Question
Function
Find the x-intercept/zero
Find the y-intercept
x=0
Evaluate
2x2×x=y
To find the x-intercept,set y=0
2x2×x=0
Multiply the terms
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Evaluate
x2×x
Use the product rule an×am=an+m to simplify the expression
x2+1
Add the numbers
x3
2x3=0
Simplify
x3=0
Solution
x=0
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Solve the equation
Solve for x
Solve for y
x=32y
Evaluate
2x2×x=y
Multiply the terms
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Evaluate
x2×x
Use the product rule an×am=an+m to simplify the expression
x2+1
Add the numbers
x3
2x3=y
Cross multiply
x3=2y
Take the 3-th root on both sides of the equation
3x3=32y
Solution
x=32y
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Testing for symmetry
Testing for symmetry about the origin
Testing for symmetry about the x-axis
Testing for symmetry about the y-axis
Symmetry with respect to the origin
Evaluate
2x2x=y
Simplify the expression
2x3=y
To test if the graph of 2x3=y is symmetry with respect to the origin,substitute -x for x and -y for y
2(−x)3=−y
Evaluate
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Evaluate
2(−x)3
Determine the sign
2−x3
Use b−a=−ba=−ba to rewrite the fraction
−2x3
−2x3=−y
Solution
Symmetry with respect to the origin
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Rewrite the equation
r=0r=2sin(θ)sec(θ)×∣sec(θ)∣r=−2sin(θ)sec(θ)×∣sec(θ)∣
Evaluate
2x2×x=y
Evaluate
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Evaluate
2x2×x
Multiply the terms
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Evaluate
x2×x
Use the product rule an×am=an+m to simplify the expression
x2+1
Add the numbers
x3
2x3
2x3=y
Multiply both sides of the equation by LCD
2x3×2=y×2
Simplify the equation
x3=y×2
Use the commutative property to reorder the terms
x3=2y
Move the expression to the left side
x3−2y=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
(cos(θ)×r)3−2sin(θ)×r=0
Factor the expression
cos3(θ)×r3−2sin(θ)×r=0
Factor the expression
r(cos3(θ)×r2−2sin(θ))=0
When the product of factors equals 0,at least one factor is 0
r=0cos3(θ)×r2−2sin(θ)=0
Solution
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Factor the expression
cos3(θ)×r2−2sin(θ)=0
Subtract the terms
cos3(θ)×r2−2sin(θ)−(−2sin(θ))=0−(−2sin(θ))
Evaluate
cos3(θ)×r2=2sin(θ)
Divide the terms
r2=cos3(θ)2sin(θ)
Simplify the expression
r2=2sin(θ)sec3(θ)
Evaluate the power
r=±2sin(θ)sec3(θ)
Simplify the expression
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Evaluate
2sin(θ)sec3(θ)
Rewrite the exponent as a sum
2sin(θ)sec2+1(θ)
Use am+n=am×an to expand the expression
2sin(θ)sec2(θ)sec(θ)
Rewrite the expression
sec2(θ)×2sin(θ)sec(θ)
Calculate
∣sec(θ)∣×2sin(θ)sec(θ)
Calculate
2sin(θ)sec(θ)×∣sec(θ)∣
r=±(2sin(θ)sec(θ)×∣sec(θ)∣)
Separate into possible cases
r=2sin(θ)sec(θ)×∣sec(θ)∣r=−2sin(θ)sec(θ)×∣sec(θ)∣
r=0r=2sin(θ)sec(θ)×∣sec(θ)∣r=−2sin(θ)sec(θ)×∣sec(θ)∣
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Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=23x2
Calculate
2x2x=y
Simplify the expression
2x3=y
Take the derivative of both sides
dxd(2x3)=dxd(y)
Calculate the derivative
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Evaluate
dxd(2x3)
Rewrite the expression
2dxd(x3)
Use dxdxn=nxn−1 to find derivative
23x2
23x2=dxd(y)
Calculate the derivative
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Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
23x2=dxdy
Solution
dxdy=23x2
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Find the second derivative
Find the second derivative with respect to x
Find the second derivative with respect to y
dx2d2y=3x
Calculate
2x2x=y
Simplify the expression
2x3=y
Take the derivative of both sides
dxd(2x3)=dxd(y)
Calculate the derivative
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Evaluate
dxd(2x3)
Rewrite the expression
2dxd(x3)
Use dxdxn=nxn−1 to find derivative
23x2
23x2=dxd(y)
Calculate the derivative
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Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
23x2=dxdy
Swap the sides of the equation
dxdy=23x2
Take the derivative of both sides
dxd(dxdy)=dxd(23x2)
Calculate the derivative
dx2d2y=dxd(23x2)
Rewrite the expression
dx2d2y=2dxd(3x2)
Evaluate the derivative
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Evaluate
dxd(3x2)
Simplify
3×dxd(x2)
Rewrite the expression
3×2x
Multiply the numbers
6x
dx2d2y=26x
Solution
dx2d2y=3x
Show Solution
