To determine the angle between the functions $f(x) = 4x^2$ and $g(x) = 3x$, we need to interpret the angle between their slopes (derivatives) at a given point. This is the angle between the tangents to the curves at a particular point, which is often called the angle between the curves at that point.

Steps:

1. Find the derivatives of both functions, since the angle between curves is related to the angle between their tangents (slopes) at a given point: $f'(x) = \frac{d}{dx}(4x^2) = 8x$ $g'(x) = \frac{d}{dx}(3x) = 3$

2. Use the formula for the angle $\theta$ between two curves, which is derived from the dot product formula for vectors. The formula is: $\tan \theta = \left| \frac{f'(x) - g'(x)}{1 + f'(x)g'(x)} \right|$ This gives the tangent of the angle between the tangents to the two curves at a specific point.

3. Choose a point where you want to find the angle. Typically, if no specific point is given, you might use $x = 1$ or another convenient point. Let's use $x = 1$ here.

   • At $x = 1$, we have: $f'(1) = 8(1) = 8, \quad g'(1) = 3$

4. Substitute into the formula for the tangent of the angle: $\tan \theta = \left| \frac{8 - 3}{1 + 8 \cdot 3} \right| = \left| \frac{5}{1 + 24} \right| = \frac{5}{25} = 0.2$

5. Find the angle: $\theta = \tan^{-1}(0.2) \approx 11.31^\circ$

Meaning:

• This is the angle between the tangents to the two curves $f(x)$ and $g(x)$ at the point $x = 1$.
• It's not the angle between vectors, but rather the angle between the slopes (tangents) of the curves at a specific point. It measures how much the two curves deviate from each other at that point.

Would you like more details or have any questions?

Here are five related questions:

1. What is the geometric interpretation of the derivative as the slope of a tangent line?
2. Can the angle between two curves be greater than $90^\circ$?
3. How does the angle between curves change as the point of tangency changes?
4. Is there a special case where two curves would have a zero angle between them?
5. How would the angle change if $g(x)$ were a constant function?

Tip: The angle between two curves can be visualized as the deviation between their tangent lines at a point of interest. This gives insight into how sharply two functions change with respect to each other at that point.