For the function $$ y=2 x^{2}+7 x+3 $$, at the point $$ x=8 $$, find the following, (a).the.slage of the tangent to the curve Enter an exact number (b) the instantaneous rate of change of the function

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To analyze the function $$ y = 2x^2 + 7x + 3 $$ at the point $$ x = 8 $$, we'll perform the following steps:

### (a) The Slope of the Tangent to the Curve at $$ x = 8 $$

The slope of the tangent to the curve at a specific point is given by the derivative of the function evaluated at that point.

1. **Find the derivative of the function:**
   $$
   \frac{dy}{dx} = \frac{d}{dx}(2x^2 + 7x + 3) = 4x + 7
   $$

2. **Evaluate the derivative at $$ x = 8 $$:**
   $$
   \left.\frac{dy}{dx}\right|_{x=8} = 4(8) + 7 = 32 + 7 = 39
   $$

**Answer for (a):**
$$
\text{Slope of the tangent at } x = 8 \text{ is } 39.
$$

### (b) The Instantaneous Rate of Change of the Function at $$ x = 8 $$

The instantaneous rate of change of a function at a particular point is essentially the slope of the tangent to the curve at that point. Therefore, it is also given by the derivative evaluated at that point.

As calculated in part (a):
$$
\text{Instantaneous rate of change at } x = 8 \text{ is } 39.
$$

**Answer for (b):**
$$
\text{The instantaneous rate of change at } x = 8 \text{ is } 39.
$$
At $$ x = 8 $$, the slope of the tangent to the curve is 39, and the instantaneous rate of change of the function is also 39.

For the function \( y=2 x^{2}+7 x+3 \), at the point \( x=8 \), find the following, (a).the.slage of the tangent to the curve Enter an exact number (b) the instantaneous rate of change of the function

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