Find the difference quotient for the following functions: 47. f(x) = 5, 48. f(x) = 1/(x+1), 49. f(x) = x/(x+1).

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 $$\frac{f(a+h) - f(a)}{h} = \frac{5 - 5}{h} = \frac{0}{h} = \boxed{\text{0}}$$

48. $f(x) = \frac{1}{x+1}$

• Step 1: Find $f(a)$ $$f(a) = \frac{1}{a+1}$$
• Step 2: Find $f(a+h)$ $$f(a+h) = \frac{1}{(a+h)+1}$$
• Step 3: Find the difference quotient $$\frac{f(a+h) - f(a)}{h} = \frac{\frac{1}{(a+h)+1} - \frac{1}{a+1}}{h}$$ Combine the fractions in the numerator: $$= \frac{\frac{(a+1) - ((a+h)+1)}{((a+h)+1)(a+1)}}{h}$$ $$= \frac{\frac{a+1 - a - h - 1}{((a+h)+1)(a+1)}}{h}$$ $$= \frac{\frac{-h}{((a+h)+1)(a+1)}}{h}$$ $$= \frac{-h}{h((a+h)+1)(a+1)}$$ Cancel $h$: $$= \boxed{\text{}\frac{-1}{(a+h+1)(a+1)}\text{}}$$

49. $f(x) = \frac{x}{x+1}$

• Step 1: Find $f(a)$ $$f(a) = \frac{a}{a+1}$$
• Step 2: Find $f(a+h)$ $$f(a+h) = \frac{a+h}{(a+h)+1}$$
• Step 3: Find the difference quotient $$\frac{f(a+h) - f(a)}{h} = \frac{\frac{a+h}{(a+h)+1} - \frac{a}{a+1}}{h}$$ Combine the fractions in the numerator: $$= \frac{\frac{(a+h)(a+1) - a((a+h)+1)}{((a+h)+1)(a+1)}}{h}$$ 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: $$= \frac{\frac{h}{((a+h)+1)(a+1)}}{h}$$ $$= \frac{h}{h((a+h)+1)(a+1)}$$ Cancel $h$: $$= \boxed{\text{}\frac{1}{(a+h+1)(a+1)}\text{}}$$

50. $f(x) = \frac{x-1}{x}$

• Step 1: Find $f(a)$ $$f(a) = \frac{a-1}{a}$$
• Step 2: Find $f(a+h)$ $$f(a+h) = \frac{(a+h)-1}{a+h}$$
• Step 3: Find the difference quotient $$\frac{f(a+h) - f(a)}{h} = \frac{\frac{a+h-1}{a+h} - \frac{a-1}{a}}{h}$$ Combine the fractions in the numerator: $$= \frac{\frac{a(a+h-1) - (a-1)(a+h)}{a(a+h)}}{h}$$ 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: $$= \frac{\frac{h}{a(a+h)}}{h}$$ $$= \frac{h}{ha(a+h)}$$ Cancel $h$: $$= \boxed{\text{}\frac{1}{a(a+h)}\text{}}$$

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 $$\frac{f(a+h) - f(a)}{h} = \frac{(a^3 + 3a^2h + 3ah^2 + h^3) - a^3}{h}$$ $$= \frac{3a^2h + 3ah^2 + h^3}{h}$$ Factor out $h$ from the numerator: $$= \frac{h(3a^2 + 3ah + h^2)}{h}$$ Cancel $h$: = \boxed{\text{3a^2 + 3ah + h^2}}