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Math 180

4.8 – Antiderivatives

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Sometimes we know the derivative of a function, and want to find the original function. (ex: finding displacement from velocity.)

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is an ______________ of on an interval if for all in . Ex 1.Find an antiderivative for each of the following functions.

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is an ______________ of on an interval if for all in . Ex 1.Find an antiderivative for each of the following functions.

antiderivative

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is an ______________ of on an interval if for all in . Ex 1.Find an antiderivative for each of the following functions.

antiderivative

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Note: The general antiderivative of is .

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Note: The general antiderivative of is .

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Ex 2.Find an antiderivative of that satisfies .

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Ex 2.Find an antiderivative of that satisfies .

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Let’s fill out the following table of antiderivatives:

Function General antiderivative

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Ex 3.Find the general antiderivative for each of the following functions.

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Notation:For the most part, the way we’ll write the general antiderivative is:

This is called the _________________ of with respect to .

is called the _____________.

is called the __________.

is called the ____________________.

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Notation:For the most part, the way we’ll write the general antiderivative is:

This is called the _________________ of with respect to .

is called the _____________.

is called the __________.

is called the ____________________.

indefinite integral

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Notation:For the most part, the way we’ll write the general antiderivative is:

This is called the _________________ of with respect to .

is called the _____________.

is called the __________.

is called the ____________________.

indefinite integral

integral sign

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Notation:For the most part, the way we’ll write the general antiderivative is:

This is called the _________________ of with respect to .

is called the _____________.

is called the __________.

is called the ____________________.

indefinite integral

integral sign

integrand

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Notation:For the most part, the way we’ll write the general antiderivative is:

This is called the _________________ of with respect to .

is called the _____________.

is called the __________.

is called the ____________________.

indefinite integral

integral sign

integrand

variable of integration

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Ex 4.Evaluate

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Ex 4.Evaluate

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Properties:1. 2. 3.

Ex 5.Evaluate

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Ex 6.Evaluate

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An equation that involves derivatives is called a _________________.For example, if is a function of , then the following is a differential equation:

A solution to the above differential equation is a function, , that satisfies the equation. To solve, we can integrate both sides to get:

This is called the _____________ since it involves an arbitrary constant, .

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An equation that involves derivatives is called a _________________.For example, if is a function of , then the following is a differential equation:

A solution to the above differential equation is a function, , that satisfies the equation. To solve, we can integrate both sides to get:

This is called the _____________ since it involves an arbitrary constant, .

differential equation

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An equation that involves derivatives is called a _________________.For example, if is a function of , then the following is a differential equation:

A solution to the above differential equation is a function, , that satisfies the equation. To solve, we can integrate both sides to get:

This is called the _____________ since it involves an arbitrary constant, .

differential equation

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An equation that involves derivatives is called a _________________.For example, if is a function of , then the following is a differential equation:

A solution to the above differential equation is a function, , that satisfies the equation. To solve, we can integrate both sides to get:

This is called the _____________ since it involves an arbitrary constant, .

differential equation

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An equation that involves derivatives is called a _________________.For example, if is a function of , then the following is a differential equation:

A solution to the above differential equation is a function, , that satisfies the equation. To solve, we can integrate both sides to get:

This is called the _____________ since it involves an arbitrary constant, .

differential equation

general solution

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If we’re given more information, like (called an ______________), we can find the particular value of that satisfies this initial condition:

This function is called the _________________ that satisfies both the differential equation and the initial condition.

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If we’re given more information, like (called an ______________), we can find the particular value of that satisfies this initial condition:

This function is called the _________________ that satisfies both the differential equation and the initial condition.

initial condition

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If we’re given more information, like (called an ______________), we can find the particular value of that satisfies this initial condition:

This function is called the _________________ that satisfies both the differential equation and the initial condition.

initial condition

𝒚=𝒙𝟒

𝟒− 𝒙𝟐+𝟒 𝒙−𝟑

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If we’re given more information, like (called an ______________), we can find the particular value of that satisfies this initial condition:

This function is called the _________________ that satisfies both the differential equation and the initial condition.

initial condition

particular solution

𝒚=𝒙𝟒

𝟒− 𝒙𝟐+𝟒 𝒙−𝟑

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Note: A differential equation combined with an initial condition is called an _______________.

Ex 7.Solve the initial value problem:,

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Note: A differential equation combined with an initial condition is called an _______________.

Ex 7.Solve the initial value problem:,

initial value problem

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Note: A differential equation combined with an initial condition is called an _______________.

Ex 7.Solve the initial value problem:,

initial value problem