To find the difference quotient [tex]\(\frac{f(x+h) - f(x)}{h}\)[/tex] for the function [tex]\(f(x) = x^2 + 3x\)[/tex], let's break down the steps.
### Step 1: Define [tex]\(f(x)\)[/tex]
Given the function:
[tex]\[ f(x) = x^2 + 3x \][/tex]
### Step 2: Compute [tex]\(f(x + h)\)[/tex]
Next, we need to find [tex]\(f(x + h)\)[/tex]. Substitute [tex]\(x + h\)[/tex] into the function [tex]\(f(x)\)[/tex]:
[tex]\[ f(x + h) = (x + h)^2 + 3(x + h) \][/tex]
Now, let's expand this expression:
[tex]\[ (x + h)^2 = x^2 + 2xh + h^2 \][/tex]
[tex]\[ 3(x + h) = 3x + 3h \][/tex]
Therefore,
[tex]\[ f(x + h) = x^2 + 2xh + h^2 + 3x + 3h \][/tex]
[tex]\[ f(x + h) = x^2 + 2xh + h^2 + 3x + 3h \][/tex]
### Step 3: Compute the Difference [tex]\(f(x + h) - f(x)\)[/tex]
We now subtract [tex]\(f(x)\)[/tex] from [tex]\(f(x + h)\)[/tex]:
[tex]\[ f(x + h) - f(x) = (x^2 + 2xh + h^2 + 3x + 3h) - (x^2 + 3x) \][/tex]
Simplifying this:
[tex]\[ f(x + h) - f(x) = x^2 + 2xh + h^2 + 3x + 3h - x^2 - 3x \][/tex]
[tex]\[ f(x + h) - f(x) = 2xh + h^2 + 3h \][/tex]
### Step 4: Form the Difference Quotient
The difference quotient is given by:
[tex]\[ \frac{f(x + h) - f(x)}{h} = \frac{2xh + h^2 + 3h}{h} \][/tex]
We can simplify this expression by factoring [tex]\(h\)[/tex] out of the numerator:
[tex]\[ \frac{2xh + h^2 + 3h}{h} = \frac{h(2x + h + 3)}{h} \][/tex]
Canceling [tex]\(h\)[/tex] in the numerator and the denominator:
[tex]\[ \frac{2xh + h^2 + 3h}{h} = 2x + h + 3 \][/tex]
### Final Answer
Hence, the simplified form of the difference quotient [tex]\(\frac{f(x+h) - f(x)}{h}\)[/tex] is:
[tex]\[ 2x + h + 3 \][/tex]
