# Calculus Examples

Find the Tangent at a Given Point Using the Limit Definition f(x)=(x^3-3+1)(x+2) , (1,-3)
,
Step 1
Check if the given point is on the graph of the given function.
Step 1.1
Evaluate at .
Step 1.1.1
Replace the variable with in the expression.
Step 1.1.2
Simplify the result.
Step 1.1.2.1
One to any power is one.
Step 1.1.2.2
Subtract from .
Step 1.1.2.3
Step 1.1.2.4
Step 1.1.2.5
Multiply by .
Step 1.1.2.6
Step 1.2
Since , the point is on the graph.
The point is on the graph
The point is on the graph
Step 2
The slope of the tangent line is the derivative of the expression.
The derivative of
Step 3
Consider the limit definition of the derivative.
Step 4
Find the components of the definition.
Step 4.1
Evaluate the function at .
Step 4.1.1
Replace the variable with in the expression.
Step 4.1.2
Simplify the result.
Step 4.1.2.1
Use the Binomial Theorem.
Step 4.1.2.2
Step 4.1.2.3
Expand by multiplying each term in the first expression by each term in the second expression.
Step 4.1.2.4
Simplify terms.
Step 4.1.2.4.1
Simplify each term.
Step 4.1.2.4.1.1
Multiply by by adding the exponents.
Step 4.1.2.4.1.1.1
Multiply by .
Step 4.1.2.4.1.1.1.1
Raise to the power of .
Step 4.1.2.4.1.1.1.2
Use the power rule to combine exponents.
Step 4.1.2.4.1.1.2
Step 4.1.2.4.1.2
Move to the left of .
Step 4.1.2.4.1.3
Multiply by by adding the exponents.
Step 4.1.2.4.1.3.1
Move .
Step 4.1.2.4.1.3.2
Multiply by .
Step 4.1.2.4.1.3.2.1
Raise to the power of .
Step 4.1.2.4.1.3.2.2
Use the power rule to combine exponents.
Step 4.1.2.4.1.3.3
Step 4.1.2.4.1.4
Multiply by by adding the exponents.
Step 4.1.2.4.1.4.1
Move .
Step 4.1.2.4.1.4.2
Multiply by .
Step 4.1.2.4.1.5
Multiply by .
Step 4.1.2.4.1.6
Multiply by by adding the exponents.
Step 4.1.2.4.1.6.1
Move .
Step 4.1.2.4.1.6.2
Multiply by .
Step 4.1.2.4.1.7
Multiply by by adding the exponents.
Step 4.1.2.4.1.7.1
Move .
Step 4.1.2.4.1.7.2
Multiply by .
Step 4.1.2.4.1.7.2.1
Raise to the power of .
Step 4.1.2.4.1.7.2.2
Use the power rule to combine exponents.
Step 4.1.2.4.1.7.3
Step 4.1.2.4.1.8
Multiply by .
Step 4.1.2.4.1.9
Multiply by by adding the exponents.
Step 4.1.2.4.1.9.1
Multiply by .
Step 4.1.2.4.1.9.1.1
Raise to the power of .
Step 4.1.2.4.1.9.1.2
Use the power rule to combine exponents.
Step 4.1.2.4.1.9.2
Step 4.1.2.4.1.10
Move to the left of .
Step 4.1.2.4.1.11
Multiply by .
Step 4.1.2.4.2
Step 4.1.2.4.2.1
Step 4.1.2.4.2.2
Step 4.1.2.5
Step 4.1.2.5.1
Move .
Step 4.1.2.5.2
Step 4.1.2.6
Step 4.2
Reorder.
Step 4.2.1
Move .
Step 4.2.2
Move .
Step 4.2.3
Move .
Step 4.2.4
Move .
Step 4.2.5
Move .
Step 4.2.6
Move .
Step 4.2.7
Move .
Step 4.2.8
Move .
Step 4.2.9
Move .
Step 4.2.10
Move .
Step 4.2.11
Move .
Step 4.2.12
Reorder and .
Step 4.3
Find the components of the definition.
Step 5
Plug in the components.
Step 6
Simplify.
Step 6.1
Simplify the numerator.
Step 6.1.1
Apply the distributive property.
Step 6.1.2
Simplify.
Step 6.1.2.1
Multiply by .
Step 6.1.2.2
Multiply by .
Step 6.1.2.3
Multiply by .
Step 6.1.3
Subtract from .
Step 6.1.4
Step 6.1.5
Subtract from .
Step 6.1.6
Step 6.1.7
Step 6.1.8
Step 6.1.9
Step 6.1.10
Step 6.1.11
Factor out of .
Step 6.1.11.1
Factor out of .
Step 6.1.11.2
Factor out of .
Step 6.1.11.3
Factor out of .
Step 6.1.11.4
Factor out of .
Step 6.1.11.5
Factor out of .
Step 6.1.11.6
Factor out of .
Step 6.1.11.7
Factor out of .
Step 6.1.11.8
Factor out of .
Step 6.1.11.9
Factor out of .
Step 6.1.11.10
Factor out of .
Step 6.1.11.11
Factor out of .
Step 6.1.11.12
Factor out of .
Step 6.1.11.13
Factor out of .
Step 6.1.11.14
Factor out of .
Step 6.1.11.15
Factor out of .
Step 6.2
Reduce the expression by cancelling the common factors.
Step 6.2.1
Cancel the common factor of .
Step 6.2.1.1
Cancel the common factor.
Step 6.2.1.2
Divide by .
Step 6.2.2
Simplify the expression.
Step 6.2.2.1
Move .
Step 6.2.2.2
Move .
Step 6.2.2.3
Move .
Step 6.2.2.4
Move .
Step 6.2.2.5
Move .
Step 6.2.2.6
Move .
Step 6.2.2.7
Move .
Step 6.2.2.8
Reorder and .
Step 7
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 8
Evaluate the limit of which is constant as approaches .
Step 9
Move the term outside of the limit because it is constant with respect to .
Step 10
Move the term outside of the limit because it is constant with respect to .
Step 11
Move the exponent from outside the limit using the Limits Power Rule.
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
Move the term outside of the limit because it is constant with respect to .
Step 15
Move the term outside of the limit because it is constant with respect to .
Step 16
Move the exponent from outside the limit using the Limits Power Rule.
Step 17
Evaluate the limit of which is constant as approaches .
Step 18
Evaluate the limits by plugging in for all occurrences of .
Step 18.1
Evaluate the limit of by plugging in for .
Step 18.2
Evaluate the limit of by plugging in for .
Step 18.3
Evaluate the limit of by plugging in for .
Step 18.4
Evaluate the limit of by plugging in for .
Step 18.5
Evaluate the limit of by plugging in for .
Step 19
Step 19.1
Simplify each term.
Step 19.1.1
Multiply .
Step 19.1.1.1
Multiply by .
Step 19.1.1.2
Multiply by .
Step 19.1.2
Raising to any positive power yields .
Step 19.1.3
Multiply .
Step 19.1.3.1
Multiply by .
Step 19.1.3.2
Multiply by .
Step 19.1.4
Raising to any positive power yields .
Step 19.1.5
Multiply .
Step 19.1.5.1
Multiply by .
Step 19.1.5.2
Multiply by .
Step 19.1.6
Raising to any positive power yields .
Step 19.1.7
Multiply by .
Step 19.1.8
Multiply by .
Step 19.2
Combine the opposite terms in .
Step 19.2.1
Step 19.2.2
Step 19.2.3
Step 19.2.4
Step 19.2.5
Step 20
Find the slope . In this case .
Step 20.1
Remove parentheses.
Step 20.2
Remove parentheses.
Step 20.3
Simplify .
Step 20.3.1
Simplify each term.
Step 20.3.1.1
One to any power is one.
Step 20.3.1.2
Multiply by .
Step 20.3.1.3
One to any power is one.
Step 20.3.1.4
Multiply by .
Step 20.3.2
Step 20.3.2.1
Step 20.3.2.2
Subtract from .
Step 21
The slope is and the point is .
Step 22
Find the value of using the formula for the equation of a line.
Step 22.1
Use the formula for the equation of a line to find .
Step 22.2
Substitute the value of into the equation.
Step 22.3
Substitute the value of into the equation.
Step 22.4
Substitute the value of into the equation.
Step 22.5
Find the value of .
Step 22.5.1
Rewrite the equation as .
Step 22.5.2
Multiply by .
Step 22.5.3
Move all terms not containing to the right side of the equation.
Step 22.5.3.1
Subtract from both sides of the equation.
Step 22.5.3.2
Subtract from .
Step 23
Now that the values of (slope) and (y-intercept) are known, substitute them into to find the equation of the line.
Step 24