What is the equation of the line normal to #f(x)= sqrt(2x^3-x) # at #x=1#?

Answer 1

#y=5/2x-5/2#

Let #y=sqrt(2x^3-x)#
#y=(2x^3-x)^(1/2)#
Applying the chain rule #rArr#
#y'=1/2(2x^3-x)^(-1/2)xx(6x^2-1)#
#y'=((6x^2-1))/(2sqrt(2x^3-x))#
This equals the gradient #m#.
So when #x=1rArr#
#m=((6-1))/(2sqrt(2-1))=5/2#

The tangent line is of the form:

#y=mx+c#
At #x=1#,
#y=sqrt(2-1)=1#
#:.1=5/2xx1 +c#
#:.c=-5/2#

So the equation of the tangent is:

#y=5/2x-5/2#
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Answer 2

To find the equation of the line normal to f(x) at x=1, we need to find the derivative of f(x) and evaluate it at x=1. Then, we can determine the slope of the normal line.

The derivative of f(x) can be found using the power rule and chain rule. Taking the derivative of f(x) = sqrt(2x^3 - x), we get:

f'(x) = (3x^2 - 1) / (2 * sqrt(2x^3 - x))

Evaluating f'(x) at x=1, we have:

f'(1) = (3(1)^2 - 1) / (2 * sqrt(2(1)^3 - 1)) = (3 - 1) / (2 * sqrt(2 - 1)) = 2 / (2 * 1) = 1

The slope of the normal line is the negative reciprocal of the derivative at x=1. Therefore, the slope of the normal line is -1.

Now, we have the slope (-1) and a point on the line (x=1, f(1)). We can use the point-slope form of a line to find the equation of the line normal to f(x) at x=1.

Using the point-slope form, the equation of the line is:

y - f(1) = -1(x - 1)

Simplifying further, we get:

y - f(1) = -x + 1

This can be rearranged to obtain the equation of the line normal to f(x) at x=1:

y = -x + f(1) + 1

Therefore, the equation of the line normal to f(x) = sqrt(2x^3 - x) at x=1 is y = -x + f(1) + 1.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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