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\n| interval | slope of the secant line |\n| ---- | ---- |\n| ( 1,2 ) | |

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\n| interval | slope of the secant line |\n| ---- | ---- |\n| ( 1,2 ) | |

Answer

Explanation:

Step1: Recall the slope formula for secant line

The slope formula for the secant line between two points ((x_1,f(x_1))) and ((x_2,f(x_2))) on the function (y = f(x)) is (m=\frac{f(x_2)-f(x_1)}{x_2 - x_1}).

Step2: Find (f(x_1)) and (f(x_2))

Given (f(x)=4x^{2}), (x_1 = 1), (x_2=2). Then (f(1)=4\times1^{2}=4), (f(2)=4\times2^{2}=16).

Step3: Calculate the slope of the secant line

Substitute into the slope formula: (m=\frac{f(2)-f(1)}{2 - 1}=\frac{16 - 4}{2-1}). Since (16-4 = 12) and (2 - 1=1), then (m = 12).

Answer:

(12)