Math Problem Statement
Solution
To solve this problem, let's break it down systematically.
The problem asks to find the -coordinate of the curve where the slope of the tangent is parallel to the -axis. This condition implies that the slope is , i.e., .
Step 1: Find the derivative
The function is:
Using the product rule for derivatives , let:
- , so ,
- , so .
Now,
Simplify:
Step 2: Set the derivative equal to zero
The slope of the tangent is parallel to the -axis when:
From the equation:
Since for any , the equation simplifies to:
Step 3: Solve
Rearrange:
Divide both sides by (valid where ):
Thus:
The general solution for is:
Step 4: Restrict the solution to
Within the interval , the solutions are:
Final Answer:
The -coordinates where the slope of the tangent is parallel to the -axis are:
Would you like a deeper explanation or a visual illustration of the solution?
Here are 5 related questions to expand your understanding:
- What is the significance of the derivative being zero in terms of the curve's behavior?
- Can we generalize this method for any function ?
- What happens if instead? How does the solution change?
- Why is periodic, and how is the period determined?
- How can you verify the solution graphically?
Tip: Always ensure to apply the product rule carefully when differentiating products of functions.
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Trigonometric Equations
Formulas
Product Rule: (uv)' = u'v + uv'
Trigonometric Identity: tan(x) = sin(x) / cos(x)
Theorems
Conditions for horizontal tangents (dy/dx = 0)
Suitable Grade Level
Grade 12 or early undergraduate