https://file.aiquickdraw.com/mathbot/file/1729054137774vdz8v42d.jpg

Let's solve the two questions in the image step-by-step.

### 1. **Finding the change in \( y \), \( \Delta y \)**:
The function given is:
\[
y = 4x^2
\]
To find the change in \( y \) when \( x = 4 \) and \( \Delta x = 0.4 \), we use the formula:
\[
\Delta y = f(x + \Delta x) - f(x)
\]
Substituting \( f(x) = 4x^2 \):
\[
\Delta y = 4(x + \Delta x)^2 - 4x^2
\]
Now plug in the values \( x = 4 \) and \( \Delta x = 0.4 \):
\[
\Delta y = 4(4 + 0.4)^2 - 4(4)^2
\]
\[
\Delta y = 4(4.4)^2 - 4(16)
\]
\[
\Delta y = 4(19.36) - 64
\]
\[
\Delta y = 77.44 - 64
\]
\[
\Delta y = 13.44
\]

So, the change in \( y \), \( \Delta y \), is **13.44**.

### 2. **Finding the differential \( dy \)**:
The differential \( dy \) is approximated using the derivative of \( y \) with respect to \( x \). First, find the derivative of \( y = 4x^2 \):
\[
\frac{dy}{dx} = 8x
\]
Now, using \( x = 4 \) and \( dx = 0.4 \), we compute:
\[
dy = \frac{dy}{dx} \cdot dx = 8(4) \cdot 0.4
\]
\[
dy = 32 \cdot 0.4 = 12.8
\]

So, the differential \( dy \) is **12.8**.

---

Would you like further clarification on any of these steps or have any specific questions?

Here are some related questions you might find useful:
1. How does the difference between \( \Delta y \) and \( dy \) change as \( \Delta x \) gets smaller?
2. How is the differential approximation related to the tangent line at a given point?
3. What is the geometric interpretation of the difference between \( \Delta y \) and \( dy \)?
4. Can we apply this process to any polynomial function? If so, how?
5. How do the concepts of limits and differentials relate in calculus?

**Tip:** When \( \Delta x \) is small, \( dy \) provides a good approximation of \( \Delta y \), which is why differentials are often used to approximate small changes in functions.

Find the change in y, Δy when x = 4 and Δx = 0.4. Find the differential dy when x = 4 and dx = 0.4.

Learn how to calculate the change in y (Δy) and the differential (dy) for the function y = 4x^2 when x = 4 and Δx = 0.4. This step-by-step guide walks through key calculus concepts, including function change and differential approximation.

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