Calculus Examples

Find the Root Using Newton's Method x^3-7=0 , a=2
x3-7=0x37=0 , a=2a=2
Step 1
Find the derivative of f(x)=x3-7f(x)=x37 for use in Newton's method.
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Step 1.1
By the Sum Rule, the derivative of x3-7x37 with respect to xx is ddx[x3]+ddx[-7]ddx[x3]+ddx[7].
ddx[x3]+ddx[-7]ddx[x3]+ddx[7]
Step 1.2
Differentiate using the Power Rule which states that ddx[xn]ddx[xn] is nxn-1nxn1 where n=3n=3.
3x2+ddx[-7]3x2+ddx[7]
Step 1.3
Since -77 is constant with respect to xx, the derivative of -77 with respect to xx is 00.
3x2+03x2+0
Step 1.4
Add 3x23x2 and 00.
3x23x2
3x23x2
Step 2
Set up the formula to find the x2x2 approximation.
x2=x1-f(x1)f(x1)
Step 3
Substitute the value of x1 into the next Newton's Method approximation.
x2=2-(2)3-73(2)2
Step 4
Simplify the right side of the equation to find x2.
x2=1.916
Step 5
Set up the formula to find the x3 approximation.
x3=x2-f(x2)f(x2)
Step 6
Substitute the value of x2 into the next Newton's Method approximation.
x3=1.916-(1.916)3-73(1.916)2
Step 7
Simplify the right side of the equation to find x3.
x3=1.91293845
Step 8
Set up the formula to find the x4 approximation.
x4=x3-f(x3)f(x3)
Step 9
Substitute the value of x3 into the next Newton's Method approximation.
x4=1.91293845-(1.91293845)3-73(1.91293845)2
Step 10
Simplify the right side of the equation to find x4.
x4=1.91293118
Step 11
Set up the formula to find the x5 approximation.
x5=x4-f(x4)f(x4)
Step 12
Substitute the value of x4 into the next Newton's Method approximation.
x5=1.91293118-(1.91293118)3-73(1.91293118)2
Step 13
Simplify the right side of the equation to find x5.
x5=1.91293118
Step 14
Set up the formula to find the x6 approximation.
x6=x5-f(x5)f(x5)
Step 15
Substitute the value of x5 into the next Newton's Method approximation.
x6=1.91293118-(1.91293118)3-73(1.91293118)2
Step 16
Simplify the right side of the equation to find x6.
x6=1.91293118
Step 17
Since the 6th and 5th approximations are equal to 6 decimal places, 1.91293118 is the approximation of the root.
1.91293118
 [x2  12  π  xdx ]