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.)

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.)

Answer

Answer:

$15.996$

Explanation:

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

The slope of the secant line between two points (x = a) and (x = b) for the function (y = f(x)) is given by (m=\frac{f(b)-f(a)}{b - a}). Here (b = 2) and (f(x)=4x^{2}), so (f(2)=4\times2^{2}=16), and for (a = 1.999), (f(1.999)=4\times(1.999)^{2}).

Step2: Calculate (f(1.999))

[ \begin{align*} f(1.999)&=4\times(1.999)^{2}\ &=4\times(2 - 0.001)^{2}\ &=4\times(4-2\times2\times0.001 + 0.001^{2})\ &=4\times(4 - 0.004+0.000001)\ &=4\times3.996001\ &=15.984004 \end{align*} ]

Step3: Calculate the slope of the secant line

[ \begin{align*} m&=\frac{f(2)-f(1.999)}{2 - 1.999}\ &=\frac{16 - 15.984004}{0.001}\ &=\frac{0.015996}{0.001}\ &=15.996 \end{align*} ]