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!