Calculus Examples

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Check if is continuous.
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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.
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is continuous on .
The function is continuous.
The function is continuous.
Check if is differentiable.
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Find the derivative.
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Find the first derivative.
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By the Sum Rule, the derivative of with respect to is .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Differentiate using the Constant Rule.
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Since is constant with respect to , the derivative of with respect to is .
Add and .
The first derivative of with respect to is .
Find if the derivative is continuous on .
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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:
is continuous on .
The function is continuous.
The function is continuous.
The function is differentiable on because the derivative is continuous on .
The function is differentiable.
The function is differentiable.
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 .
find the derivative of .
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By the Sum Rule, the derivative of with respect to is .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Differentiate using the Constant Rule.
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Since is constant with respect to , the derivative of with respect to is .
Add and .
To find the arc length of a function, use the formula .
Evaluate the integral.
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Since is constant with respect to , move out of the integral.
Substitute and simplify.
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Evaluate at and at .
Move to the left of .
Multiply by .
Multiply by .
Add and .
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