Consider the following function
                                q:R→Rq(t)=∣t(t−2)(t−3)∣.*q*:R→R*q*(*t*)=∣*t*(*t*−2)(*t*−3)∣.

If m*m* is slope of the tangent of the function q(t)*q*(*t*) at point t=32*t*=23​, find the value m−274*m*−427​


To find the slope \( m \) of the tangent line to the function \( q(t) = |t(t-2)(t-3)| \) at the point \( t = \frac{3}{2} \), and then determine the value of \( m - \frac{274}{427} \), we proceed as follows:

1. **Simplify the function \( q(t) \)**:
   \[
   q(t) = |t(t-2)(t-3)|
   \]
   The absolute value complicates the differentiation, so we need to consider the sign of the expression inside the absolute value around \( t = \frac{3}{2} \).

2. **Analyze the sign of \( t(t-2)(t-3) \) at \( t = \frac{3}{2} \)**:
   \[
   t(t-2)(t-3) \text{ at } t = \frac{3}{2} = \frac{3}{2} \left( \frac{3}{2} - 2 \right) \left( \frac{3}{2} - 3 \right)
   \]
   \[
   = \frac{3}{2} \left( -\frac{1}{2} \right) \left( -\frac{3}{2} \right)
   \]
   \[
   = \frac{3}{2} \cdot -\frac{1}{2} \cdot -\frac{3}{2} = \frac{9}{8}
   \]
   Since \( t(t-2)(t-3) \) is positive at \( t = \frac{3}{2} \), we can remove the absolute value in this interval.

3. **Simplify \( q(t) \) for \( t \) around \( \frac{3}{2} \)**:
   \[
   q(t) = t(t-2)(t-3)
   \]

4. **Differentiate \( q(t) \)**:
   Use the product rule for differentiation:
   \[
   q(t) = t(t^2 - 5t + 6)
   \]
   Expand and differentiate:
   \[
   q(t) = t^3 - 5t^2 + 6t
   \]
   \[
   q'(t) = 3t^2 - 10t + 6
   \]

5. **Evaluate \( q'(t) \) at \( t = \frac{3}{2} \)**:
   \[
   q'\left(\frac{3}{2}\right) = 3\left(\frac{3}{2}\right)^2 - 10\left(\frac{3}{2}\right) + 6
   \]
   \[
   = 3 \cdot \frac{9}{4} - 10 \cdot \frac{3}{2} + 6
   \]
   \[
   = \frac{27}{4} - 15 + 6
   \]
   \[
   = \frac{27}{4} - \frac{60}{4} + \frac{24}{4}
   \]
   \[
   = \frac{27 - 60 + 24}{4}
   \]
   \[
   = \frac{-9}{4}
   \]
   So, the slope \( m = -\frac{9}{4} \).

6. **Calculate \( m - \frac{274}{427} \)**:
   \[
   m - \frac{274}{427} = -\frac{9}{4} - \frac{274}{427}
   \]
   Convert \(-\frac{9}{4}\) to a fraction with denominator 427:
   \[
   -\frac{9}{4} = -\frac{9 \times 427}{4 \times 427} = -\frac{3843}{1708}
   \]
   Convert \(-\frac{274}{427}\) to a fraction with denominator 1708:
   \[
   -\frac{274}{427} = -\frac{274 \times 4}{427 \times 4} = -\frac{1096}{1708}
   \]

   Combine the fractions:
   \[
   -\frac{3843}{1708} - \frac{1096}{1708} = -\frac{3843 + 1096}{1708} = -\frac{4939}{1708}
   \]

Thus, the value of \( m - \frac{274}{427} \) is:
\[
\boxed{-\frac{4939}{1708}}
\]

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 handle absolute value functions when finding derivatives?
2. Why is the sign of the expression inside the absolute value important?
3. How can the product rule help in differentiating complex functions?
4. What are the steps to simplify the fraction to a common denominator?
5. How do you interpret the slope of the tangent line at a given point?
6. What are the properties of polynomial functions regarding their derivatives?
7. How does the slope affect the graph of the function?
8. Can you use numerical methods to verify the calculated slope?

**Tip:** When dealing with absolute values in functions, carefully analyze the sign changes to correctly remove the absolute value for differentiation purposes.

Learn how to find the slope of the tangent line to the function q(t) = |t(t-2)(t-3)| at the point t = 3/2. Detailed step-by-step solution involving absolute value functions, derivatives, and the product rule. Suitable for advanced high school students and college-level calculus enthusiasts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation