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Calculus Examples
Step 1
Step 1.1
Find the second derivative.
Step 1.1.1
Find the first derivative.
Step 1.1.1.1
Apply basic rules of exponents.
Step 1.1.1.1.1
Rewrite as .
Step 1.1.1.1.2
Multiply the exponents in .
Step 1.1.1.1.2.1
Apply the power rule and multiply exponents, .
Step 1.1.1.1.2.2
Multiply by .
Step 1.1.1.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.3
Simplify.
Step 1.1.1.3.1
Rewrite the expression using the negative exponent rule .
Step 1.1.1.3.2
Combine terms.
Step 1.1.1.3.2.1
Combine and .
Step 1.1.1.3.2.2
Move the negative in front of the fraction.
Step 1.1.2
Find the second derivative.
Step 1.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.2
Apply basic rules of exponents.
Step 1.1.2.2.1
Rewrite as .
Step 1.1.2.2.2
Multiply the exponents in .
Step 1.1.2.2.2.1
Apply the power rule and multiply exponents, .
Step 1.1.2.2.2.2
Multiply by .
Step 1.1.2.3
Differentiate using the Power Rule which states that is where .
Step 1.1.2.4
Multiply by .
Step 1.1.2.5
Simplify.
Step 1.1.2.5.1
Rewrite the expression using the negative exponent rule .
Step 1.1.2.5.2
Combine and .
Step 1.1.3
The second derivative of with respect to is .
Step 1.2
Set the second derivative equal to then solve the equation .
Step 1.2.1
Set the second derivative equal to .
Step 1.2.2
Set the numerator equal to zero.
Step 1.2.3
Since , there are no solutions.
No solution
No solution
No solution
Step 2
Step 2.1
Set the denominator in equal to to find where the expression is undefined.
Step 2.2
Solve for .
Step 2.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2.2.2
Simplify .
Step 2.2.2.1
Rewrite as .
Step 2.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 2.2.2.3
Plus or minus is .
Step 2.3
The domain is all values of that make the expression defined.
Interval Notation:
Set-Builder Notation:
Interval Notation:
Set-Builder Notation:
Step 3
Create intervals around the -values where the second derivative is zero or undefined.
Step 4
Step 4.1
Replace the variable with in the expression.
Step 4.2
Simplify the result.
Step 4.2.1
Raise to the power of .
Step 4.2.2
Cancel the common factor of and .
Step 4.2.2.1
Factor out of .
Step 4.2.2.2
Cancel the common factors.
Step 4.2.2.2.1
Factor out of .
Step 4.2.2.2.2
Cancel the common factor.
Step 4.2.2.2.3
Rewrite the expression.
Step 4.2.3
The final answer is .
Step 4.3
The graph is concave up on the interval because is positive.
Concave up on since is positive
Concave up on since is positive
Step 5
Step 5.1
Replace the variable with in the expression.
Step 5.2
Simplify the result.
Step 5.2.1
Raise to the power of .
Step 5.2.2
Cancel the common factor of and .
Step 5.2.2.1
Factor out of .
Step 5.2.2.2
Cancel the common factors.
Step 5.2.2.2.1
Factor out of .
Step 5.2.2.2.2
Cancel the common factor.
Step 5.2.2.2.3
Rewrite the expression.
Step 5.2.3
The final answer is .
Step 5.3
The graph is concave up on the interval because is positive.
Concave up on since is positive
Concave up on since is positive
Step 6
The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.
Concave up on since is positive
Concave up on since is positive
Step 7