# What is the equation of the line that is normal to #f(x)= sqrt( x^3-3x+2) # at # x=5 #?

The straight line equation is: y=mx+x. The gradient m is gained from differentiating f(x), then m when x=5 is f'(x=5). The value of y is gained from f(x=5). Now you have values for y,x and m, so can find c. Take the negative reciprocal of the tangent gradient to get the normal gradient, and the normal line equation is:

Now we can find the y-intercept, c.

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To find the equation of the line that is normal to the function f(x) = sqrt(x^3 - 3x + 2) at x = 5, we need to determine the slope of the tangent line at x = 5 and then find the negative reciprocal of that slope to obtain the slope of the normal line.

To find the slope of the tangent line, we can take the derivative of the function f(x) with respect to x and evaluate it at x = 5.

The derivative of f(x) = sqrt(x^3 - 3x + 2) is given by:

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

Evaluating f'(x) at x = 5, we get:

f'(5) = (3 * 5^2 - 3) / (2 * sqrt(5^3 - 3 * 5 + 2))

Simplifying further:

f'(5) = (3 * 25 - 3) / (2 * sqrt(125 - 15 + 2)) = (72) / (2 * sqrt(112)) = 36 / sqrt(112)

The slope of the tangent line at x = 5 is 36 / sqrt(112).

To find the slope of the normal line, we take the negative reciprocal of the slope of the tangent line:

Slope of normal line = -1 / (36 / sqrt(112)) = -sqrt(112) / 36

Now that we have the slope of the normal line, we can use the point-slope form of a line to find the equation. We know that the line passes through the point (5, f(5)).

Substituting the values into the point-slope form:

y - f(5) = (-sqrt(112) / 36) * (x - 5)

Simplifying further:

y - f(5) = (-sqrt(112) / 36) * x + (5 * sqrt(112) / 36)

This is the equation of the line that is normal to f(x) = sqrt(x^3 - 3x + 2) at x = 5.

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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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