Math / Calculus

QuestionQuestion 7
Let $f(x)=\sqrt{3 x+4}$ . Calculate the difference quotient: $\frac{f(x+h)-f(x)}{h}$

Studdy Solution

STEP 1

What is this asking? We're taking a function, tweaking it a tiny bit, and seeing how much it changes.
It's like poking a bouncy ball and seeing how far it jumps! Watch out! Don't forget to handle those square roots carefully!
Also, remember that h isn't zero.

STEP 2

1. Define the function
2. Set up the difference quotient
3. Simplify the expression

STEP 3

Alright, let's define our function!
We have $f(x) = \sqrt{3x + 4}$ .
This tells us what happens to x – it gets multiplied by 3, then we add 4, and finally, we take the square root of the whole thing.

STEP 4

Now, let's build our difference quotient: $\frac{f(x+h) - f(x)}{h}$ This looks scary, but it's just a fancy way of calculating the change in $f$ divided by the change in x.

STEP 5

Let's plug in our function: $\frac{\sqrt{3(x+h) + 4} - \sqrt{3x + 4}}{h}$ We've replaced x with (x+h) in the first square root.

STEP 6

Time to simplify!
We'll use a trick called multiplying by the conjugate.
The conjugate of $a - b$ is $a + b$ .
Multiplying by the conjugate helps us get rid of those pesky square roots.
Remember, we can multiply by one without changing the value of the expression.
Our "one" will be the conjugate divided by itself.

STEP 7

Here's the conjugate: $\sqrt{3(x+h) + 4} + \sqrt{3x + 4}$ .
Let's multiply:
$\frac{\sqrt{3(x+h) + 4} - \sqrt{3x + 4}}{h} \cdot \frac{\sqrt{3(x+h) + 4} + \sqrt{3x + 4}}{\sqrt{3(x+h) + 4} + \sqrt{3x + 4}}$

STEP 8

This gives us: $\frac{(3(x+h) + 4) - (3x + 4)}{h(\sqrt{3(x+h) + 4} + \sqrt{3x + 4})}$ Notice how the square roots disappeared in the numerator!
That's the magic of the conjugate!

STEP 9

Let's distribute and simplify the numerator: $\frac{3x + 3h + 4 - 3x - 4}{h(\sqrt{3(x+h) + 4} + \sqrt{3x + 4})}$ The 3x and the 4 add to zero with -3x and -4 respectively.

STEP 10

We're left with: $\frac{3h}{h(\sqrt{3(x+h) + 4} + \sqrt{3x + 4})}$ Now, we can divide the numerator and denominator by h (since $h \ne 0$ ): $\frac{3}{\sqrt{3(x+h) + 4} + \sqrt{3x + 4}}$

STEP 11

Our final simplified difference quotient is: $\frac{3}{\sqrt{3(x+h) + 4} + \sqrt{3x + 4}}$

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