Initial Value in Calculus: Definition, Method & Example

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  • 0:02 What Is Initial Value?
  • 0:41 The Constant of Integration
  • 0:57 The Method
  • 1:15 Sample Problem
  • 2:49 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this lesson, you'lll learn about how the initial value of calculus is related to the integration process. You will see how crucial having an initial condition is and how useful it is in solving your problems.

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.

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

The method to solving initial value problems requires just a couple of steps:

  1. Integrate the derivative to find the function.
  2. 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.

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.

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