What Is Initial Value?
Initial value in calculus is a type of problem involving the use of an initial condition. This type of problem produces an unknown constant that requires the use of an initial condition or known point to solve. When you are given an initial condition, it will look like this:
This tells you that when x = 0, your y = 2. The initial condition does not have to be when x = 0. It can be any point. It's called an initial condition because it is the first bit of information you know about the newly integrated function. You need this information to fully integrate functions. Let's see why.
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The Constant of Integration
When you calculate the indefinite integral, you end up with something called the constant of integration. It looks like this when you write it out.
Because you have this unknown constant, you need a known point to plug into your equation to figure it out. This known point is your initial condition.
The method to solving initial value problems requires just a couple of steps:
- Integrate the derivative to find the function.
- Use your initial condition to find your constant of integration to get your full answer.
Let's see how these two steps play out with a sample problem.
The integral is also called the antiderivative because if you take the derivative of the answer, you would get the function you just integrated. With initial value problems, you're often given the derivative of the function you are trying to find along with an initial condition.
Our first step tells us to integrate, or take the antiderivative of the derivative. To do that, we need to move the dx over to the other side. So, let's integrate both sides.
We have combined the constant of integration from both sides into just the one because they are both constants and are therefore, like terms. They'll be combined later. Our answer isn't complete without knowing this constant. So, let's go onto step two.
Step two tells us to use the initial condition to help us find the constant of integration. Our initial condition tells us that when x = 0, our y = 2. Let's plug that information into our integration result and solve for the constant.
Our constant of integration is a 2. Now we can plug this information into our integration result to get our full answer.
Wow! We are done. That wasn't so bad. Just two steps to solve this type of problem!
Let's review. Initial value problems involve the use of an initial condition to help you solve integration problems where you have a constant of integration. There are two steps involved in solving such problems. The two steps are to integrate and then to use the initial condition to find the constant of integration to get the full answer.
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Initial Value when Acceleration Function if Given
If we know the acceleration of a moving particle, as a function of time, we can determine the position function, but we need to know two initial conditions to obtain the unique position function.
Because velocity is an antiderivative of acceleration, we integrate the acceleration function to obtain a velocity function. Integrating the velocity function, we obtain a position function.
With each integration, we have a different constant of integration.
If the acceleration is given as a(t) , then
When we integrate the acceleration function, we obtain a velocity function that contains a constant of integration, which is determined by using the velocity at a given point in time.
Integrating the velocity function, we obtain a position function that contains a constant of integration, which is determined uniquely by knowing the position's initial value.
Find the velocity and position functions of a particle whose acceleration is a(t)=6t if the particle starts from rest and initially the position is 2 m from the origin.
The velocity is
because the particle starts from rest, v(0)=0 and position is
because, initially s(0)=2.
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Initial Value in Calculus: Definition, Method & Example
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