for the function ( f(x)=4x^{2} ), make a table of slopes of secant lines and make a conjecture about the…

for the function ( f(x)=4x^{2} ), make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at ( x = 2 ).\ncomplete the table.\n(do not round until the final answer. then round to the nearest thousandth as needed.)\n| interval | slope of the secant line |\n| ---- | ---- |\n| ( 1,2 ) | 12 |\n| ( 1.5,2 ) | |

for the function ( f(x)=4x^{2} ), make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at ( x = 2 ).\ncomplete the table.\n(do not round until the final answer. then round to the nearest thousandth as needed.)\n| interval | slope of the secant line |\n| ---- | ---- |\n| ( 1,2 ) | 12 |\n| ( 1.5,2 ) | |

Answer

Answer:

$14$

Explanation:

Step1: Recall the formula for the slope of the secant line

The formula for the slope of the secant line between two points (x = a) and (x = b) on the function (y = f(x)) is (m=\frac{f(b)-f(a)}{b - a}).

Step2: Identify (a), (b), and (f(x))

Given (f(x)=4x^{2}), (a = 1.5), and (b = 2). First, find (f(1.5)) and (f(2)). For (x = 1.5), (f(1.5)=4\times(1.5)^{2}=4\times2.25 = 9). For (x = 2), (f(2)=4\times(2)^{2}=16).

Step3: Calculate the slope of the secant line

Substitute into the formula (m=\frac{f(2)-f(1.5)}{2 - 1.5}). (m=\frac{16 - 9}{2-1.5}=\frac{7}{0.5}=14).