# How do you find the equation of the tangent and normal line to the curve #y=tanx# at #x=-pi/4#?

Tangent:

Normal:

The gradient tangent to a curve at any particular point is given by the derivative.

If

When

So the tangent passes through

Using

# y-(-1) = (2)(x-(-pi/4)) #

# :. y+1 = 2x+pi/2 #

# :. y = 2x+pi/2-1 #

The normal is perpendicular to the tangent, so the product of their gradients is -1 hence normal passes through

so the equation of the normal is:

# y-(-1) = -1/2(x-(-pi/4)) #

# :. y+1 = -1/2x-pi/2 #

# :. y = -1/2x-pi/8 -1 #

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To find the equation of the tangent line to the curve (y = \tan(x)) at (x = -\frac{\pi}{4}), we first find the derivative of (y = \tan(x)), which is (y' = \sec^2(x)). Then, we evaluate (y') at (x = -\frac{\pi}{4}) to find the slope of the tangent line.

(y'(-\frac{\pi}{4}) = \sec^2(-\frac{\pi}{4}) = \frac{1}{\cos^2(-\frac{\pi}{4})} = 1)

So, the slope of the tangent line is (m = 1) at (x = -\frac{\pi}{4}).

Now, we use the point-slope form of a line to find the equation of the tangent line:

(y - y_1 = m(x - x_1))

Plugging in (x_1 = -\frac{\pi}{4}), (y_1 = \tan(-\frac{\pi}{4}) = -1), and (m = 1), we get:

(y - (-1) = 1(x - (-\frac{\pi}{4})))

which simplifies to:

(y + 1 = x + \frac{\pi}{4})

To find the equation of the normal line, we note that the slope of the normal line is the negative reciprocal of the slope of the tangent line. Therefore, the slope of the normal line is (-1). Using the point-slope form again, we find the equation of the normal line:

(y - (-1) = -1(x - (-\frac{\pi}{4})))

which simplifies to:

(y + 1 = -x - \frac{\pi}{4})

So, the equations of the tangent and normal lines to the curve (y = \tan(x)) at (x = -\frac{\pi}{4}) are (y = x + \frac{\pi}{4} - 1) and (y = -x - \frac{\pi}{4} - 1), respectively.

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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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