Calculus Examples

Determine if Continuous f(x)=sec((pix)/4)
f(x)=sec(πx4)
Step 1
Find the domain to determine if the expression is continuous.
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Step 1.1
Set the argument in sec(πx4) equal to π2+πn to find where the expression is undefined.
πx4=π2+πn, for any integer n
Step 1.2
Solve for x.
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Step 1.2.1
Multiply both sides of the equation by 4π.
4ππx4=4π(π2+πn)
Step 1.2.2
Simplify both sides of the equation.
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Step 1.2.2.1
Simplify the left side.
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Step 1.2.2.1.1
Simplify 4ππx4.
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Step 1.2.2.1.1.1
Cancel the common factor of 4.
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Step 1.2.2.1.1.1.1
Cancel the common factor.
4ππx4=4π(π2+πn)
Step 1.2.2.1.1.1.2
Rewrite the expression.
1π(πx)=4π(π2+πn)
1π(πx)=4π(π2+πn)
Step 1.2.2.1.1.2
Cancel the common factor of π.
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Step 1.2.2.1.1.2.1
Factor π out of πx.
1π(π(x))=4π(π2+πn)
Step 1.2.2.1.1.2.2
Cancel the common factor.
1π(πx)=4π(π2+πn)
Step 1.2.2.1.1.2.3
Rewrite the expression.
x=4π(π2+πn)
x=4π(π2+πn)
x=4π(π2+πn)
x=4π(π2+πn)
Step 1.2.2.2
Simplify the right side.
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Step 1.2.2.2.1
Simplify 4π(π2+πn).
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Step 1.2.2.2.1.1
Apply the distributive property.
x=4ππ2+4π(πn)
Step 1.2.2.2.1.2
Cancel the common factor of 2.
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Step 1.2.2.2.1.2.1
Factor 2 out of 4.
x=2(2)ππ2+4π(πn)
Step 1.2.2.2.1.2.2
Cancel the common factor.
x=22ππ2+4π(πn)
Step 1.2.2.2.1.2.3
Rewrite the expression.
x=2ππ+4π(πn)
x=2ππ+4π(πn)
Step 1.2.2.2.1.3
Cancel the common factor of π.
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Step 1.2.2.2.1.3.1
Cancel the common factor.
x=2ππ+4π(πn)
Step 1.2.2.2.1.3.2
Rewrite the expression.
x=2+4π(πn)
x=2+4π(πn)
Step 1.2.2.2.1.4
Cancel the common factor of π.
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Step 1.2.2.2.1.4.1
Factor π out of πn.
x=2+4π(π(n))
Step 1.2.2.2.1.4.2
Cancel the common factor.
x=2+4π(πn)
Step 1.2.2.2.1.4.3
Rewrite the expression.
x=2+4n
x=2+4n
x=2+4n
x=2+4n
x=2+4n
Step 1.2.3
Reorder 2 and 4n.
x=4n+2
x=4n+2
Step 1.3
The domain is all values of x that make the expression defined.
Set-Builder Notation:
{x|x4n+2}, for any integer n
Set-Builder Notation:
{x|x4n+2}, for any integer n
Step 2
Since the domain is not all real numbers, sec(πx4) is not continuous over all real numbers.
Not continuous
Step 3
 [x2  12  π  xdx ]