Math Problem Statement
calculate the equation of the tangent line to f(t)=3t^2 × cos(t) at t=pi
Solution
To find the equation of the tangent line to the function at , we need to follow these steps:
- Calculate to find the point where the tangent touches the curve.
- Differentiate with respect to to get , the slope of the tangent line.
- Evaluate to find the slope at .
- Use the point-slope form of the tangent line: .
Step 1: Calculate
So, the point on the curve at is .
Step 2: Differentiate to find
Since , we apply the product rule:
Step 3: Evaluate
Substitute into : Since and : The slope of the tangent line at is .
Step 4: Write the equation of the tangent line
Using the point-slope form , where , , and : Simplifying,
Final Answer
The equation of the tangent line to at is:
Would you like further details or have any questions?
Related Questions
- How do we find the tangent line of at ?
- What is the derivative of using the product rule?
- How do we find the tangent line of a trigonometric function at a specific point?
- What is the geometric interpretation of the slope of the tangent line?
- How does the product rule apply when differentiating ?
Tip: Remember that the product rule states: if and are functions of , then .
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Trigonometric Functions
Tangent Line
Formulas
Product Rule: (uv)' = u'v + uv'
Point-slope form of a line: y - y1 = m(x - x1)
Theorems
Product Rule for Differentiation
Suitable Grade Level
Grades 11-12
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