To find the area of the region bounded by the parabola $y = 3x^2$, the tangent line to this parabola at the point $(4, 48)$, and the x-axis, we can follow these steps:

Step 1: Find the equation of the tangent line

The equation of the tangent line at a point on a curve is given by: $y - y_1 = m(x - x_1)$ where $(x_1, y_1)$ is the point of tangency and $m$ is the slope of the tangent line.

The parabola is given by $y = 3x^2$. To find the slope $m$, we first differentiate $y = 3x^2$ to get the derivative, which gives the slope of the tangent at any point $x$: $\frac{dy}{dx} = 6x$ At the point $(4, 48)$, we substitute $x = 4$ into the derivative to find the slope: $m = 6(4) = 24$

Now, using the point $(4, 48)$ and the slope $m = 24$, the equation of the tangent line is: $y - 48 = 24(x - 4)$ Simplifying this equation: $y - 48 = 24x - 96$ $y = 24x - 48$

Step 2: Find the x-intercepts of the tangent line and the parabola

x-intercept of the tangent line:

Set $y = 0$ in the equation of the tangent line $y = 24x - 48$ to find the x-intercept: $0 = 24x - 48$ $24x = 48$ $x = 2$ So, the x-intercept of the tangent line is $x = 2$.

x-intercept of the parabola:

Set $y = 0$ in the equation of the parabola $y = 3x^2$: $0 = 3x^2$ $x = 0$ So, the x-intercept of the parabola is $x = 0$.

Step 3: Set up the integral for the area

The region of interest is bounded by the x-axis, the parabola $y = 3x^2$, and the tangent line $y = 24x - 48$ from $x = 0$ to $x = 2$. To find the area, we subtract the area under the parabola from the area under the tangent line.

The area under the tangent line from $x = 0$ to $x = 2$ is: $\text{Area}_\text{tangent} = \int_0^2 (24x - 48) \, dx$

The area under the parabola from $x = 0$ to $x = 2$ is: $\text{Area}_\text{parabola} = \int_0^2 3x^2 \, dx$

The total area between the two curves is: $\text{Area} = \text{Area}_\text{tangent} - \text{Area}_\text{parabola}$

Step 4: Compute the integrals

First, calculate the area under the tangent line: $\text{Area}_\text{tangent} = \int_0^2 (24x - 48) \, dx = \left[ 12x^2 - 48x \right]_0^2$ Substitute the limits: $\text{Area}_\text{tangent} = (12(2)^2 - 48(2)) - (12(0)^2 - 48(0)) = (12 \times 4 - 48 \times 2) = (48 - 96) = -48$

Next, calculate the area under the parabola: $\text{Area}_\text{parabola} = \int_0^2 3x^2 \, dx = \left[ x^3 \right]_0^2$ Substitute the limits: $\text{Area}_\text{parabola} = (2^3) - (0^3) = 8$

Now, calculate the total area: $\text{Area} = \text{Area}_\text{tangent} - \text{Area}_\text{parabola} = 48 - 8 = 40$

Step 5: Final answer

Thus, the area of the region bounded by the parabola, the tangent line, and the x-axis is 40 square units.

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Would you like further details on any of the steps, or do you have additional questions?

Here are some related questions to consider:

1. How can we find the area between two curves more generally?
2. What is the significance of the derivative in finding the tangent line to a curve?
3. Can this method be applied to higher-order polynomials and more complex curves?
4. How does the concept of the tangent line relate to linear approximations?
5. How can you use the second derivative to determine the concavity of the parabola?

Tip: When solving area problems involving curves, always ensure you have the correct bounds and pay close attention to where the curves intersect the x-axis.