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Calculus Examples
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Step 1
Step 1.1
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Interval Notation:
Set-Builder Notation:
Step 1.2
is continuous on .
The function is continuous.
The function is continuous.
Step 2
Step 2.1
Find the derivative.
Step 2.1.1
Find the first derivative.
Step 2.1.1.1
Differentiate.
Step 2.1.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 2.1.1.1.2
Differentiate using the Power Rule which states that is where .
Step 2.1.1.2
Evaluate .
Step 2.1.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 2.1.1.2.3
Multiply by .
Step 2.1.2
The first derivative of with respect to is .
Step 2.2
Find if the derivative is continuous on .
Step 2.2.1
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Interval Notation:
Set-Builder Notation:
Step 2.2.2
is continuous on .
The function is continuous.
The function is continuous.
Step 2.3
The function is differentiable on because the derivative is continuous on .
The function is differentiable.
The function is differentiable.
Step 3
For arc length to be guaranteed, the function and its derivative must both be continuous on the closed interval .
The function and its derivative are continuous on the closed interval .
Step 4
Step 4.1
Differentiate.
Step 4.1.1
By the Sum Rule, the derivative of with respect to is .
Step 4.1.2
Differentiate using the Power Rule which states that is where .
Step 4.2
Evaluate .
Step 4.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.2.2
Differentiate using the Power Rule which states that is where .
Step 4.2.3
Multiply by .
Step 5
To find the arc length of a function, use the formula .
Step 6
Step 6.1
Complete the square.
Step 6.1.1
Use the form , to find the values of , , and .
Step 6.1.2
Consider the vertex form of a parabola.
Step 6.1.3
Find the value of using the formula .
Step 6.1.3.1
Substitute the values of and into the formula .
Step 6.1.3.2
Simplify the right side.
Step 6.1.3.2.1
Cancel the common factor of and .
Step 6.1.3.2.1.1
Factor out of .
Step 6.1.3.2.1.2
Cancel the common factors.
Step 6.1.3.2.1.2.1
Factor out of .
Step 6.1.3.2.1.2.2
Cancel the common factor.
Step 6.1.3.2.1.2.3
Rewrite the expression.
Step 6.1.3.2.2
Cancel the common factor of .
Step 6.1.3.2.2.1
Cancel the common factor.
Step 6.1.3.2.2.2
Rewrite the expression.
Step 6.1.4
Find the value of using the formula .
Step 6.1.4.1
Substitute the values of , and into the formula .
Step 6.1.4.2
Simplify the right side.
Step 6.1.4.2.1
Simplify each term.
Step 6.1.4.2.1.1
Raise to the power of .
Step 6.1.4.2.1.2
Multiply by .
Step 6.1.4.2.1.3
Divide by .
Step 6.1.4.2.1.4
Multiply by .
Step 6.1.4.2.2
Subtract from .
Step 6.1.5
Substitute the values of , , and into the vertex form .
Step 6.2
Let . Then . Rewrite using and .
Step 6.2.1
Let . Find .
Step 6.2.1.1
Differentiate .
Step 6.2.1.2
By the Sum Rule, the derivative of with respect to is .
Step 6.2.1.3
Differentiate using the Power Rule which states that is where .
Step 6.2.1.4
Since is constant with respect to , the derivative of with respect to is .
Step 6.2.1.5
Add and .
Step 6.2.2
Substitute the lower limit in for in .
Step 6.2.3
Add and .
Step 6.2.4
Substitute the upper limit in for in .
Step 6.2.5
Add and .
Step 6.2.6
The values found for and will be used to evaluate the definite integral.
Step 6.2.7
Rewrite the problem using , , and the new limits of integration.
Step 6.3
Let , where . Then . Note that since , is positive.
Step 6.4
Simplify terms.
Step 6.4.1
Simplify .
Step 6.4.1.1
Simplify each term.
Step 6.4.1.1.1
Combine and .
Step 6.4.1.1.2
Apply the product rule to .
Step 6.4.1.1.3
Raise to the power of .
Step 6.4.1.1.4
Cancel the common factor of .
Step 6.4.1.1.4.1
Cancel the common factor.
Step 6.4.1.1.4.2
Rewrite the expression.
Step 6.4.1.2
Apply pythagorean identity.
Step 6.4.1.3
Pull terms out from under the radical, assuming positive real numbers.
Step 6.4.2
Simplify.
Step 6.4.2.1
Combine and .
Step 6.4.2.2
Multiply by by adding the exponents.
Step 6.4.2.2.1
Multiply by .
Step 6.4.2.2.1.1
Raise to the power of .
Step 6.4.2.2.1.2
Use the power rule to combine exponents.
Step 6.4.2.2.2
Add and .
Step 6.5
Since is constant with respect to , move out of the integral.
Step 6.6
Apply the reduction formula.
Step 6.7
The integral of with respect to is .
Step 6.8
Simplify.
Step 6.8.1
Combine and .
Step 6.8.2
To write as a fraction with a common denominator, multiply by .
Step 6.8.3
Combine and .
Step 6.8.4
Combine the numerators over the common denominator.
Step 6.8.5
Move to the left of .
Step 6.8.6
Multiply by .
Step 6.8.7
Multiply by .
Step 6.9
Substitute and simplify.
Step 6.9.1
Evaluate at and at .
Step 6.9.2
Evaluate at and at .
Step 6.9.3
Remove unnecessary parentheses.
Step 6.10
Use the quotient property of logarithms, .
Step 6.11
Simplify.
Step 6.11.1
Simplify the numerator.
Step 6.11.1.1
Evaluate .
Step 6.11.1.2
Evaluate .
Step 6.11.2
Multiply by .
Step 6.11.3
Divide by .
Step 6.11.4
Multiply by .
Step 6.11.5
Simplify each term.
Step 6.11.5.1
Simplify the numerator.
Step 6.11.5.1.1
Evaluate .
Step 6.11.5.1.2
Evaluate .
Step 6.11.5.2
Multiply by .
Step 6.11.5.3
Divide by .
Step 6.11.6
Subtract from .
Step 6.11.7
Multiply by .
Step 6.11.8
is approximately which is positive so remove the absolute value
Step 6.11.9
is approximately which is positive so remove the absolute value
Step 7
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 8