a. for the function and point below, find f(a). f(x)=2x^{2}+3x, a = - 3 b. determine an equation of the line…

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)=2x^{2}+3x), (f^\prime(x)=4x + 3).

Step2: Evaluate f'(a)

Substitute (a=-3) into (f^\prime(x)). So (f^\prime(-3)=4\times(-3)+3). [ \begin{align*} f^\prime(-3)&=-12 + 3\ &=-9 \end{align*} ]

Step3: Find f(a)

Substitute (x = a=-3) into (f(x)). (f(-3)=2\times(-3)^{2}+3\times(-3)=2\times9-9=18 - 9=9).

Step4: Find the equation of the tangent line

The point - slope form of a line is (y - y_1=m(x - x_1)), where ((x_1,y_1)=(a,f(a))=(-3,9)) and (m = f^\prime(a)=-9). [ \begin{align*} y-9&=-9(x + 3)\ y-9&=-9x-27\ y&=-9x-18 \end{align*} ]

Answer:

a. (f^\prime(-3)=-9) b. (y=-9x - 18)