Question

fill in the blanks.
the expression
\\( \frac { f ( x + h ) - f ( x ) } { h } \\), \\( h \
eq 0 \\), is called the
\\( \square \\) of the function f. we find this expression by replacing x with
\\( \square \\) each time x appears in the functions equation. then we
subtract
\\( \square \\). after simplifying, we factor
\\( \square \\) from the numerator and divide out identical factors of
\\( \square \\) in the numerator and denominator.
the expression
\\( \frac { f ( x + h ) - f ( x ) } { h } \\), \\( h \
eq 0 \\), is called the
\\( \square \\) of the function f. we find this expression by replacing x with
\\( \square \\) each time x appears in the functions equation. then we
subtract
\\( \square \\). after simplifying, we factor
\\( \square \\) from the numerator and divide out identical factors of
\\( \square \\) in the numerator and denominator.

Explanation:

The expression \(\frac{f(x + h)-f(x)}{h},h
eq0\) is a fundamental concept in calculus. When dealing with this expression, we first substitute \(x+h\) into the function \(f\) (wherever \(x\) appears in the function's equation). Then we subtract \(f(x)\) from \(f(x + h)\). After simplifying the numerator, we often find that \(h\) is a common factor. We factor out \(h\) from the numerator and divide out the identical factors of \(h\) (since \(h
eq0\)) in the numerator and denominator.

Answer:

The expression \(\frac{f(x + h)-f(x)}{h},h
eq0\) is called the difference quotient of the function \(f\). We find this expression by replacing \(x\) with \(x + h\) each time \(x\) appears in the function's equation. Then we subtract \(f(x)\). After simplifying, we factor \(h\) from the numerator and divide out identical factors of \(h\) in the numerator and denominator.