Question

use the four - step process to find f(x) and then find f(1), f(2), and f(3). f(x)=2x^{2}+3x - 2 f(x)= (type an integer or a simplified fraction.) f(1)= (type an integer or a simplified fraction.) f(2)= (type an integer or a simplified fraction.) f(3)= (type an integer or a simplified fraction.)

Explanation:

Step1: Recall power - rule for differentiation

The power - rule states that if \(y = ax^n\), then \(y^\prime=anx^{n - 1}\), and the derivative of a constant \(c\) is \(0\). Given \(f(x)=2x^{2}+3x - 2\).
The derivative of \(2x^{2}\) is \(2\times2x^{2 - 1}=4x\), the derivative of \(3x\) is \(3\times1x^{1 - 1}=3\), and the derivative of \(-2\) is \(0\). So \(f^\prime(x)=4x + 3\).

Step2: Find \(f^\prime(1)\)

Substitute \(x = 1\) into \(f^\prime(x)\). \(f^\prime(1)=4\times1+3=7\).

Step3: Find \(f^\prime(2)\)

Substitute \(x = 2\) into \(f^\prime(x)\). \(f^\prime(2)=4\times2+3=8 + 3=11\).

Step4: Find \(f^\prime(3)\)

Substitute \(x = 3\) into \(f^\prime(x)\). \(f^\prime(3)=4\times3+3=12 + 3=15\).

Answer:

\(f^\prime(x)=4x + 3\)
\(f^\prime(1)=7\)
\(f^\prime(2)=11\)
\(f^\prime(3)=15\)