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3. If [tex]$y = x^2 - 3x$[/tex], find [tex]$\Delta y - dy$[/tex] in terms of [tex][tex]$x$[/tex][/tex] and [tex]$\Delta x$[/tex].

Sagot :

GinnyAnswer
Sure, let's solve this step by step.

### Step 1: Define the function and the change in [tex]\( y \)[/tex]
Given the function [tex]\( y = x^2 - 3x \)[/tex]:
1. The change in [tex]\( y \)[/tex] when [tex]\( x \)[/tex] changes by [tex]\( \Delta x \)[/tex] is denoted as [tex]\( \Delta y \)[/tex] and is given by:
[tex]\[ \Delta y = y(x + \Delta x) - y(x) \][/tex]

### Step 2: Calculate [tex]\( y(x + \Delta x) \)[/tex]
First, we need to find [tex]\( y(x + \Delta x) \)[/tex]:
[tex]\[ y(x + \Delta x) = (x + \Delta x)^2 - 3(x + \Delta x) \][/tex]
Expanding this:
[tex]\[ y(x + \Delta x) = x^2 + 2x(\Delta x) + (\Delta x)^2 - 3x - 3(\Delta x) \][/tex]
So:
[tex]\[ y(x + \Delta x) = x^2 + 2x(\Delta x) + (\Delta x)^2 - 3x - 3(\Delta x) \][/tex]

### Step 3: Calculate [tex]\( \Delta y \)[/tex]
Now, compute [tex]\( \Delta y \)[/tex]:
[tex]\[ \Delta y = y(x + \Delta x) - y(x) \][/tex]
[tex]\[ \Delta y = [x^2 + 2x(\Delta x) + (\Delta x)^2 - 3x - 3(\Delta x)] - [x^2 - 3x] \][/tex]
[tex]\[ \Delta y = x^2 + 2x(\Delta x) + (\Delta x)^2 - 3x - 3(\Delta x) - x^2 + 3x \][/tex]
[tex]\[ \Delta y = 2x(\Delta x) + (\Delta x)^2 - 3(\Delta x) \][/tex]
[tex]\[ \Delta y = 2x(\Delta x) - 3(\Delta x) + (\Delta x)^2 \][/tex]

### Step 4: Calculate the differential [tex]\( dy \)[/tex]
The differential [tex]\( dy \)[/tex] is found by taking the derivative of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex] and multiplying by [tex]\( \Delta x \)[/tex].

First, compute the derivative of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{dy}{dx} = 2x - 3 \][/tex]

So:
[tex]\[ dy = \left(\frac{dy}{dx}\right) \Delta x \][/tex]
[tex]\[ dy = (2x - 3) \Delta x \][/tex]

### Step 5: Calculate [tex]\( \Delta y - dy \)[/tex]
Finally, we compute the difference:
[tex]\[ \Delta y - dy \][/tex]
[tex]\[ \Delta y - dy = [2x(\Delta x) - 3(\Delta x) + (\Delta x)^2] - [(2x - 3) \Delta x] \][/tex]
[tex]\[ \Delta y - dy = 2x(\Delta x) - 3(\Delta x) + (\Delta x)^2 - 2x(\Delta x) + 3(\Delta x) \][/tex]
[tex]\[ \Delta y - dy = (\Delta x)^2 \][/tex]

### Conclusion
The expression for [tex]\( \Delta y - dy \)[/tex] in terms of [tex]\( x \)[/tex] and [tex]\( \Delta x \)[/tex] is:
[tex]\[ \Delta y - dy = (\Delta x)^2 \][/tex]

Using the example values [tex]\( x = 1 \)[/tex] and [tex]\( \Delta x = 0.1 \)[/tex]:
[tex]\[ \Delta y - dy = (0.1)^2 = 0.01 \][/tex]

So, with the given values, the final answer is [tex]\( 0.01 \)[/tex].
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