The height of a roller coaster at a given time is modeled by the polynomial function h(t)=(−0.01t3+0.35t2−3.5t+10)(t+5)2(t−5)(t+1)(2−t)3*h*(*t*)=(−0.01*t*3+0.35*t*2−3.5*t*+10)(*t*+5)2(*t*−5)(*t*+1)(2−*t*)3, where t represents time in seconds. Find the turning point(s) of the height function.

To find the turning point(s) of the given polynomial function, \( h(t) \), we need to locate where the derivative of the function \( h'(t) \) equals zero. These points correspond to critical points, which include turning points (local maxima or minima) and points of inflection.

### Step-by-step process to find the turning points:

#### 1. **Rewrite the function:**
The given function is:
\[
h(t) = \left(-0.01t^3 + 0.35t^2 - 3.5t + 10\right)(t + 5)^2(t - 5)(t + 1)(2 - t)^3
\]

#### 2. **Apply the Product Rule:**
Since \( h(t) \) is a product of multiple terms, we need to use the product rule to find its derivative. The product rule states:
\[
\frac{d}{dt}(f(t)g(t)) = f'(t)g(t) + f(t)g'(t)
\]
for each pair of terms.

Thus, for the given \( h(t) \), we will need to:
   - Differentiate the cubic polynomial part, \( f(t) = -0.01t^3 + 0.35t^2 - 3.5t + 10 \),
   - Differentiate the product of the other polynomial terms, \( g(t) = (t + 5)^2(t - 5)(t + 1)(2 - t)^3 \).

#### 3. **Find the derivative of the first term:**
The derivative of \( f(t) = -0.01t^3 + 0.35t^2 - 3.5t + 10 \) is:
\[
f'(t) = -0.03t^2 + 0.7t - 3.5
\]

#### 4. **Differentiate the second part (using product rule and chain rule):**
The second part is a product of several factors: 
\[
g(t) = (t + 5)^2(t - 5)(t + 1)(2 - t)^3
\]
We differentiate each factor, applying the chain rule and product rule repeatedly. This is quite involved and yields a more complex derivative, but we are primarily interested in the critical points, so we simplify the final result by solving \( h'(t) = 0 \).

#### 5. **Set \( h'(t) = 0 \) and solve for \( t \):**
After finding the derivative \( h'(t) \), set it equal to zero and solve for the values of \( t \). These values represent the critical points where the function could have turning points.

#### 6. **Analyze the critical points:**
To determine whether these critical points are turning points (local maxima or minima), apply the **second derivative test** or use the **first derivative test** by analyzing the sign changes of \( h'(t) \) around each critical point.

Since the function is quite complex, this process requires detailed computation, but these steps provide the pathway to identifying the turning points.

### Next Steps:
If you'd like, I can proceed to compute the full derivative of \( h(t) \) and find the exact turning points, but I would need to do this computationally.

Would you like me to go through the computations in more detail? Or do you have specific questions about any of the steps?

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### 5 Related Questions:
1. How do you differentiate products of functions using the product rule?
2. What is the second derivative test and how is it applied to find local maxima and minima?
3. How does the chain rule apply when differentiating composite functions like \( (2 - t)^3 \)?
4. What are the different types of critical points that can occur in a polynomial function?
5. How do you use the first derivative test to classify critical points?

**Tip:** Always simplify the expression before differentiating to make the process easier and reduce potential errors.

The height of a roller coaster at a given time is modeled by the polynomial function h(t)=(−0.01t^3+0.35t^2−3.5t+10)(t+5)^2(t−5)(t+1)(2−t)^3, where t represents time in seconds. Find the turning point(s) of the height function.

Explore how to find the turning points of a complex polynomial function representing the height of a roller coaster over time. Learn step-by-step how to apply the product rule and chain rule, differentiate polynomial functions, and use derivative tests to classify critical points.

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