Question

1. find the difference quotient, \\(\frac{f(x+h)-f(x)}{h}\\), given \\(f(x) = x^2 - 3x\\).

a) \\(h - 3\\)

b) \\(2x + h - 3\\)

c) \\(-3\\)

d) \\(2x + h + 3\\)

Explanation:

Step1: Compute \( f(x + h) \)

Substitute \( x + h \) into \( f(x) = x^2 - 3x \). So, \( f(x + h)=(x + h)^2 - 3(x + h) \). Expand \( (x + h)^2 \) as \( x^2 + 2xh + h^2 \) and distribute \( -3 \) in \( -3(x + h) \) to get \( -3x - 3h \). Thus, \( f(x + h)=x^2 + 2xh + h^2 - 3x - 3h \).

Step2: Compute \( f(x + h)-f(x) \)

Subtract \( f(x)=x^2 - 3x \) from \( f(x + h) \). So, \( (x^2 + 2xh + h^2 - 3x - 3h)-(x^2 - 3x) \). Remove the parentheses: \( x^2 + 2xh + h^2 - 3x - 3h - x^2 + 3x \). Combine like terms: \( x^2 - x^2 + 2xh + h^2 - 3x + 3x - 3h = 2xh + h^2 - 3h \).

Step3: Compute the difference quotient \( \frac{f(x + h)-f(x)}{h} \)

Divide \( 2xh + h^2 - 3h \) by \( h \) (assuming \( h
eq0 \)). Factor out \( h \) from the numerator: \( \frac{h(2x + h - 3)}{h} \). Cancel out \( h \) from numerator and denominator, we get \( 2x + h - 3 \).

Answer:

B) \( 2x + h - 3 \)