Given the following data about a function f(x), the equation of the tangent line at x = 5 is approximated by $\begin{array}{cccccccc}
x & 3 & 3.5 & 4 & 4.5 & 5 & 5.5 & 6 \
f(x) & 10 & 8 & 7 & 4 & 2 & 0 & -1
\end{array}$

A) $y-5=-4(x-2)$

B) $y-5=-8(x-2)$

C) $y-2=-4(x-5)$

D) $y-2=-8(x-5)$

Question

Given the following data about a function f(x), the equation of the tangent line at x = 5 is approximated by $\begin{array}{cccccccc}
x & 3 & 3.5 & 4 & 4.5 & 5 & 5.5 & 6 \
f(x) & 10 & 8 & 7 & 4 & 2 & 0 & -1
\end{array}$

A) $y-5=-4(x-2)$

B) $y-5=-8(x-2)$

C) $y-2=-4(x-5)$

D) $y-2=-8(x-5)$

Quizplus · Accepted Answer

The slope of the tangent line at $x = 5$ can be approximated using the average rate of change around $x = 5$. Using the points $x = 4.5$ and $x = 5.5$, where $f(4.5) = 4$ and $f(5.5) = 0$, the slope $m$ is $\frac{0 - 4}{5.5 - 4.5} = \frac{-4}{1} = -4$. The equation of the tangent line in point-slope form is $y - f(a) = m(x - a)$, where $a = 5$ and $f(a) = 2$, so the equation is $y - 2 = -4(x - 5)$.

Given the Following Data About a Function F(x), the Equation x33.544.555.56f(x)1087420−1\begin{array}{cccccccc}x & 3 & 3.5 & 4 & 4.5 & 5 & 5.5 & 6 \\f(x) & 10 & 8 & 7 & 4 & 2 & 0 & -1\end{array}xf(x)​310​3.58​47​4.54​52​5.50​6−1​

Given the Following Data About a Function F(x), the Equation $\begin{array}{cccccccc}x & 3 & 3.5 & 4 & 4.5 & 5 & 5.5 & 6 \\f(x) & 10 & 8 & 7 & 4 & 2 & 0 & -1\end{array}$