# Calculus Examples

Find the Tangent at a Given Point Using the Limit Definition y=5x^3-3x , (1,2)
,
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
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
Subtract from .
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
Use the Binomial Theorem.
Step 5.1.2.1.2
Apply the distributive property.
Step 5.1.2.1.3
Simplify.
Step 5.1.2.1.3.1
Multiply by .
Step 5.1.2.1.3.2
Multiply by .
Step 5.1.2.1.4
Remove parentheses.
Step 5.1.2.1.5
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
Move .
Step 5.2.4
Move .
Step 5.2.5
Move .
Step 5.2.6
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
Multiply by .
Step 7.1.3
Multiply by .
Step 7.1.4
Subtract from .
Step 7.1.5
Step 7.1.6
Step 7.1.7
Step 7.1.8
Factor out of .
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 7.1.8.6
Factor out of .
Step 7.1.8.7
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
Simplify the expression.
Step 7.2.2.1
Move .
Step 7.2.2.2
Move .
Step 7.2.2.3
Reorder and .
Step 8
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 9
Evaluate the limit of which is constant as approaches .
Step 10
Move the term outside of the limit because it is constant with respect to .
Step 11
Move the term outside of the limit because it is constant with respect to .
Step 12
Move the exponent from outside the limit using the Limits Power Rule.
Step 13
Evaluate the limit of which is constant as approaches .
Step 14
Evaluate the limits by plugging in for all occurrences of .
Step 14.1
Evaluate the limit of by plugging in for .
Step 14.2
Evaluate the limit of by plugging in for .
Step 15
Step 15.1
Simplify each term.
Step 15.1.1
Multiply .
Step 15.1.1.1
Multiply by .
Step 15.1.1.2
Multiply by .
Step 15.1.2
Raising to any positive power yields .
Step 15.1.3
Multiply by .
Step 15.1.4
Multiply by .
Step 15.2
Combine the opposite terms in .
Step 15.2.1
Step 15.2.2
Step 16
Find the slope . In this case .
Step 16.1
Remove parentheses.
Step 16.2
Simplify .
Step 16.2.1
Simplify each term.
Step 16.2.1.1
One to any power is one.
Step 16.2.1.2
Multiply by .
Step 16.2.2
Subtract from .
Step 17
The slope is and the point is .
Step 18
Find the value of using the formula for the equation of a line.
Step 18.1
Use the formula for the equation of a line to find .
Step 18.2
Substitute the value of into the equation.
Step 18.3
Substitute the value of into the equation.
Step 18.4
Substitute the value of into the equation.
Step 18.5
Find the value of .
Step 18.5.1
Rewrite the equation as .
Step 18.5.2
Multiply by .
Step 18.5.3
Move all terms not containing to the right side of the equation.
Step 18.5.3.1
Subtract from both sides of the equation.
Step 18.5.3.2
Subtract from .
Step 19
Now that the values of (slope) and (y-intercept) are known, substitute them into to find the equation of the line.
Step 20