To find the tangent line approximation [tex]\( T \)[/tex] to the graph of [tex]\( f \)[/tex] at the given point [tex]\((7, 49)\)[/tex], we follow these steps:
1. Find the function [tex]\( f(x) \)[/tex] and its value at [tex]\( x = 7 \)[/tex]:
[tex]\[
f(x) = x^2
\][/tex]
At [tex]\( x = 7 \)[/tex]:
[tex]\[
f(7) = 7^2 = 49
\][/tex]
2. Calculate the derivative [tex]\( f'(x) \)[/tex]:
[tex]\[
f'(x) = 2x
\][/tex]
The derivative at [tex]\( x = 7 \)[/tex] is:
[tex]\[
f'(7) = 2 \times 7 = 14
\][/tex]
3. Write the equation of the tangent line:
The equation of the tangent line at point [tex]\((7, 49)\)[/tex] is given by:
[tex]\[
T(x) = f(7) + f'(7) \cdot (x - 7)
\][/tex]
Substituting the values:
[tex]\[
T(x) = 49 + 14 \cdot (x - 7)
\][/tex]
4. Simplify the equation of the tangent line:
[tex]\[
T(x) = 49 + 14x - 98
\][/tex]
[tex]\[
T(x) = 14x - 49
\][/tex]
So, the tangent line approximation [tex]\( T \)[/tex] is:
[tex]\[
T(x) = 14x - 49
\][/tex]
Next, we complete the table by evaluating [tex]\( f(x) \)[/tex] and [tex]\( T(x) \)[/tex] at the given points [tex]\( x = 6.9, 6.99, 7, 7.01, \)[/tex] and [tex]\( 7.1 \)[/tex] and rounding the answers to four decimal places:
1. Calculating [tex]\( f(x) \)[/tex]:
- For [tex]\( x = 6.9 \)[/tex]:
[tex]\[
f(6.9) = (6.9)^2 = 47.61
\][/tex]
- For [tex]\( x = 6.99 \)[/tex]:
[tex]\[
f(6.99) = (6.99)^2 = 48.8601
\][/tex]
- For [tex]\( x = 7 \)[/tex]:
[tex]\[
f(7) = (7)^2 = 49
\][/tex]
- For [tex]\( x = 7.01 \)[/tex]:
[tex]\[
f(7.01) = (7.01)^2 = 49.1401
\][/tex]
- For [tex]\( x = 7.1 \)[/tex]:
[tex]\[
f(7.1) = (7.1)^2 = 50.41
\][/tex]
2. Calculating [tex]\( T(x) \)[/tex]:
- For [tex]\( x = 6.9 \)[/tex]:
[tex]\[
T(6.9) = 14(6.9) - 49 = 96.6 - 49 = 47.6
\][/tex]
- For [tex]\( x = 6.99 \)[/tex]:
[tex]\[
T(6.99) = 14(6.99) - 49 = 97.86 - 49 = 48.86
\][/tex]
- For [tex]\( x = 7 \)[/tex]:
[tex]\[
T(7) = 14(7) - 49 = 98 - 49 = 49
\][/tex]
- For [tex]\( x = 7.01 \)[/tex]:
[tex]\[
T(7.01) = 14(7.01) - 49 = 98.14 - 49 = 49.14
\][/tex]
- For [tex]\( x = 7.1 \)[/tex]:
[tex]\[
T(7.1) = 14(7.1) - 49 = 99.4 - 49 = 50.4
\][/tex]
Therefore, the completed table is:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 6.9 & 6.99 & 7 & 7.01 & 7.1 \\
\hline
f(x) & 47.61 & 48.8601 & 49 & 49.1401 & 50.41 \\
\hline
T(x) & 47.6 & 48.86 & 49 & 49.14 & 50.4 \\
\hline
\end{array}
\][/tex]
