# AP Calculus AB : Applications of antidifferentiation

## Example Questions

### Example Question #11 : Applications Of Antidifferentiation

If  what is ?

Explanation:

Taking the derivative of  gives you .

Taking the derivative of  gives you .

Finally taking the derivative of  gives you .

Therefore .

### Example Question #5 : Solving Separable Differential Equations And Using Them In Modeling

Solve the following separable differential equation  with initial condition .

Explanation:

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 #6 : Solving Separable Differential Equations And Using Them In Modeling

Solve the separable, first-order differential equation for

Explanation:

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 : Applications Of Antidifferentiation

Solve the separable differential equation

given the condition

Explanation:

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:

### Example Question #12 : Applications Of Antidifferentiation

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

Explanation:

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 : Applications Of Antidifferentiation

Solve the separable differential equation

where

Explanation:

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:

### Example Question #14 : Applications Of Antidifferentiation

Solve the following separable differential equation:

given the condition that at

Explanation:

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:

### Example Question #15 : Applications Of Antidifferentiation

Solve the following separable differential equation:

Explanation:

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:

### Example Question #16 : Applications Of Antidifferentiation

Solve the separable differential equation

given the initial condition

Explanation:

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 constant.

Now, exponentiate both sides to isolate y, and use the properties of exponents to rearrange the integration constant:

(The exponential of the constant is another constant.)

Finally, we solve for the integration constant using the initial condition:

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

Solve the separable differential equation:

Explanation:

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 solve for y:

Because the problem statement said that y is negative - and y cannot be zero - our final answer is