Calculus Examples

Find the Absolute Max and Min over the Interval g(x)=- square root of 5-x^2 , - square root of 5<=x<=0
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Step 1
Find the critical points.
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Step 1.1
Find the first derivative.
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Step 1.1.1
Find the first derivative.
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Step 1.1.1.1
Differentiate using the Constant Multiple Rule.
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Step 1.1.1.1.1
Use to rewrite as .
Step 1.1.1.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2
Differentiate using the chain rule, which states that is where and .
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Step 1.1.1.2.1
To apply the Chain Rule, set as .
Step 1.1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2.3
Replace all occurrences of with .
Step 1.1.1.3
To write as a fraction with a common denominator, multiply by .
Step 1.1.1.4
Combine and .
Step 1.1.1.5
Combine the numerators over the common denominator.
Step 1.1.1.6
Simplify the numerator.
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Step 1.1.1.6.1
Multiply by .
Step 1.1.1.6.2
Subtract from .
Step 1.1.1.7
Combine fractions.
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Step 1.1.1.7.1
Move the negative in front of the fraction.
Step 1.1.1.7.2
Combine and .
Step 1.1.1.7.3
Move to the denominator using the negative exponent rule .
Step 1.1.1.8
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.9
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.10
Add and .
Step 1.1.1.11
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.12
Multiply.
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Step 1.1.1.12.1
Multiply by .
Step 1.1.1.12.2
Multiply by .
Step 1.1.1.13
Differentiate using the Power Rule which states that is where .
Step 1.1.1.14
Simplify terms.
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Step 1.1.1.14.1
Combine and .
Step 1.1.1.14.2
Combine and .
Step 1.1.1.14.3
Cancel the common factor.
Step 1.1.1.14.4
Rewrite the expression.
Step 1.1.1.14.5
Reorder terms.
Step 1.1.2
The first derivative of with respect to is .
Step 1.2
Set the first derivative equal to then solve the equation .
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Step 1.2.1
Set the first derivative equal to .
Step 1.2.2
Set the numerator equal to zero.
Step 1.3
Find the values where the derivative is undefined.
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Step 1.3.1
Convert expressions with fractional exponents to radicals.
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Step 1.3.1.1
Apply the rule to rewrite the exponentiation as a radical.
Step 1.3.1.2
Anything raised to is the base itself.
Step 1.3.2
Set the denominator in equal to to find where the expression is undefined.
Step 1.3.3
Solve for .
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Step 1.3.3.1
To remove the radical on the left side of the equation, square both sides of the equation.
Step 1.3.3.2
Simplify each side of the equation.
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Step 1.3.3.2.1
Use to rewrite as .
Step 1.3.3.2.2
Simplify the left side.
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Step 1.3.3.2.2.1
Simplify .
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Step 1.3.3.2.2.1.1
Multiply the exponents in .
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Step 1.3.3.2.2.1.1.1
Apply the power rule and multiply exponents, .
Step 1.3.3.2.2.1.1.2
Cancel the common factor of .
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Step 1.3.3.2.2.1.1.2.1
Cancel the common factor.
Step 1.3.3.2.2.1.1.2.2
Rewrite the expression.
Step 1.3.3.2.2.1.2
Simplify.
Step 1.3.3.2.3
Simplify the right side.
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Step 1.3.3.2.3.1
Raising to any positive power yields .
Step 1.3.3.3
Solve for .
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Step 1.3.3.3.1
Subtract from both sides of the equation.
Step 1.3.3.3.2
Divide each term in by and simplify.
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Step 1.3.3.3.2.1
Divide each term in by .
Step 1.3.3.3.2.2
Simplify the left side.
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Step 1.3.3.3.2.2.1
Dividing two negative values results in a positive value.
Step 1.3.3.3.2.2.2
Divide by .
Step 1.3.3.3.2.3
Simplify the right side.
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Step 1.3.3.3.2.3.1
Divide by .
Step 1.3.3.3.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 1.3.3.3.4
The complete solution is the result of both the positive and negative portions of the solution.
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Step 1.3.3.3.4.1
First, use the positive value of the to find the first solution.
Step 1.3.3.3.4.2
Next, use the negative value of the to find the second solution.
Step 1.3.3.3.4.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 1.3.4
Set the radicand in less than to find where the expression is undefined.
Step 1.3.5
Solve for .
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Step 1.3.5.1
Subtract from both sides of the inequality.
Step 1.3.5.2
Divide each term in by and simplify.
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Step 1.3.5.2.1
Divide each term in by . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
Step 1.3.5.2.2
Simplify the left side.
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Step 1.3.5.2.2.1
Dividing two negative values results in a positive value.
Step 1.3.5.2.2.2
Divide by .
Step 1.3.5.2.3
Simplify the right side.
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Step 1.3.5.2.3.1
Divide by .
Step 1.3.5.3
Take the specified root of both sides of the inequality to eliminate the exponent on the left side.
Step 1.3.5.4
Simplify the left side.
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Step 1.3.5.4.1
Pull terms out from under the radical.
Step 1.3.5.5
Write as a piecewise.
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Step 1.3.5.5.1
To find the interval for the first piece, find where the inside of the absolute value is non-negative.
Step 1.3.5.5.2
In the piece where is non-negative, remove the absolute value.
Step 1.3.5.5.3
To find the interval for the second piece, find where the inside of the absolute value is negative.
Step 1.3.5.5.4
In the piece where is negative, remove the absolute value and multiply by .
Step 1.3.5.5.5
Write as a piecewise.
Step 1.3.5.6
Find the intersection of and .
Step 1.3.5.7
Divide each term in by and simplify.
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Step 1.3.5.7.1
Divide each term in by . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
Step 1.3.5.7.2
Simplify the left side.
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Step 1.3.5.7.2.1
Dividing two negative values results in a positive value.
Step 1.3.5.7.2.2
Divide by .
Step 1.3.5.7.3
Simplify the right side.
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Step 1.3.5.7.3.1
Move the negative one from the denominator of .
Step 1.3.5.7.3.2
Rewrite as .
Step 1.3.5.8
Find the union of the solutions.
or
or
Step 1.3.6
The equation is undefined where the denominator equals , the argument of a square root is less than , or the argument of a logarithm is less than or equal to .
Step 1.4
Evaluate at each value where the derivative is or undefined.
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Step 1.4.1
Evaluate at .
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Step 1.4.1.1
Substitute for .
Step 1.4.1.2
Simplify.
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Step 1.4.1.2.1
Raising to any positive power yields .
Step 1.4.1.2.2
Multiply by .
Step 1.4.1.2.3
Add and .
Step 1.4.2
Evaluate at .
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Step 1.4.2.1
Substitute for .
Step 1.4.2.2
Simplify.
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Step 1.4.2.2.1
Apply the product rule to .
Step 1.4.2.2.2
Multiply by by adding the exponents.
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Step 1.4.2.2.2.1
Move .
Step 1.4.2.2.2.2
Multiply by .
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Step 1.4.2.2.2.2.1
Raise to the power of .
Step 1.4.2.2.2.2.2
Use the power rule to combine exponents.
Step 1.4.2.2.2.3
Add and .
Step 1.4.2.2.3
Raise to the power of .
Step 1.4.2.2.4
Rewrite as .
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Step 1.4.2.2.4.1
Use to rewrite as .
Step 1.4.2.2.4.2
Apply the power rule and multiply exponents, .
Step 1.4.2.2.4.3
Combine and .
Step 1.4.2.2.4.4
Cancel the common factor of .
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Step 1.4.2.2.4.4.1
Cancel the common factor.
Step 1.4.2.2.4.4.2
Rewrite the expression.
Step 1.4.2.2.4.5
Evaluate the exponent.
Step 1.4.2.2.5
Simplify the expression.
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Step 1.4.2.2.5.1
Multiply by .
Step 1.4.2.2.5.2
Subtract from .
Step 1.4.2.2.5.3
Rewrite as .
Step 1.4.2.2.5.4
Pull terms out from under the radical, assuming positive real numbers.
Step 1.4.2.2.5.5
Multiply by .
Step 1.4.3
Evaluate at .
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Step 1.4.3.1
Substitute for .
Step 1.4.3.2
Simplify.
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Step 1.4.3.2.1
Rewrite as .
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Step 1.4.3.2.1.1
Use to rewrite as .
Step 1.4.3.2.1.2
Apply the power rule and multiply exponents, .
Step 1.4.3.2.1.3
Combine and .
Step 1.4.3.2.1.4
Cancel the common factor of .
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Step 1.4.3.2.1.4.1
Cancel the common factor.
Step 1.4.3.2.1.4.2
Rewrite the expression.
Step 1.4.3.2.1.5
Evaluate the exponent.
Step 1.4.3.2.2
Simplify the expression.
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Step 1.4.3.2.2.1
Multiply by .
Step 1.4.3.2.2.2
Subtract from .
Step 1.4.3.2.2.3
Rewrite as .
Step 1.4.3.2.2.4
Pull terms out from under the radical, assuming positive real numbers.
Step 1.4.3.2.2.5
Multiply by .
Step 1.4.4
List all of the points.
Step 2
Exclude the points that are not on the interval.
Step 3
Evaluate at the included endpoints.
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Step 3.1
Evaluate at .
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Step 3.1.1
Substitute for .
Step 3.1.2
Simplify.
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Step 3.1.2.1
Apply the product rule to .
Step 3.1.2.2
Multiply by by adding the exponents.
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Step 3.1.2.2.1
Move .
Step 3.1.2.2.2
Multiply by .
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Step 3.1.2.2.2.1
Raise to the power of .
Step 3.1.2.2.2.2
Use the power rule to combine exponents.
Step 3.1.2.2.3
Add and .
Step 3.1.2.3
Raise to the power of .
Step 3.1.2.4
Rewrite as .
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Step 3.1.2.4.1
Use to rewrite as .
Step 3.1.2.4.2
Apply the power rule and multiply exponents, .
Step 3.1.2.4.3
Combine and .
Step 3.1.2.4.4
Cancel the common factor of .
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Step 3.1.2.4.4.1
Cancel the common factor.
Step 3.1.2.4.4.2
Rewrite the expression.
Step 3.1.2.4.5
Evaluate the exponent.
Step 3.1.2.5
Simplify the expression.
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Step 3.1.2.5.1
Multiply by .
Step 3.1.2.5.2
Subtract from .
Step 3.1.2.5.3
Rewrite as .
Step 3.1.2.5.4
Pull terms out from under the radical, assuming positive real numbers.
Step 3.1.2.5.5
Multiply by .
Step 3.2
Evaluate at .
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Step 3.2.1
Substitute for .
Step 3.2.2
Simplify.
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Step 3.2.2.1
Raising to any positive power yields .
Step 3.2.2.2
Multiply by .
Step 3.2.2.3
Add and .
Step 3.3
List all of the points.
Step 4
Compare the values found for each value of in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest value and the minimum will occur at the lowest value.
Absolute Maximum:
Absolute Minimum:
Step 5