Calculus Examples

Find the Absolute Max and Min over the Interval g(x)=1/20x^2-25 square root of x on 0 , 49
on ,
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
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.2
Evaluate .
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Step 1.1.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2.3
Combine and .
Step 1.1.1.2.4
Combine and .
Step 1.1.1.2.5
Cancel the common factor of and .
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Step 1.1.1.2.5.1
Factor out of .
Step 1.1.1.2.5.2
Cancel the common factors.
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Step 1.1.1.2.5.2.1
Factor out of .
Step 1.1.1.2.5.2.2
Cancel the common factor.
Step 1.1.1.2.5.2.3
Rewrite the expression.
Step 1.1.1.3
Evaluate .
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Step 1.1.1.3.1
Use to rewrite as .
Step 1.1.1.3.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.3.3
Differentiate using the Power Rule which states that is where .
Step 1.1.1.3.4
To write as a fraction with a common denominator, multiply by .
Step 1.1.1.3.5
Combine and .
Step 1.1.1.3.6
Combine the numerators over the common denominator.
Step 1.1.1.3.7
Simplify the numerator.
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Step 1.1.1.3.7.1
Multiply by .
Step 1.1.1.3.7.2
Subtract from .
Step 1.1.1.3.8
Move the negative in front of the fraction.
Step 1.1.1.3.9
Combine and .
Step 1.1.1.3.10
Combine and .
Step 1.1.1.3.11
Move to the denominator using the negative exponent rule .
Step 1.1.1.3.12
Move the negative in front of the fraction.
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
Find the LCD of the terms in the equation.
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Step 1.2.2.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 1.2.2.2
Since contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part then find LCM for the variable part .
Step 1.2.2.3
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
Step 1.2.2.4
has factors of and .
Step 1.2.2.5
Since has no factors besides and .
is a prime number
Step 1.2.2.6
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
Step 1.2.2.7
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
Step 1.2.2.8
Multiply by .
Step 1.2.2.9
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
Step 1.2.2.10
The LCM for is the numeric part multiplied by the variable part.
Step 1.2.3
Multiply each term in by to eliminate the fractions.
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Step 1.2.3.1
Multiply each term in by .
Step 1.2.3.2
Simplify the left side.
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Step 1.2.3.2.1
Simplify each term.
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Step 1.2.3.2.1.1
Rewrite using the commutative property of multiplication.
Step 1.2.3.2.1.2
Cancel the common factor of .
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Step 1.2.3.2.1.2.1
Cancel the common factor.
Step 1.2.3.2.1.2.2
Rewrite the expression.
Step 1.2.3.2.1.3
Multiply by by adding the exponents.
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Step 1.2.3.2.1.3.1
Multiply by .
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Step 1.2.3.2.1.3.1.1
Raise to the power of .
Step 1.2.3.2.1.3.1.2
Use the power rule to combine exponents.
Step 1.2.3.2.1.3.2
Write as a fraction with a common denominator.
Step 1.2.3.2.1.3.3
Combine the numerators over the common denominator.
Step 1.2.3.2.1.3.4
Add and .
Step 1.2.3.2.1.4
Cancel the common factor of .
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Step 1.2.3.2.1.4.1
Move the leading negative in into the numerator.
Step 1.2.3.2.1.4.2
Factor out of .
Step 1.2.3.2.1.4.3
Cancel the common factor.
Step 1.2.3.2.1.4.4
Rewrite the expression.
Step 1.2.3.2.1.5
Multiply by .
Step 1.2.3.3
Simplify the right side.
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Step 1.2.3.3.1
Multiply .
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Step 1.2.3.3.1.1
Multiply by .
Step 1.2.3.3.1.2
Multiply by .
Step 1.2.4
Solve the equation.
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Step 1.2.4.1
Add to both sides of the equation.
Step 1.2.4.2
Raise each side of the equation to the power of to eliminate the fractional exponent on the left side.
Step 1.2.4.3
Simplify the exponent.
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Step 1.2.4.3.1
Simplify the left side.
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Step 1.2.4.3.1.1
Simplify .
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Step 1.2.4.3.1.1.1
Multiply the exponents in .
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Step 1.2.4.3.1.1.1.1
Apply the power rule and multiply exponents, .
Step 1.2.4.3.1.1.1.2
Cancel the common factor of .
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Step 1.2.4.3.1.1.1.2.1
Cancel the common factor.
Step 1.2.4.3.1.1.1.2.2
Rewrite the expression.
Step 1.2.4.3.1.1.1.3
Cancel the common factor of .
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Step 1.2.4.3.1.1.1.3.1
Cancel the common factor.
Step 1.2.4.3.1.1.1.3.2
Rewrite the expression.
Step 1.2.4.3.1.1.2
Simplify.
Step 1.2.4.3.2
Simplify the right side.
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Step 1.2.4.3.2.1
Simplify .
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Step 1.2.4.3.2.1.1
Simplify the expression.
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Step 1.2.4.3.2.1.1.1
Rewrite as .
Step 1.2.4.3.2.1.1.2
Apply the power rule and multiply exponents, .
Step 1.2.4.3.2.1.2
Cancel the common factor of .
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Step 1.2.4.3.2.1.2.1
Cancel the common factor.
Step 1.2.4.3.2.1.2.2
Rewrite the expression.
Step 1.2.4.3.2.1.3
Raise to the power of .
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
Apply the product rule to .
Step 1.3.3.2.2.1.2
Raise to the power of .
Step 1.3.3.2.2.1.3
Multiply the exponents in .
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Step 1.3.3.2.2.1.3.1
Apply the power rule and multiply exponents, .
Step 1.3.3.2.2.1.3.2
Cancel the common factor of .
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Step 1.3.3.2.2.1.3.2.1
Cancel the common factor.
Step 1.3.3.2.2.1.3.2.2
Rewrite the expression.
Step 1.3.3.2.2.1.4
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
Divide each term in by and simplify.
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Step 1.3.3.3.1
Divide each term in by .
Step 1.3.3.3.2
Simplify the left side.
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Step 1.3.3.3.2.1
Cancel the common factor of .
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Step 1.3.3.3.2.1.1
Cancel the common factor.
Step 1.3.3.3.2.1.2
Divide by .
Step 1.3.3.3.3
Simplify the right side.
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Step 1.3.3.3.3.1
Divide by .
Step 1.3.4
Set the radicand in less than to find where the expression is undefined.
Step 1.3.5
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
Simplify each term.
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Step 1.4.1.2.1.1
Raise to the power of .
Step 1.4.1.2.1.2
Cancel the common factor of .
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Step 1.4.1.2.1.2.1
Factor out of .
Step 1.4.1.2.1.2.2
Factor out of .
Step 1.4.1.2.1.2.3
Cancel the common factor.
Step 1.4.1.2.1.2.4
Rewrite the expression.
Step 1.4.1.2.1.3
Combine and .
Step 1.4.1.2.1.4
Rewrite as .
Step 1.4.1.2.1.5
Pull terms out from under the radical, assuming positive real numbers.
Step 1.4.1.2.1.6
Multiply by .
Step 1.4.1.2.2
To write as a fraction with a common denominator, multiply by .
Step 1.4.1.2.3
Combine and .
Step 1.4.1.2.4
Combine the numerators over the common denominator.
Step 1.4.1.2.5
Simplify the numerator.
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Step 1.4.1.2.5.1
Multiply by .
Step 1.4.1.2.5.2
Subtract from .
Step 1.4.1.2.6
Move the negative in front of the fraction.
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
Simplify each term.
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Step 1.4.2.2.1.1
Raising to any positive power yields .
Step 1.4.2.2.1.2
Multiply by .
Step 1.4.2.2.1.3
Rewrite as .
Step 1.4.2.2.1.4
Pull terms out from under the radical, assuming positive real numbers.
Step 1.4.2.2.1.5
Multiply by .
Step 1.4.2.2.2
Add and .
Step 1.4.3
List all of the points.
Step 2
Evaluate at the included endpoints.
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Step 2.1
Evaluate at .
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Step 2.1.1
Substitute for .
Step 2.1.2
Simplify.
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Step 2.1.2.1
Simplify each term.
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Step 2.1.2.1.1
Raising to any positive power yields .
Step 2.1.2.1.2
Multiply by .
Step 2.1.2.1.3
Rewrite as .
Step 2.1.2.1.4
Pull terms out from under the radical, assuming positive real numbers.
Step 2.1.2.1.5
Multiply by .
Step 2.1.2.2
Add and .
Step 2.2
Evaluate at .
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Step 2.2.1
Substitute for .
Step 2.2.2
Simplify.
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Step 2.2.2.1
Simplify each term.
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Step 2.2.2.1.1
Raise to the power of .
Step 2.2.2.1.2
Combine and .
Step 2.2.2.1.3
Rewrite as .
Step 2.2.2.1.4
Pull terms out from under the radical, assuming positive real numbers.
Step 2.2.2.1.5
Multiply by .
Step 2.2.2.2
To write as a fraction with a common denominator, multiply by .
Step 2.2.2.3
Combine and .
Step 2.2.2.4
Combine the numerators over the common denominator.
Step 2.2.2.5
Simplify the numerator.
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Step 2.2.2.5.1
Multiply by .
Step 2.2.2.5.2
Subtract from .
Step 2.2.2.6
Move the negative in front of the fraction.
Step 2.3
List all of the points.
Step 3
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 4