a. what is the slope of the tangent line to the graph of f at the given point? b. what is the slope…

Explanation:

Step1: Find the derivative of f(x)

Using the power - rule, if (y = ax^n), then (y^\prime=anx^{n - 1}). For (f(x)=7x^{2}+x - 8), (f^\prime(x)=\frac{d}{dx}(7x^{2})+\frac{d}{dx}(x)-\frac{d}{dx}(8)). So (f^\prime(x)=14x + 1).

Step2: Evaluate the derivative at the given x - value

We want to find the slope of the tangent line at the point ((0,-8)), so we substitute (x = 0) into (f^\prime(x)). When (x = 0), (f^\prime(0)=14\times0+1=1).

Step3: Find the equation of the tangent line

The slope - intercept form of a line is (y=mx + b), where (m) is the slope and (b) is the y - intercept. We know (m = 1) (from part a) and the line passes through the point ((0,-8)). Substituting (x = 0), (y=-8) and (m = 1) into (y=mx + b), we get (-8=1\times0 + b), so (b=-8). The equation of the tangent line is (y=x - 8).

Answer:

a. 1 b. (y=x - 8)