Question

use the four-step process to find r’(x) and then find r’(1), r’(2), and r’(3). r(x) = 1 + 4x² r’(x) = □

Explanation:

Step1: Compute r(x+h)

$r(x+h) = 1 + 4(x+h)^2 = 1 + 4(x^2 + 2xh + h^2) = 1 + 4x^2 + 8xh + 4h^2$

Step2: Find r(x+h)-r(x)

$r(x+h)-r(x) = [1 + 4x^2 + 8xh + 4h^2] - [1 + 4x^2] = 8xh + 4h^2$

Step3: Calculate difference quotient

$\frac{r(x+h)-r(x)}{h} = \frac{8xh + 4h^2}{h} = 8x + 4h$

Step4: Take limit as h→0

$r'(x) = \lim_{h \to 0}(8x + 4h) = 8x$

Step5: Evaluate r'(1), r'(2), r'(3)

$r'(1)=8(1)=8$, $r'(2)=8(2)=16$, $r'(3)=8(3)=24$

Answer:

$r'(x) = 8x$
$r'(1) = 8$, $r'(2) = 16$, $r'(3) = 24$