Answer :
Sure! Let's find the difference quotient for the given functions step by step.
### Problem 83: [tex]\( f(x) = 4x + 3 \)[/tex]
1. Write the definition of the difference quotient:
The difference quotient of a function [tex]\( f \)[/tex] is given by:
[tex]\[
\frac{f(x+h) - f(x)}{h}
\][/tex]
2. Substitute [tex]\( f(x) = 4x + 3 \)[/tex] into the formula:
[tex]\[
f(x + h) = 4(x + h) + 3 = 4x + 4h + 3
\][/tex]
Now, substitute back into the difference quotient formula:
[tex]\[
\frac{f(x+h) - f(x)}{h} = \frac{(4x + 4h + 3) - (4x + 3)}{h}
\][/tex]
3. Simplify the numerator:
[tex]\[
\frac{(4x + 4h + 3) - (4x + 3)}{h} = \frac{4x + 4h + 3 - 4x - 3}{h} = \frac{4h}{h}
\][/tex]
4. Simplify the fraction:
[tex]\[
\frac{4h}{h} = 4
\][/tex]
Therefore, the difference quotient for [tex]\( f(x) = 4x + 3 \)[/tex] is [tex]\( 4 \)[/tex].
### Problem 84: [tex]\( f(x) = -3x + 1 \)[/tex]
1. Write the definition of the difference quotient:
The difference quotient of a function [tex]\( f \)[/tex] is given by:
[tex]\[
\frac{f(x+h) - f(x)}{h}
\][/tex]
2. Substitute [tex]\( f(x) = -3x + 1 \)[/tex] into the formula:
[tex]\[
f(x + h) = -3(x + h) + 1 = -3x - 3h + 1
\][/tex]
Now, substitute back into the difference quotient formula:
[tex]\[
\frac{f(x+h) - f(x)}{h} = \frac{(-3x - 3h + 1) - (-3x + 1)}{h}
\][/tex]
3. Simplify the numerator:
[tex]\[
\frac{(-3x - 3h + 1) - (-3x + 1)}{h} = \frac{-3x - 3h + 1 + 3x - 1}{h} = \frac{-3h}{h}
\][/tex]
4. Simplify the fraction:
[tex]\[
\frac{-3h}{h} = -3
\][/tex]
Therefore, the difference quotient for [tex]\( f(x) = -3x + 1 \)[/tex] is [tex]\( -3 \)[/tex].
To summarize:
- For [tex]\( f(x) = 4x + 3 \)[/tex], the difference quotient is [tex]\( 4 \)[/tex].
- For [tex]\( f(x) = -3x + 1 \)[/tex], the difference quotient is [tex]\( -3 \)[/tex].
### Problem 83: [tex]\( f(x) = 4x + 3 \)[/tex]
1. Write the definition of the difference quotient:
The difference quotient of a function [tex]\( f \)[/tex] is given by:
[tex]\[
\frac{f(x+h) - f(x)}{h}
\][/tex]
2. Substitute [tex]\( f(x) = 4x + 3 \)[/tex] into the formula:
[tex]\[
f(x + h) = 4(x + h) + 3 = 4x + 4h + 3
\][/tex]
Now, substitute back into the difference quotient formula:
[tex]\[
\frac{f(x+h) - f(x)}{h} = \frac{(4x + 4h + 3) - (4x + 3)}{h}
\][/tex]
3. Simplify the numerator:
[tex]\[
\frac{(4x + 4h + 3) - (4x + 3)}{h} = \frac{4x + 4h + 3 - 4x - 3}{h} = \frac{4h}{h}
\][/tex]
4. Simplify the fraction:
[tex]\[
\frac{4h}{h} = 4
\][/tex]
Therefore, the difference quotient for [tex]\( f(x) = 4x + 3 \)[/tex] is [tex]\( 4 \)[/tex].
### Problem 84: [tex]\( f(x) = -3x + 1 \)[/tex]
1. Write the definition of the difference quotient:
The difference quotient of a function [tex]\( f \)[/tex] is given by:
[tex]\[
\frac{f(x+h) - f(x)}{h}
\][/tex]
2. Substitute [tex]\( f(x) = -3x + 1 \)[/tex] into the formula:
[tex]\[
f(x + h) = -3(x + h) + 1 = -3x - 3h + 1
\][/tex]
Now, substitute back into the difference quotient formula:
[tex]\[
\frac{f(x+h) - f(x)}{h} = \frac{(-3x - 3h + 1) - (-3x + 1)}{h}
\][/tex]
3. Simplify the numerator:
[tex]\[
\frac{(-3x - 3h + 1) - (-3x + 1)}{h} = \frac{-3x - 3h + 1 + 3x - 1}{h} = \frac{-3h}{h}
\][/tex]
4. Simplify the fraction:
[tex]\[
\frac{-3h}{h} = -3
\][/tex]
Therefore, the difference quotient for [tex]\( f(x) = -3x + 1 \)[/tex] is [tex]\( -3 \)[/tex].
To summarize:
- For [tex]\( f(x) = 4x + 3 \)[/tex], the difference quotient is [tex]\( 4 \)[/tex].
- For [tex]\( f(x) = -3x + 1 \)[/tex], the difference quotient is [tex]\( -3 \)[/tex].
