Question

1. - / 5 points find the difference quotient and simplify your answer. ( f(x) = x^2 - 2x + 7 ), ( \frac{f(3 + h) - f(3)}{h} ), ( h

eq 0 )

Explanation:

Step1: Find \( f(3 + h) \)

Substitute \( x = 3 + h \) into \( f(x)=x^{2}-2x + 7 \).
\( f(3 + h)=(3 + h)^{2}-2(3 + h)+7 \)
Expand \( (3 + h)^{2}=9 + 6h+h^{2} \) and \( -2(3 + h)=-6 - 2h \).
So \( f(3 + h)=9 + 6h+h^{2}-6 - 2h + 7 \)
Simplify: \( f(3 + h)=h^{2}+4h + 10 \)

Step2: Find \( f(3) \)

Substitute \( x = 3 \) into \( f(x)=x^{2}-2x + 7 \).
\( f(3)=3^{2}-2\times3 + 7=9 - 6 + 7 = 10 \)

Step3: Compute \( f(3 + h)-f(3) \)

Substitute the values of \( f(3 + h) \) and \( f(3) \):
\( f(3 + h)-f(3)=(h^{2}+4h + 10)-10=h^{2}+4h \)

Step4: Compute the difference quotient \( \frac{f(3 + h)-f(3)}{h} \)

Divide \( f(3 + h)-f(3) \) by \( h \) (\( h
eq0 \)):
\( \frac{h^{2}+4h}{h}=\frac{h(h + 4)}{h}=h + 4 \) (since \( h
eq0 \), we can cancel \( h \))

Answer:

\( h + 4 \)