Question

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

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. \(y=-9x - 18\)