math 180 4.8 – antiderivatives 1. sometimes we know the derivative of a function, and want to find...
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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