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.
Step 2.1
Evaluate at .
Step 2.1.1
Replace the variable with in the expression.
Step 2.1.2
Simplify the result.
Step 2.1.2.1
Simplify each term.
Step 2.1.2.1.1
Raising to any positive power yields .
Step 2.1.2.1.2
Multiply by .
Step 2.1.2.2
Step 2.1.2.2.1
Step 2.1.2.2.2
Step 2.1.2.3
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.
Step 5.1
Evaluate the function at .
Step 5.1.1
Replace the variable with in the expression.
Step 5.1.2
Simplify the result.
Step 5.1.2.1
Simplify each term.
Step 5.1.2.1.1
Rewrite as .
Step 5.1.2.1.2
Expand using the FOIL Method.
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.
Step 5.1.2.1.3.1
Simplify each term.
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
Step 5.1.2.1.3.2.1
Reorder and .
Step 5.1.2.1.3.2.2
Step 5.1.2.1.4
Apply the distributive property.
Step 5.1.2.2
Step 5.2
Reorder.
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.
Step 7.1
Simplify the numerator.
Step 7.1.1
Apply the distributive property.
Step 7.1.2
Simplify.
Step 7.1.2.1
Multiply by .
Step 7.1.2.2
Multiply by .
Step 7.1.3
Subtract from .
Step 7.1.4
Step 7.1.5
Subtract from .
Step 7.1.6
Step 7.1.7
Subtract from .
Step 7.1.8
Step 7.1.9
Factor out of .
Step 7.1.9.1
Factor out of .
Step 7.1.9.2
Factor out of .
Step 7.1.9.3
Factor out of .
Step 7.1.9.4
Factor out of .
Step 7.1.9.5
Factor out of .
Step 7.2
Reduce the expression by cancelling the common factors.
Step 7.2.1
Cancel the common factor of .
Step 7.2.1.1
Cancel the common factor.
Step 7.2.1.2
Divide by .
Step 7.2.2
Reorder and .
Step 8
Evaluate the limit.
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
Evaluate the limit of which is constant as approaches .
Step 9
Evaluate the limit of by plugging in for .
Step 10
Step 11
Simplify .
Step 11.1
Multiply by .
Step 11.2
Step 12
The slope is and the point is .
Step 13
Find the value of using the formula for the equation of a line.
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 .
Step 13.5.1
Rewrite the equation as .
Step 13.5.2
Simplify .
Step 13.5.2.1
Multiply by .
Step 13.5.2.2