How do you find equations for the lines that are tangent and normal to the graph of #y=secx# at #x=pi/4#?

Answer 1
The corresponding y coordinate is #y = sec(pi/4) = 1/cos(pi/4) = sqrt(2)#

We now find the derivative.

#y = secx#
#y= 1/cosx#
#y' = (0 xx cosx - (-sinx xx 1))/(cosx)^2#
#y' = (sinx)/(cos^2x)#
#y' = tanxsecx#

We can now find the slope of the tangent.

#m_"tangent" = tan(pi/4)sec(pi/4)#
#m_"tangent" = 1(sqrt(2))#
#m_"tangent" =sqrt(2)#

The equation is therefore:

#y - y_1 = m(x - x_1)#
#y- sqrt(2) = sqrt(2)(x- pi/4)#
#y - sqrt(2) = sqrt(2)x - (sqrt(2)pi)/4#
#y= sqrt(2)x - (sqrt(2)pi)/4 + sqrt(2)#
#y= sqrt(2x) - (sqrt(2)pi + 4sqrt(2))/4#
#y = sqrt(2x) - (sqrt(2)(pi + 4))/4#

Hopefully this helps!

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Answer 2

To find the equations for the lines that are tangent and normal to the graph of y=secx at x=pi/4, we need to determine the slope of the tangent line and the normal line at that point.

First, we find the derivative of y=secx using the chain rule, which is dy/dx = secx * tanx.

Next, we substitute x=pi/4 into the derivative to find the slope of the tangent line at that point. The slope is dy/dx = sec(pi/4) * tan(pi/4).

To find the slope of the normal line, we take the negative reciprocal of the tangent line's slope. The slope of the normal line is -1 / (sec(pi/4) * tan(pi/4)).

Using the point-slope form of a line, we can write the equations for the tangent and normal lines. The equation for the tangent line is y - sec(pi/4) = (sec(pi/4) * tan(pi/4))(x - pi/4), and the equation for the normal line is y - sec(pi/4) = (-1 / (sec(pi/4) * tan(pi/4)))(x - pi/4).

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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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