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

for the function ( f(x)=4 x^{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.)

for the function ( f(x)=4 x^{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.)

Answer

Answer:

$15.6$

Explanation:

Step1: Recall the slope formula for secant line

The slope formula for the secant line of a function (y = f(x)) over the interval ([a,b]) is (m=\frac{f(b)-f(a)}{b - a}).

Step2: Calculate (f(1.9)) and (f(2))

Given (f(x)=4x^{2}), then (f(2)=4\times2^{2}=16), and (f(1.9)=4\times(1.9)^{2}=4\times3.61 = 14.44).

Step3: Apply the slope formula

Substitute (a = 1.9), (b = 2), (f(a)=14.44), (f(b)=16) into the slope formula (m=\frac{f(b)-f(a)}{b - a}). So (m=\frac{16 - 14.44}{2-1.9}=\frac{1.56}{0.1}=15.6).