# AP Calculus AB : Applications of antidifferentiation

## Example Questions

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### Example Question #1 : Solving Separable Differential Equations And Using Them In Modeling

Find (dy/dx).

sin(xy) = x + cos(y)

dy/dx = (1 – ycos(xy))/(xcos(xy) + sin(y))

dy/dx = (1 – cos(xy))/(cos(xy) + sin(y))

dy/dx = (xcos(xy) + sin(y))/(1 – ycos(xy))

None of the above

dy/dx = (cos(xy) + sin(y))/(1 – cos(xy))

dy/dx = (1 – ycos(xy))/(xcos(xy) + sin(y))

Explanation:

The first step of the problem is to differentiate with respect to (dy/dx):

cos(xy)[(x)(dy/dx) + y(1)] = 1 – sin(y)(dy/dx)

*Note: When differentiating cos(xy) remember to use the product rule. (xy' + x'y)

Step 2: Clean the differentiated problem up

cos(xy)(x)(dy/dx) + cos(xy)y = 1 – sin(y)(dy/dx)

cos(xy)(x)(dy/dx) + sin(y)(dy/dx) = 1 – cos(xy)y

Step 3: Solve for (dy/dx)

dy/dx = (1 – ycos(xy))/(xcos(xy) + sin(y))

### Example Question #2 : Solving Separable Differential Equations And Using Them In Modeling

Find the equation of the normal line at  on the graph .

Explanation:

Now plug in .

now we know 6 is the slope for the tangent line. However, we aren't looking for the slope of the tangent line. The slope of the normal line is the negative reciprocal of the tangent's slope; meaning the slope of the normal is . Now find the equation of the normal line.

### Example Question #3 : Solving Separable Differential Equations And Using Them In Modeling

What is the derivative of

Explanation:

Use the quotient rule.

Find  if

Explanation:

### Example Question #1 : Solving Separable Differential Equations And Using Them In Modeling

Find the derivative:

Explanation:

To find the derivative, multiply the exponent by the coefficent in front of the x term and then decrease the exponent by 1:

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

Find the solution to the equation at with initial condition .

Explanation:

First, we need to solve the differential equation of .

, where is a constant

, where is a constant

To find , use the initial condition, , and solve:

Therefore, .

Finally, at , .

### Example Question #7 : Solving Separable Differential Equations And Using Them In Modeling

Solve the differential equation:

Note that  is on the curve.

Explanation:

In order to solve differential equations, you must separate the variables first.

Since point  is on the curve, .

To get rid of the log, raise every term to the power of e:

### Example Question #23 : Integrals

Suppose \$1000 is invested in an account that pays 4.3% interest compounded continuously. Find an expression for the amount in the account after time .

Explanation:

The differential equation is , with boundary condition .

This is a separable first order differential equation.

Integrate both sides.

Plug in the initial condition above to see that .

### Example Question #1 : Applications Of Antidifferentiation

Find the solution to the differential equation

when .

Explanation:

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

.

Differentiate .