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Can you do all the steps and show the work please

The given problem provides the rate of change of temperature \( T'(t) \) as a function of time \( t \):

\[
T'(t) = (4t - 2)e^{t^2 - t}
\]

We are asked to find the change in temperature, \( \Delta T \), between \( t = 1 \) minute and \( t = 2 \) minutes. The change in temperature \( \Delta T \) is given by the integral of the rate of change of temperature:

\[
\Delta T = \int_{1}^{2} T'(t) \, dt
\]

### Step 1: Set up the integral
The integral we need to solve is:

\[
\Delta T = \int_{1}^{2} (4t - 2)e^{t^2 - t} \, dt
\]

### Step 2: Use substitution method
To solve this integral, we can use substitution. Let:

\[
u = t^2 - t
\]

Then, differentiate \( u \) with respect to \( t \):

\[
\frac{du}{dt} = 2t - 1
\]
or equivalently:
\[
du = (2t - 1) \, dt
\]

Now, we need to express \( (4t - 2) \, dt \) in terms of \( du \). Notice that:

\[
4t - 2 = 2(2t - 1)
\]

So, we can rewrite the integral as:

\[
\Delta T = \int_{1}^{2} 2(2t - 1)e^{t^2 - t} \, dt
\]

This simplifies using \( du = (2t - 1) \, dt \):

\[
\Delta T = 2 \int_{1}^{2} e^{u} \, du
\]

### Step 3: Solve the integral
The integral of \( e^u \) is straightforward:

\[
\int e^u \, du = e^u
\]

Now, we need to substitute back for \( u = t^2 - t \). Thus:

\[
\Delta T = 2 \left[ e^{t^2 - t} \right]_{1}^{2}
\]

### Step 4: Evaluate the bounds
First, evaluate at \( t = 2 \):

\[
e^{(2^2 - 2)} = e^{4 - 2} = e^2
\]

Next, evaluate at \( t = 1 \):

\[
e^{(1^2 - 1)} = e^{1 - 1} = e^0 = 1
\]

Now, substitute these values into the expression for \( \Delta T \):

\[
\Delta T = 2 \left( e^2 - 1 \right)
\]

### Step 5: Final Answer
Therefore, the change in temperature between \( t = 1 \) and \( t = 2 \) minutes is:

\[
\Delta T = 2 \left( e^2 - 1 \right) \approx 2 \left( 7.389 - 1 \right) = 2 \times 6.389 = 12.778 \, \text{°C}
\]

### Conclusion
The change in temperature \( \Delta T \) between \( t = 1 \) minute and \( t = 2 \) minutes is approximately \( 12.778 \, \text{°C} \).

---

Do you want further details on any step or have any questions?

Here are 5 related questions to consider:
1. How do you choose a substitution for solving integrals?
2. What is the significance of the exponential function in the rate of change?
3. Can this method be applied to any rate of change problem?
4. How would the result change if the bounds were different, say from \( t = 0 \) to \( t = 3 \)?
5. Why does solving integrals help in finding the total change of a quantity?

**Tip**: Always check whether substitution or integration by parts is the best method for complex integrals like this one!

Find the change in temperature, ΔT, between t=1 and t=2 min, given that the rate of change of temperature is T'(t) = (4t - 2)e^{t^2 - t} °C/min.

Learn how to calculate the change in temperature between t=1 min and t=2 min using integration. The rate of change of temperature is given by T'(t) = (4t - 2)e^{t^2 - t}. Follow a step-by-step guide to solving this calculus problem using substitution and integration methods.

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