For each of the following functions find the expression for

For each of the following functions, find the expression for f(x + h) - f(x)/h Simplify each of your expressions far enough so that plugging in h = 0 would be allowed (a) f(x) = x^2 - 7x (b) f(x) = 2x + 3 (c) f(x) = x^2 - 7 (d) f(x) = 2 - x^2 (e) f(x) = - pi x^2 - pi^2 (f) f(x) = squareroot x - 9

Solution

f(x) = x2-7x. Then [f(x+h)-f(x)]/h = [ {(x+h)2 -7(x+h) }-{ x2-7x}]/h = [x2+2hx+h2- 7x-7h–x2+7x]/h = (h2+2hx -7h)/h = (h+2x -7). f(x) = 2x+3. Then [f(x+h)-f(x)]/h =[ {2(x+h)+3} –(2x+3)]/h = (2x+2h+3-2x-3)/h = 2h/h = 2. f(x) = x2-7. Then [f(x+h)-f(x)]/h = [ {(x+h)2-7}-{ x2-7}]/h = [x2+2hx+h2-7-x2+7]/h = (2hx+h2)/h = 2x+h. f(x) = 2- x2 . Then [f(x+h)-f(x)]/h = [{ 2-(x+h)2}-(2- x2)]/h = [2-( x2+2hx+h2)-2+x2]/h = (-2hx-h2)/h = -(2x+h). f(x) = -x2-2. Then [f(x+h)-f(x)]/h= [ {- (x+h)2-2}-(-x2-2)]/h = [-(x2+2hx+h2)- 2+ x2+2]/h = (-2 hx- h2)/h = -(2 x+ h). f(x) = (x-9)1/2 =[ (x-9)1/2 (x-9)1/2]/ (x-9)1/2 = (x-9)/ (x-9)1/2 . Then [f(x+h)-f(x)] = [ (x+h-9)/(x+h-9)1/2 ]- [(x-9)/(x-9)1/2] = [(x+h-9) (x-9)1/2 –(x-9) (x+h-9)1/2]/ (x+h-9)1/2(x-9)1/2 = (x-9)1/2(x+h-9)1/2 [(x+h-9)1/2-(x-9)1/2]/ (x+h-9)1/2(x-9)1/2 = [(x+h-9)1/2-(x-9)1/2]. Hence, [f(x+h)-f(x)]/h = [(x+h-9)1/2-(x-9)1/2]/h= [(x+h-9)- (x-9)]/h