What is the equation of the tangent to the curve:𝑦=2𝑥3+𝑥2y=2x 3 +x 2 y, equals, 2, x, cubed, plus, x, squaredat the point where 𝑥=2x=2x, equals, 2?

To find the equation of the tangent to the curve $y = 2x^3 + x^2$ at the point where $x = 2$, we can follow these steps:

1. Break Down the Problem

1. Calculate $y$ at $x = 2$.
2. Find the derivative of the curve to get the slope of the tangent at that point.
3. Use the point-slope form to find the equation of the tangent line.

2. Relevant Concepts

• The derivative $\frac{dy}{dx}$ gives the slope of the tangent line at any point on the curve.
• The point-slope form of a line's equation is given by: $y - y_1 = m(x - x_1)$ where $m$ is the slope and $(x_1, y_1)$ is the point through which the line passes.

3. Analysis and Detail

1. Calculate $y$ at $x = 2$: $y = 2(2^3) + (2^2) = 2(8) + 4 = 16 + 4 = 20$ So the point is $(2, 20)$.

2. Find the derivative of the curve: $\frac{dy}{dx} = \frac{d}{dx}(2x^3 + x^2) = 6x^2 + 2x$

   Now, substitute $x = 2$ to find the slope: $\frac{dy}{dx}\bigg|_{x=2} = 6(2^2) + 2(2) = 6(4) + 4 = 24 + 4 = 28$ So the slope $m = 28$.

3. Use the point-slope form: $y - 20 = 28(x - 2)$ Expanding this gives: $y - 20 = 28x - 56$ Thus, $y = 28x - 36$

4. Verify and Summarize

We calculated $y$ at $x = 2$ and found the point to be $(2, 20)$. The slope of the tangent was found to be 28. Substituting into the point-slope form, we derived the equation of the tangent line.

Final Answer

The equation of the tangent to the curve at the point where $x = 2$ is: $y = 28x - 36$