calculate the equation of the tangent line to f(t)=3t^2 × cos(t) at t=pi

To find the equation of the tangent line to the function \( f(t) = 3t^2 \cos(t) \) at \( t = \pi \), we need to follow these steps:

1. **Calculate \( f(\pi) \)** to find the point where the tangent touches the curve.
2. **Differentiate \( f(t) \)** with respect to \( t \) to get \( f'(t) \), the slope of the tangent line.
3. **Evaluate \( f'(\pi) \)** to find the slope at \( t = \pi \).
4. Use the point-slope form of the tangent line: \( y - f(\pi) = f'(\pi) (t - \pi) \).

### Step 1: Calculate \( f(\pi) \)
\[
f(\pi) = 3\pi^2 \cos(\pi) = 3\pi^2 \cdot (-1) = -3\pi^2
\]
So, the point on the curve at \( t = \pi \) is \( (\pi, -3\pi^2) \).

### Step 2: Differentiate \( f(t) \) to find \( f'(t) \)
Since \( f(t) = 3t^2 \cos(t) \), we apply the product rule:
\[
f'(t) = \frac{d}{dt}(3t^2) \cdot \cos(t) + 3t^2 \cdot \frac{d}{dt}(\cos(t))
\]
\[
= (6t) \cos(t) + 3t^2 \cdot (-\sin(t))
\]
\[
= 6t \cos(t) - 3t^2 \sin(t)
\]

### Step 3: Evaluate \( f'(\pi) \)
Substitute \( t = \pi \) into \( f'(t) \):
\[
f'(\pi) = 6\pi \cos(\pi) - 3\pi^2 \sin(\pi)
\]
Since \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \):
\[
f'(\pi) = 6\pi \cdot (-1) - 3\pi^2 \cdot 0 = -6\pi
\]
The slope of the tangent line at \( t = \pi \) is \( -6\pi \).

### Step 4: Write the equation of the tangent line
Using the point-slope form \( y - y_1 = m(t - t_1) \), where \( m = -6\pi \), \( t_1 = \pi \), and \( y_1 = -3\pi^2 \):
\[
y + 3\pi^2 = -6\pi (t - \pi)
\]
Simplifying,
\[
y = -6\pi t + 6\pi^2 - 3\pi^2
\]
\[
y = -6\pi t + 3\pi^2
\]

### Final Answer
The equation of the tangent line to \( f(t) = 3t^2 \cos(t) \) at \( t = \pi \) is:
\[
y = -6\pi t + 3\pi^2
\]

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### Related Questions
1. How do we find the tangent line of \( f(t) = t^3 \sin(t) \) at \( t = \pi/2 \)?
2. What is the derivative of \( g(t) = e^t \cos(t) \) using the product rule?
3. How do we find the tangent line of a trigonometric function at a specific point?
4. What is the geometric interpretation of the slope of the tangent line?
5. How does the product rule apply when differentiating \( h(t) = t^2 \sin(t^2) \)?

**Tip:** Remember that the product rule states: if \( u(t) \) and \( v(t) \) are functions of \( t \), then \( \frac{d}{dt} [u(t)v(t)] = u'(t)v(t) + u(t)v'(t) \).

Calculate the equation of the tangent line to f(t) = 3t^2 × cos(t) at t = π.

Discover how to calculate the tangent line equation for f(t) = 3t^2 cos(t) at t = π. This problem involves calculus concepts such as differentiation using the product rule and writing tangent lines using point-slope form. Suitable for high school students in Grades 11-12.

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