Answer :
To estimate the values of [tex]\( f(23) \)[/tex], [tex]\( f(21) \)[/tex], and [tex]\( f(27) \)[/tex] for the function [tex]\( f(x) \)[/tex], we can use the concept of linear approximation. Linear approximation involves using the tangent line at a given point to estimate the values of the function near that point. This method uses the known value of the function and its derivative at a certain point.
Here's how you can calculate these estimates step-by-step:
1. Given Information:
- [tex]\( f(22) = 66 \)[/tex]
- [tex]\( f'(22) = -2 \)[/tex]
2. Linear Approximation Formula:
The linear approximation of a function [tex]\( f(x) \)[/tex] near [tex]\( x = a \)[/tex] is given by:
[tex]\[
f(x) \approx f(a) + f'(a)(x - a)
\][/tex]
Here, [tex]\( a = 22 \)[/tex].
3. Estimate [tex]\( f(23) \)[/tex]:
- Use [tex]\( a = 22 \)[/tex], [tex]\( x = 23 \)[/tex]:
[tex]\[
f(23) \approx f(22) + f'(22) \times (23 - 22)
\][/tex]
- Plug in the known values:
[tex]\[
f(23) \approx 66 + (-2) \times (23 - 22)
\][/tex]
[tex]\[
f(23) \approx 66 - 2 \times 1 = 64
\][/tex]
4. Estimate [tex]\( f(21) \)[/tex]:
- Use [tex]\( a = 22 \)[/tex], [tex]\( x = 21 \)[/tex]:
[tex]\[
f(21) \approx f(22) + f'(22) \times (21 - 22)
\][/tex]
- Plug in the known values:
[tex]\[
f(21) \approx 66 + (-2) \times (21 - 22)
\][/tex]
[tex]\[
f(21) \approx 66 + 2 \times 1 = 68
\][/tex]
5. Estimate [tex]\( f(27) \)[/tex]:
- Use [tex]\( a = 22 \)[/tex], [tex]\( x = 27 \)[/tex]:
[tex]\[
f(27) \approx f(22) + f'(22) \times (27 - 22)
\][/tex]
- Plug in the known values:
[tex]\[
f(27) \approx 66 + (-2) \times (27 - 22)
\][/tex]
[tex]\[
f(27) \approx 66 - 2 \times 5 = 56
\][/tex]
Therefore, the estimated values are:
- [tex]\( f(23) = 64 \)[/tex]
- [tex]\( f(21) = 68 \)[/tex]
- [tex]\( f(27) = 56 \)[/tex]
Here's how you can calculate these estimates step-by-step:
1. Given Information:
- [tex]\( f(22) = 66 \)[/tex]
- [tex]\( f'(22) = -2 \)[/tex]
2. Linear Approximation Formula:
The linear approximation of a function [tex]\( f(x) \)[/tex] near [tex]\( x = a \)[/tex] is given by:
[tex]\[
f(x) \approx f(a) + f'(a)(x - a)
\][/tex]
Here, [tex]\( a = 22 \)[/tex].
3. Estimate [tex]\( f(23) \)[/tex]:
- Use [tex]\( a = 22 \)[/tex], [tex]\( x = 23 \)[/tex]:
[tex]\[
f(23) \approx f(22) + f'(22) \times (23 - 22)
\][/tex]
- Plug in the known values:
[tex]\[
f(23) \approx 66 + (-2) \times (23 - 22)
\][/tex]
[tex]\[
f(23) \approx 66 - 2 \times 1 = 64
\][/tex]
4. Estimate [tex]\( f(21) \)[/tex]:
- Use [tex]\( a = 22 \)[/tex], [tex]\( x = 21 \)[/tex]:
[tex]\[
f(21) \approx f(22) + f'(22) \times (21 - 22)
\][/tex]
- Plug in the known values:
[tex]\[
f(21) \approx 66 + (-2) \times (21 - 22)
\][/tex]
[tex]\[
f(21) \approx 66 + 2 \times 1 = 68
\][/tex]
5. Estimate [tex]\( f(27) \)[/tex]:
- Use [tex]\( a = 22 \)[/tex], [tex]\( x = 27 \)[/tex]:
[tex]\[
f(27) \approx f(22) + f'(22) \times (27 - 22)
\][/tex]
- Plug in the known values:
[tex]\[
f(27) \approx 66 + (-2) \times (27 - 22)
\][/tex]
[tex]\[
f(27) \approx 66 - 2 \times 5 = 56
\][/tex]
Therefore, the estimated values are:
- [tex]\( f(23) = 64 \)[/tex]
- [tex]\( f(21) = 68 \)[/tex]
- [tex]\( f(27) = 56 \)[/tex]
