Let $f ( x ) = x ^ { 2 / 3 }.$ (a) Find a linear approximation of f at x = 8:
(b) Use this linear approximation to estimate the value of the function at 7, 9, 7.99, and 8.01.(c) Make a table comparing your estimates with the actual function values. Discuss what this tells you about the linear approximation.(d) Graph the original function and its linear approximation over the interval [7; 9]. What does the graph tell you about the size of the difference between the function values and the linear approximation values?

Question

Quizplus · Accepted Answer

The Answer is (a)

y = \frac { 1 } { 3 } x + \frac { 4 } { 3 }

(b), (c)

\begin{array} { | c | c | c | } \hline x & x ^ { 2 / 3 } & \frac { 1 } { 3 } x + \frac { 4 } { 3 } \\hline 7 & 3.65930 & 3.6 \9 & 4.32674 & 4 . \overline { 3 } \7.99 & 3.99666 & 3.99666 \8.01 & 4.00333 & 4.00333 \\hline\end{array}

The linear approximation

\frac { 1 } { 3 } x + \frac { 4 } { 3 }

is a good approximation to

x ^ { 2 / 3 }

when x is near 8.
(d) $(a) y = \frac { 1 } { 3 } x + \frac { 4 } { 3 } (b), (c) \begin{array} { | c | c | c | } \hline x & x ^ { 2 / 3 } & \frac { 1 } { 3 } x + \frac { 4 } { 3 } \ \hline 7 & 3.65930 & 3.6 \ 9 & 4.32674 & 4 . \overline { 3 } \ 7.99 & 3.99666 & 3.99666 \ 8.01 & 4.00333 & 4.00333 \ \hline \end{array} The linear approximation \frac { 1 } { 3 } x + \frac { 4 } { 3 } is a good approximation to x ^ { 2 / 3 } when x is near 8. (d)$

Let f(x)=x2/3.f ( x ) = x ^ { 2 / 3 }.f(x)=x2/3.

Let $f ( x ) = x ^ { 2 / 3 }.$