Question

Compute the average rate of change of g(x)=x^2-1/4 between x=1 and x=3

Answer 1

Average Rate Of Change- It's Really Just The Slope, And We Can Use The Fancy Formula Below, But That's Unnecessary, We Can Just Find Two Coordinate Points

$\frac{g(x + h) - g(x)}{h}$
X=1
$g(x) = {x}^{2} - \frac{1}{4} \\ g(1) = (1)^{2} - \frac{1}{4} \\ g(1) = \frac{3}{4} \\ (1, \frac{3}{4} )$
x=3
$g(x) = {x}^{2} - \frac{1}{4} \\ g(3) = (3)^{2} - \frac{1}{4} \\ g(3) = 9 - \frac{1}{4} \\ g(3) = \frac{35}{4} \\ (3, \frac{35}{4} )$
Slope
$\frac{y_2- y_1}{x_2- x_1}$
$(1, \frac{3}{4} ) \\ (3, \frac{35}{4} ) \:$

$\frac{ \frac{35}{4} - \frac{3}{4} }{3- 1} \: = \frac{ \frac{32}{4} }{2} = \frac{32}{4} \times \frac{1}{2} = \frac{32}{4} = 8$

ANSWER

$8 \: avg \: rate \: of \: change \: between \: points$