Question
Function
Find the x-intercept/zero
Find the y-intercept
x=0
Evaluate
y=21x3
To find the x-intercept,set y=0
0=21x3
Swap the sides of the equation
21x3=0
Rewrite the expression
x3=0
Solution
x=0
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Solve the equation
x=32y
Evaluate
y=21x3
Swap the sides of the equation
21x3=y
Multiply by the reciprocal
21x3×2=y×2
Multiply
x3=y×2
Use the commutative property to reorder the terms
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
(y=21x3)
To test if the graph of y=21x3 is symmetry with respect to the origin,substitute -x for x and -y for y
−y=21(−x)3
Evaluate
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Evaluate
21(−x)3
Rewrite the expression
21(−x3)
Multiplying or dividing an odd number of negative terms equals a negative
−21x3
−y=−21x3
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
y=21x3
Multiply both sides of the equation by LCD
y×2=21x3×2
Use the commutative property to reorder the terms
2y=21x3×2
Simplify the equation
2y=x3
Move the expression to the left side
2y−x3=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
2sin(θ)×r−(cos(θ)×r)3=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=0−cos3(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)∣
Show Solution

Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=23x2
Calculate
(y=21x3)
Take the derivative of both sides
dxd(y)=dxd(21x3)
Calculate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
dxdy=dxd(21x3)
Solution
More Steps

Evaluate
dxd(21x3)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
21×dxd(x3)
Use dxdxn=nxn−1 to find derivative
21×3x2
Multiply the terms
23x2
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
(y=21x3)
Take the derivative of both sides
dxd(y)=dxd(21x3)
Calculate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
dxdy=dxd(21x3)
Calculate the derivative
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Evaluate
dxd(21x3)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
21×dxd(x3)
Use dxdxn=nxn−1 to find derivative
21×3x2
Multiply the terms
23x2
dxdy=23x2
Take the derivative of both sides
dxd(dxdy)=dxd(23x2)
Calculate the derivative
dx2d2y=dxd(23x2)
Simplify
dx2d2y=23×dxd(x2)
Rewrite the expression
dx2d2y=23×2x
Solution
More Steps

Evaluate
23×2
Reduce the numbers
3×1
Simplify
3
dx2d2y=3x
Show Solution
