Calculus Examples

Find the Linearization at a=1 f(x)=x^4+2x^2 , a=1
,
Step 1
Consider the function used to find the linearization at .
Step 2
Substitute the value of into the linearization function.
Step 3
Evaluate .
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Step 3.1
Replace the variable with in the expression.
Step 3.2
Simplify .
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Step 3.2.1
Remove parentheses.
Step 3.2.2
Simplify each term.
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Step 3.2.2.1
One to any power is one.
Step 3.2.2.2
One to any power is one.
Step 3.2.2.3
Multiply by .
Step 3.2.3
Add and .
Step 4
Find the derivative and evaluate it at .
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Step 4.1
Find the derivative of .
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Step 4.1.1
Differentiate.
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Step 4.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 4.1.1.2
Differentiate using the Power Rule which states that is where .
Step 4.1.2
Evaluate .
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Step 4.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.2.2
Differentiate using the Power Rule which states that is where .
Step 4.1.2.3
Multiply by .
Step 4.2
Replace the variable with in the expression.
Step 4.3
Simplify.
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Step 4.3.1
Simplify each term.
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Step 4.3.1.1
One to any power is one.
Step 4.3.1.2
Multiply by .
Step 4.3.1.3
Multiply by .
Step 4.3.2
Add and .
Step 5
Substitute the components into the linearization function in order to find the linearization at .
Step 6
Simplify.
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Step 6.1
Simplify each term.
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Step 6.1.1
Apply the distributive property.
Step 6.1.2
Multiply by .
Step 6.2
Subtract from .
Step 7