This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions for problems 47, 48, 49, 50, and 52. 47. f(x) = 5 Step 1: Find f(a) f(a) = 5 Step 2: Find f(a+h) f(a+h) = 5 Step 3: Find the difference quotient (f(a+h) - f(a))/(h) = (5 - 5)/(h) = (0)/(h) = 0 48. f(x) = (1)/(x+1) Step 1: Find f(a) f(a) = (1)/(a+1) Step 2: Find f(a+h) f(a+h) = (1)/((a+h)+1) Step 3: Find the difference quotient (f(a+h) - f(a))/(h) = (1)/((a+h)+1) - (1)/(a+1)h Combine the fractions in the numerator: = ((a+1) - ((a+h)+1))/(((a+h)+1)(a+1))h = (a+1 - a - h - 1)/(((a+h)+1)(a+1))h = (-h)/(((a+h)+1)(a+1))h = (-h)/(h((a+h)+1)(a+1)) Cancel h: = (-1)/((a+h+1)(a+1)) 49. f(x) = (x)/(x+1) Step 1: Find f(a) f(a) = (a)/(a+1) Step 2: Find f(a+h) f(a+h) = (a+h)/((a+h)+1) Step 3: Find the difference quotient (f(a+h) - f(a))/(h) = (a+h)/((a+h)+1) - (a)/(a+1)h Combine the fractions in the numerator: = ((a+h)(a+1) - a((a+h)+1))/(((a+h)+1)(a+1))h Expand the numerator: (a+h)(a+1) - a(a+h+1) = (a^2 + a + ah + h) - (a^2 + ah + a) = a^2 + a + ah + h - a^2 - ah - a = h Substitute back into the expression: = (h)/(((a+h)+1)(a+1))h = (h)/(h((a+h)+1)(a+1)) Cancel h: = (1)/((a+h+1)(a+1)) 50. f(x) = (x-1)/(x) Step 1: Find f(a) f(a) = (a-1)/(a) Step 2: Find f(a+h) f(a+h) = ((a+h)-1)/(a+h) Step 3: Find the difference quotient (f(a+h) - f(a))/(h) = (a+h-1)/(a+h) - (a-1)/(a)h Combine the fractions in the numerator: = (a(a+h-1) - (a-1)(a+h))/(a(a+h))h Expand the numerator: a(a+h-1) - (a-1)(a+h) = (a^2 + ah - a) - (a^2 + ah - a - h) = a^2 + ah - a - a^2 - ah + a + h = h Substitute back into the expression: = (h)/(a(a+h))h = (h)/(ha(a+h)) Cancel h: = (1)/(a(a+h)) 52. f(x) = x^3 Step 1: Find f(a) f(a) = a^3 Step 2: Find f(a+h) f(a+h) = (a+h)^3 Expand (a+h)^3: f(a+h) = a^3 + 3a^2h + 3ah^2 + h^3 Step 3: Find the difference quotient (f(a+h) - f(a))/(h) = ((a^3 + 3a^2h + 3ah^2 + h^3) - a^3)/(h) = (3a^2h + 3ah^2 + h^3)/(h) Factor out h from the numerator: = (h(3a^2 + 3ah + h^2))/(h) Cancel h: = 3a^2 + 3ah + h^2