0 Subtract and add f(x + h)g(x) = lim ((f( x + h)g(x + h) - f(x + h)g(x) + f(x + h) - g(x) - f(x)g(x))/h) h 0 = lim ((f( x + h)(g( x + h) - g(x)) + g(x)(f( x + h) - f(x))/h) h 0 = lim ((f ( x+ h)(g( x + h) - g(x))/h) + (g(x)f(x + h) - f(x)/h) h 0 = lim (f( x + h) lim ((g(x + h) - g(x))/h