# What is the equation of the line normal to # f(x)=sin(2x+pi)# at # x=pi/3#?

Step 1: Determine the corresponding y coordinate that the function and the tangent pass through

This can be done by evaluating

Evaluate using the sum and difference formulae:

Note we could have also combined

So, the function, the tangent and the normal pass through the point

Step 2: Determine the derivative

We will use the chain rule to find the derivative of this function.

Let

The derivative of

The derivative of the entire function is then

Step 3: Determine the slope of the tangent

The slope of the tangent is given by evaluating

This time I will use the combination inside the parentheses method.

Hence, the slope of the tangent is

Step 4: Use point-slope form to determine the equation of the normal line

We know the point that the function, tangent and normal pass through, and the slope of normal line. We also know that the normal is a line or a linear function. Hence, we can use point-slope form to determine the normal line's equation.

If you insist on an approximation, the equation of the line is

Since this is my

Practice exercises:

- Determine the equations of the tangents to the following relations at the given points:
a)

#y = sqrt(x^3 + 9)# , at#x = -2# b)

#y = 2^(x - 3)# at#x = 5# c)

#y = x^5 + 3x^4 - 2x^3 - 8x^2 + 2x - 1# at#x = 1# d)

#y = sin(2x)# at#x = pi/4# #2# . Determine the equations of the normal lines to the following relations at the given points:a)

#y = root(4)(1/3x^3 + 2x + 1)# at the point#x = 3# b)

#y = log_3(2x + 11)# at the point#x = 8# c)

#y = tan(4x + pi/2)# at the point#pi/8# d)

#(x - 3)^2 + (y - 5)^2 = 25# at the point#(6, 1)# Hopefully this helps, and good luck!

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The equation of the line normal to f(x)=sin(2x+pi) at x=pi/3 is y = -2√3(x - π/3) + sin(2π/3 + π).

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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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