Question

attempt 1: 10 attempts remaining. given ( f(x) = 5x - 7 ), use the difference quotient in a table to estimate ( f(3) ). (in your calculations, do not round, but for your final answer, round to the nearest whole number.) ( f(3) = ) (\boxed{}) submit answer next item

Explanation:

Step1: Recall the difference quotient formula

The difference quotient to estimate the derivative \( f'(a) \) is \( \frac{f(a + h)-f(a)}{h} \), where \( h \) is a small number (we can use \( h = 0.0001 \) for a good estimate). Here, \( a = 3 \) and \( f(x)=5x - 7 \).

First, calculate \( f(3 + h) \) and \( f(3) \).

For \( f(3) \): Substitute \( x = 3 \) into \( f(x) \), we get \( f(3)=5(3)-7=15 - 7 = 8 \).

For \( f(3 + h) \): Substitute \( x = 3+h \) into \( f(x) \), we get \( f(3 + h)=5(3 + h)-7=15+5h - 7=8 + 5h \).

Step2: Apply the difference quotient formula

The difference quotient is \( \frac{f(3 + h)-f(3)}{h}=\frac{(8 + 5h)-8}{h}=\frac{5h}{h} \).

Simplify the expression: \( \frac{5h}{h}=5 \) (for \( h
eq0 \)). As \( h \) approaches 0, this value approaches the derivative. So regardless of the small \( h \) we choose (as long as \( h
eq0 \)), the difference quotient gives 5.

Answer:

5