To find the tangent line approximation [tex]\( T(x) \)[/tex] to the graph of [tex]\( f \)[/tex] at the point [tex]\( (5, \csc 5) \)[/tex], we need to follow a few steps.
First, let's recap the formula provided for the tangent line approximation:
[tex]\[ T(x) = \csc(5) - \csc(5) \cot(5) (x - 5) \][/tex]
### Step-by-Step Solution:
1. Evaluate [tex]\( \csc(5) \)[/tex]
The cosecant function is the reciprocal of the sine function:
[tex]\[ \csc(x) = \frac{1}{\sin(x)} \][/tex]
So,
[tex]\[ \csc(5) = \frac{1}{\sin(5)} \][/tex]
Using a calculator to find the sine of 5 radians approximately, we get:
[tex]\[ \sin(5) \approx -0.958924 \][/tex]
Therefore,
[tex]\[ \csc(5) \approx \frac{1}{-0.958924} \approx -1.0429 \][/tex]
2. Evaluate [tex]\( \cot(5) \)[/tex]
The cotangent function is the reciprocal of the tangent function:
[tex]\[ \cot(x) = \frac{\cos(x)}{\sin(x)} \][/tex]
Using a calculator to find the cosine and sine of 5 radians, we get:
[tex]\[ \cos(5) \approx 0.283662 \][/tex]
[tex]\[ \sin(5) \approx -0.958924 \][/tex]
Therefore,
[tex]\[ \cot(5) \approx \frac{0.283662}{-0.958924} \approx -0.2958 \][/tex]
3. Construct the Tangent Line [tex]\( T(x) \)[/tex]
Substitute the values of [tex]\( \csc(5) \)[/tex] and [tex]\( \cot(5) \)[/tex] into the tangent line equation:
[tex]\[ T(x) = -1.0429 - (-1.0429)(-0.2958)(x - 5) \][/tex]
Simplify the equation:
[tex]\[ T(x) = -1.0429 - 0.3084(x - 5) \][/tex]
4. Calculate [tex]\( T(x) \)[/tex] for the given values of [tex]\( x \)[/tex]
Plug in the values for [tex]\( x \)[/tex] given in the table:
- For [tex]\( x = 4.9 \)[/tex]:
[tex]\[ T(4.9) = -1.0429 - 0.3084(4.9 - 5) \][/tex]
[tex]\[ T(4.9) = -1.0429 - 0.3084(-0.1) \][/tex]
[tex]\[ T(4.9) = -1.0429 + 0.0308 \][/tex]
[tex]\[ T(4.9) \approx -1.0121 \][/tex]
- For [tex]\( x = 4.99 \)[/tex]:
[tex]\[ T(4.99) = -1.0429 - 0.3084(4.99 - 5) \][/tex]
[tex]\[ T(4.99) = -1.0429 - 0.3084(-0.01) \][/tex]
[tex]\[ T(4.99) = -1.0429 + 0.0031 \][/tex]
[tex]\[ T(4.99) \approx -1.0398 \][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ T(5) = -1.0429 - 0.3084(5 - 5) \][/tex]
[tex]\[ T(5) = -1.0429 \][/tex]
- For [tex]\( x = 5.01 \)[/tex]:
[tex]\[ T(5.01) = -1.0429 - 0.3084(5.01 - 5) \][/tex]
[tex]\[ T(5.01) = -1.0429 - 0.3084(0.01) \][/tex]
[tex]\[ T(5.01) = -1.0429 - 0.0031 \][/tex]
[tex]\[ T(5.01) \approx -1.0460 \][/tex]
- For [tex]\( x = 5.1 \)[/tex]:
[tex]\[ T(5.1) = -1.0429 - 0.3084(5.1 - 5) \][/tex]
[tex]\[ T(5.1) = -1.0429 - 0.3084(0.1) \][/tex]
[tex]\[ T(5.1) = -1.0429 - 0.0308 \][/tex]
[tex]\[ T(5.1) \approx -1.0737 \][/tex]
### Complete the Table:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 4.9 & 4.99 & 5 & 5.01 & 5.1 \\
\hline
f(x) & -1.0179 & & & & \\
\hline
T(x) & -1.0121 & -1.0398 & -1.0429 & -1.0460 & -1.0737 \\
\hline
\end{array}
\][/tex]
So, the tangent line approximation values [tex]\( T(x) \)[/tex] have been filled in as calculated above.
