### All AP Calculus AB Resources

## Example Questions

### Example Question #8 : Solving Separable Differential Equations And Using Them In Modeling

Find the solution to the differential equation

when .

**Possible Answers:**

**Correct answer:**

First, separate the variables of the original differential equation:

.

Then, take the antiderivative of both sides, which gives

.

Use the given condition , plugging in

and , to solve for . This gives , so the correct answer is

.

### Example Question #31 : Integrals

Differentiate .

**Possible Answers:**

**Correct answer:**

While differentiating, multiply the exponent with the coefficient then subtract the exponent by one.

### Example Question #31 : Integrals

If what is ?

**Possible Answers:**

**Correct answer:**

Taking the derivative of gives you .

Taking the derivative of gives you .

Finally taking the derivative of gives you .

Therefore .

### Example Question #9 : Solving Separable Differential Equations And Using Them In Modeling

Solve the following separable differential equation with initial condition .

**Possible Answers:**

**Correct answer:**

We proceed as follows

. Start

. Rewrite as .

. Multiply both sides by , and divide both sides by .

. Integrate both sides. Do not forget the on one of the sides.

Substitute the initial condition .

.

. Solve for .

. Exponentiate both sides .

. Rule of exponents.

### Example Question #10 : Solving Separable Differential Equations And Using Them In Modeling

Solve the separable, first-order differential equation for :

**Possible Answers:**

**Correct answer:**

Solve the separable, first-order differential equation for :

First collect all the terms with the derivative to one side of the equation.

*Important Conceptual Note: often in texts on differential equations differentials often appear to have been rearranged algebraically as if is a "fraction," making it appear as if we "multiplied both sides" by to get: . This is not the case. The derivative is a limit by definition and, when the limit exists, can take on any real number which includes irrational numbers i.e. numbers which cannot be written as a ratio of two integers.*

* For instance, we cannot represent as a ratio, but some functions may have a derivative at a point such that the derivative is equal to , or a funciton may simply have an irrational number like as a derivative. For instance, if we write the derivative . Claiming that and are representative of a "numerator" and "denominator" respectively, we would essentially be claiming to have found a way to write an irrational number, such as as a ratio, which is preposterous. The expression is simply notation. *

Here is what we are **really **doing.

*Note that the constants of integration can just be combined into one constant by defining . *

Solve for :

Applying the initial condition:

Here we have two possible solutions. However, because of the initial condition, we can easily rule out the negative solution. must be equal to positive .

### Example Question #11 : Solving Separable Differential Equations And Using Them In Modeling

Solve the separable differential equation

given the condition

**Possible Answers:**

None of the other answers

**Correct answer:**

To solve this equation, we must separate the variables such that terms containing x and y are on the same side as dx and dy, respectively:

Integrating both sides of the equation, we get

The integrals were found using the following rules:

,

After combining the constants of integration into a single C, exponentiating both sides, and using the properties of exponents to simplify, we get

To solve for C, we use the condition given:

Our final answer is

### Example Question #12 : Solving Separable Differential Equations And Using Them In Modeling

is a function of . Solve for in this differential equation:

**Possible Answers:**

**Correct answer:**

First, rewrite the expression on the right as the power of the radicand:

The expressions with can be separated from those with by multiplying both sides by :

Find the indefinite integral of both sides:

The expression on the right can be integrated using the Power Rule. On the right, use some -substitution, setting ; this makes and :

Apply some algebra to solve for :

Substitute back for , and apply some algebra:

### Example Question #13 : Solving Separable Differential Equations And Using Them In Modeling

Solve the separable differential equation

where

**Possible Answers:**

**Correct answer:**

To solve the separable differential equation, we must separate x and y to the same sides as their respective derivatives:

Next, we integrate both sides:

The integrals were solved using the following rules:

,

The two constants of integration were combined to make a single one.

Now, we exponentiate both sides to solve for y, keeping in mind rules for exponents which allow us to move the integration constant to the front:

To solve for the constant of integration, we use the condition given:

Our final answer is

### Example Question #14 : Solving Separable Differential Equations And Using Them In Modeling

Solve the following separable differential equation:

given the condition that at

**Possible Answers:**

**Correct answer:**

To solve the separable differential equation, we must separate x and y and their respective derivatives to either side of the equal sign:

Now, we integrate both sides of the equation:

The integrals were found using their identical rules.

Exponentiating both sides of the equation to solve for y - and keeping in mind the rules of exponents - we get

Now, we solve for the integration constant by using the condition given:

Our final answer is

### Example Question #15 : Solving Separable Differential Equations And Using Them In Modeling

Solve the following separable differential equation:

**Possible Answers:**

**Correct answer:**

To solve the separable differential equation, we must separate x and y to the same sides as their respective derivatives:

Next, we integrate both sides, where on the lefthand side, the following substitution is made:

The integrals were solved using the following rules:

The two constants of integration were combined to make a single one.

Now, we solve for y:

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