Calculus Examples

Find the Tangent at a Given Point Using the Limit Definition
,
Step 1
Write as a function.
Step 2
Check if the given point is on the graph of the given function.
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Step 2.1
Evaluate at .
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Step 2.1.1
Replace the variable with in the expression.
Step 2.1.2
Simplify the result.
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Step 2.1.2.1
Simplify each term.
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Step 2.1.2.1.1
One to any power is one.
Step 2.1.2.1.2
Multiply by .
Step 2.1.2.1.3
Multiply by .
Step 2.1.2.2
Add and .
Step 2.1.2.3
The final answer is .
Step 2.2
Since , the point is on the graph.
The point is on the graph
The point is on the graph
Step 3
The slope of the tangent line is the derivative of the expression.
The derivative of
Step 4
Consider the limit definition of the derivative.
Step 5
Find the components of the definition.
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Step 5.1
Evaluate the function at .
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Step 5.1.1
Replace the variable with in the expression.
Step 5.1.2
Simplify the result.
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Step 5.1.2.1
Simplify each term.
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Step 5.1.2.1.1
Rewrite as .
Step 5.1.2.1.2
Expand using the FOIL Method.
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Step 5.1.2.1.2.1
Apply the distributive property.
Step 5.1.2.1.2.2
Apply the distributive property.
Step 5.1.2.1.2.3
Apply the distributive property.
Step 5.1.2.1.3
Simplify and combine like terms.
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Step 5.1.2.1.3.1
Simplify each term.
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Step 5.1.2.1.3.1.1
Multiply by .
Step 5.1.2.1.3.1.2
Multiply by .
Step 5.1.2.1.3.2
Add and .
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Step 5.1.2.1.3.2.1
Reorder and .
Step 5.1.2.1.3.2.2
Add and .
Step 5.1.2.1.4
Apply the distributive property.
Step 5.1.2.1.5
Multiply by .
Step 5.1.2.1.6
Apply the distributive property.
Step 5.1.2.2
The final answer is .
Step 5.2
Reorder.
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Step 5.2.1
Move .
Step 5.2.2
Move .
Step 5.2.3
Reorder and .
Step 5.3
Find the components of the definition.
Step 6
Plug in the components.
Step 7
Simplify.
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Step 7.1
Simplify the numerator.
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Step 7.1.1
Apply the distributive property.
Step 7.1.2
Multiply by .
Step 7.1.3
Multiply by .
Step 7.1.4
Subtract from .
Step 7.1.5
Add and .
Step 7.1.6
Subtract from .
Step 7.1.7
Add and .
Step 7.1.8
Factor out of .
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Step 7.1.8.1
Factor out of .
Step 7.1.8.2
Factor out of .
Step 7.1.8.3
Factor out of .
Step 7.1.8.4
Factor out of .
Step 7.1.8.5
Factor out of .
Step 7.2
Simplify terms.
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Step 7.2.1
Cancel the common factor of .
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Step 7.2.1.1
Cancel the common factor.
Step 7.2.1.2
Divide by .
Step 7.2.2
Apply the distributive property.
Step 7.3
Simplify.
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Step 7.3.1
Multiply by .
Step 7.3.2
Multiply by .
Step 7.4
Reorder and .
Step 8
Evaluate the limit.
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Step 8.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 8.2
Evaluate the limit of which is constant as approaches .
Step 8.3
Move the term outside of the limit because it is constant with respect to .
Step 8.4
Evaluate the limit of which is constant as approaches .
Step 9
Evaluate the limit of by plugging in for .
Step 10
Simplify the answer.
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Step 10.1
Multiply by .
Step 10.2
Add and .
Step 11
Simplify .
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Step 11.1
Multiply by .
Step 11.2
Add and .
Step 12
The slope is and the point is .
Step 13
Find the value of using the formula for the equation of a line.
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Step 13.1
Use the formula for the equation of a line to find .
Step 13.2
Substitute the value of into the equation.
Step 13.3
Substitute the value of into the equation.
Step 13.4
Substitute the value of into the equation.
Step 13.5
Find the value of .
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Step 13.5.1
Rewrite the equation as .
Step 13.5.2
Multiply by .
Step 13.5.3
Move all terms not containing to the right side of the equation.
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Step 13.5.3.1
Subtract from both sides of the equation.
Step 13.5.3.2
Subtract from .
Step 14
Now that the values of (slope) and (y-intercept) are known, substitute them into to find the equation of the line.
Step 15
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