Question

standard 1 homework score: 22.5/28 answered: 17/20 question 18 determine and fully simplify the difference quotient, (\frac{f(a + h) - f(a)}{h}), for the function (f(x) = x^2 + 4x). (\frac{f(a + h) - f(a)}{h} = ) question help: video message instructor post to forum submit question

Explanation:

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

Substitute \( x = a + h \) into \( f(x)=x^{2}+4x \). So \( f(a + h)=(a + h)^{2}+4(a + h) \). Expand \( (a + h)^{2} \) using the formula \( (m + n)^{2}=m^{2}+2mn + n^{2} \), we get \( (a + h)^{2}=a^{2}+2ah+h^{2} \). Then \( f(a + h)=a^{2}+2ah + h^{2}+4a + 4h \).

Step2: Find \( f(a) \)

Substitute \( x = a \) into \( f(x)=x^{2}+4x \), we have \( f(a)=a^{2}+4a \).

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

Subtract \( f(a) \) from \( f(a + h) \): \( f(a + h)-f(a)=(a^{2}+2ah + h^{2}+4a + 4h)-(a^{2}+4a) \). Simplify this expression by removing the parentheses and combining like terms: \( a^{2}+2ah + h^{2}+4a + 4h - a^{2}-4a=2ah+h^{2}+4h \).

Step4: Divide by \( h \) ( \( h

eq0 \))
Now divide \( f(a + h)-f(a) \) by \( h \): \( \frac{f(a + h)-f(a)}{h}=\frac{2ah + h^{2}+4h}{h} \). Factor out \( h \) from the numerator: \( \frac{h(2a + h + 4)}{h} \). Cancel out the common factor \( h \) (since \( h
eq0 \)), we get \( 2a+h + 4 \).

Answer:

\( 2a + h + 4 \)