Question

find each of the following for ( f(x) = 6x^2 - 8x + 5 ).
(a) ( f(x+h) )
(b) ( f(x+h) - f(x) )
(c) ( \frac{f(x+h) - f(x)}{h} )

(a) ( f(x+h) = square )

(b) ( f(x+h) - f(x) = square )

(c) ( \frac{f(x+h) - f(x)}{h} = square )

Explanation:

Part (a)

Step1: Substitute \( x+h \) into \( f(x) \)

We have \( f(x) = 6x^2 - 8x + 5 \). To find \( f(x + h) \), we replace every \( x \) in the function with \( x + h \). So we get:
\( f(x + h)=6(x + h)^2-8(x + h)+5 \)

Step2: Expand the expressions

First, expand \( (x + h)^2 \) using the formula \( (a + b)^2=a^2 + 2ab + b^2 \), where \( a=x \) and \( b = h \). So \( (x + h)^2=x^2+2xh + h^2 \). Then:
\( 6(x + h)^2=6(x^2 + 2xh+h^2)=6x^2+12xh + 6h^2 \)
And \( -8(x + h)=-8x-8h \)
Now, combine all the terms:
\( f(x + h)=6x^2+12xh + 6h^2-8x-8h + 5 \)

Step1: Substitute \( f(x + h) \) and \( f(x) \)

We know that \( f(x + h)=6x^2+12xh + 6h^2-8x-8h + 5 \) and \( f(x)=6x^2-8x + 5 \). Then \( f(x + h)-f(x)=(6x^2+12xh + 6h^2-8x-8h + 5)-(6x^2-8x + 5) \)

Step2: Simplify the expression

Distribute the negative sign:
\( 6x^2+12xh + 6h^2-8x-8h + 5-6x^2 + 8x-5 \)
Now, combine like terms:
The \( 6x^2 \) and \( -6x^2 \) cancel out, the \( -8x \) and \( 8x \) cancel out, and the \( 5 \) and \( -5 \) cancel out. We are left with \( 12xh+6h^2 - 8h \)

Step1: Substitute \( f(x + h)-f(x) \)

We know that \( f(x + h)-f(x)=12xh + 6h^2-8h \). Then \( \frac{f(x + h)-f(x)}{h}=\frac{12xh + 6h^2-8h}{h} \)

Step2: Factor out \( h \) from the numerator

Factor \( h \) from each term in the numerator: \( \frac{h(12x + 6h-8)}{h} \)

Step3: Cancel out \( h \) (assuming \( h

eq0 \))
Cancel the common factor \( h \) in the numerator and the denominator: \( 12x + 6h-8 \)

Answer:

\( 6x^2 + 12xh + 6h^2-8x - 8h + 5 \)

Part (b)