### All Calculus 2 Resources

## Example Questions

### Example Question #23 : Fundamental Theorem Of Calculus

**Possible Answers:**

**Correct answer:**

This is a Second Fundamental Theorem of Calculus problem. Since the derivative cancels out the integral, we just need to plug in the bounds into the function (top bound - bottom bottom) and multiply each by their derivative.

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### Example Question #24 : Fundamental Theorem Of Calculus

Differentiate:

**Possible Answers:**

**Correct answer:**

Differentiate:

**Use the Second Fundamental Theorem of Calculus:**

**________________________________________________________**

*If the function is continuous on an open interval and if is in , then for a function defined by , we have . *

* ________________________________________________________________*

To understand how to apply the second FTOC, notice that in our case is a function of a function. The chainrule will therefore have to be applied as well. To see how this works, let . The function can now be written in the form:

Now we can go back to writing in terms of , the derivative is,

### Example Question #25 : Fundamental Theorem Of Calculus

In exponentially decaying systems, often times the solutions to differential equations take on the form of an integral called Duhamel's Integral. This is given by:

Where is a constant and is a function that represents an external force.

Equations of the form are sinusoidal functions, where is the imaginary constant. Determine the external force needed to produce solutions of sinusoidal behavior

**Possible Answers:**

**Correct answer:**

We plug in into our formula and determine the external force .

Taking the derivative of each side with respect to , and applying the fundamental theorem of calculus:

Solving for ,

### Example Question #26 : Fundamental Theorem Of Calculus

In harmonic systems, often times the solutions to differential equations take on the form of an integral called Duhamel's Integral. This is given by:

Where is a constant and is a function that represents an external force.

What external force is needed in order to obtain if ?

**Possible Answers:**

**Correct answer:**

In order to solve this, we substitute into our equation and solve for :

Taking the derivative of each side with respect to , we get that:

Using the fundamental theorem of calculus:

Solving for :

Since ,

### Example Question #27 : Fundamental Theorem Of Calculus

Let

Find .

**Possible Answers:**

**Correct answer:**

The Fundamental Theorem of Calculus tells us that

therefore,

From here plug in 0 into this equation.

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