Consider the following function

p
:
R
→
R
p:R→R

p
(
t
)
=
{
8
e
(
t
−
2
)
−
8
t
−
2
if
0
≤
t
<
2
,
8
t
=
2
8
(
t
2
−
4
)
1
ln
⁡
(
t
−
2
)
if
t
>
2
p(t)=
⎩
⎨
⎧
​

t−2
8e
(t−2)
−8
​

8
8(t
2
−4)
ln(t−2)
1
​

​

if 0≤t<2,
t=2
if t>2
​

If linear function
L
p
(
t
)
=
A
t
+
B
L
p
​
(t)=At+B denotes the best linear approximation of the function
p
(
t
)
p(t) at the point
t
=
1
t=1, find the value of
−
2
e
−
1
−
1
(
A
+
B
)
e
−1
−1
−2
​
(A+B)

To solve for \(-2e^{-1} - 1 (A + B)\) where \(L_p(t) = At + B\) denotes the best linear approximation of the function \(p(t)\) at the point \(t = 1\), we will go through the following steps:

1. **Define the function \(p(t)\) and identify the correct piece for \(t = 1\)**:
   \[
   p(t) = 
   \begin{cases} 
   8e^{t-2} - 8 & \text{if } 0 \le t < 2 \\
   8 & \text{if } t = 2 \\
   8(t^2 - 4) \ln(t - 2) & \text{if } t > 2 
   \end{cases}
   \]
   Since \(t = 1\) falls in the interval \(0 \le t < 2\), we use the piece:
   \[
   p(t) = 8e^{t-2} - 8
   \]

2. **Calculate \(p(1)\)**:
   \[
   p(1) = 8e^{1-2} - 8 = 8e^{-1} - 8
   \]

3. **Calculate the derivative \(p'(t)\) for \(0 \le t < 2\)**:
   \[
   p(t) = 8e^{t-2} - 8 \implies p'(t) = 8e^{t-2}
   \]
   Evaluate at \(t = 1\):
   \[
   p'(1) = 8e^{1-2} = 8e^{-1}
   \]

4. **Identify the best linear approximation \(L_p(t)\) at \(t = 1\)**:
   The best linear approximation \(L_p(t)\) at \(t = 1\) is given by the tangent line at that point:
   \[
   L_p(t) = p'(1)(t-1) + p(1)
   \]
   Substitute \(p(1)\) and \(p'(1)\):
   \[
   L_p(t) = 8e^{-1}(t - 1) + (8e^{-1} - 8)
   \]
   Simplify the equation:
   \[
   L_p(t) = 8e^{-1}t - 8e^{-1} + 8e^{-1} - 8 = 8e^{-1}t - 8
   \]
   Therefore, \(A = 8e^{-1}\) and \(B = -8\).

5. **Calculate the expression \(-2e^{-1} - 1 (A + B)\)**:
   \[
   A + B = 8e^{-1} - 8
   \]
   \[
   -2e^{-1} - 1 (8e^{-1} - 8) = -2e^{-1} - 8e^{-1} + 8 = -10e^{-1} + 8
   \]

Thus, the value of the given expression is:
\[
\boxed{-10e^{-1} + 8}
\]

Would you like more details on any part of this solution or have any further questions?

Here are 8 related questions to deepen your understanding:
1. How do you determine which piece of a piecewise function to use?
2. What is the significance of the derivative in finding the linear approximation?
3. Why is the tangent line used for linear approximation?
4. How do you interpret the constants \(A\) and \(B\) in the context of the tangent line?
5. How does \(e^{-1}\) simplify in calculations?
6. What are the steps to differentiate exponential functions?
7. How do piecewise functions affect continuity and differentiability?
8. Can you use linear approximation to estimate values of the function near \(t = 1\)?

**Tip:** When finding linear approximations, ensure you correctly identify and use the derivative at the point of interest to get accurate approximations.

Explore the piecewise function and linear approximation in mathematics with a detailed example solving for -2e^-1 - 1 (A + B). This problem covers piecewise functions, exponential functions, and their applications in linear approximation. Suitable for advanced high school students and anyone interested in calculus and mathematical analysis.

Math Problem Statement

p ( t )

{ 8 e ( t − 2 ) − 8 t − 2 if 0 ≤ t < 2 , 8 t

If linear function L p ( t )

A t + B L p (t)=At+B denotes the best linear approximation of the function p ( t ) p(t) at the point t

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation