How do you find the equation of the tangent line to the curve #f(x)=(secx)(tanx)# at x=pi/6?

Answer 1

Equation of tangent is #10x-3sqrt3y-(5pi)/3+2sqrt3=0#

To find tangent line, we need to find the point at which tangent is drawn and its slope and then use point slope form of equation.

Former is easily available from #f(x)# as #f(pi/6)=sec(pi/6)tan(pi/6)=2/sqrt3xx1/sqrt3=2/3# Hence tangent is desired at #(pi/6,2/3)#.
For slope we need #(df)/(dx)=secx xxsec^2x+tanx xxsecxtanx#
= #sec^3x+secxtan^2x# and at #x=pi/6#, it is #(2/sqrt3)^3+2/sqrt3xx(1/sqrt3)^2=8/(3sqrt3)+2/(3sqrt3)=10/(3sqrt3)#
As equation of a line of slope #m# passing through #(x_1,y_1)# is
#(y-y_1)=m(x-x_1)# and hence equation of tangent is
#(y-2/3)=10/(3sqrt3)(x-pi/6)# or
#3sqrt3y-2sqrt3=10(x-pi/6)# or
#10x-3sqrt3y-(5pi)/3+2sqrt3=0#
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Answer 2

To find the equation of the tangent line to the curve f(x) = (secx)(tanx) at x = pi/6, we need to find the derivative of the function and evaluate it at x = pi/6.

First, let's find the derivative of f(x). Using the product rule, we have:

f'(x) = (secx)(secx)(tanx) + (secx)(tanx)(tanx)

Next, we substitute x = pi/6 into f'(x) to find the slope of the tangent line at x = pi/6:

f'(pi/6) = (sec(pi/6))(sec(pi/6))(tan(pi/6)) + (sec(pi/6))(tan(pi/6))(tan(pi/6))

Now, we simplify the expression:

f'(pi/6) = (2)(2)(√3/3) + (2)(√3/3)(√3/3)

Finally, we simplify further to get the slope of the tangent line:

f'(pi/6) = 4√3/3 + 2/3

Therefore, the slope of the tangent line to the curve f(x) = (secx)(tanx) at x = pi/6 is 4√3/3 + 2/3.

To find the equation of the tangent line, we use the point-slope form of a line. We know that the point (pi/6, f(pi/6)) lies on the tangent line. Substituting x = pi/6 and f(x) = (sec(pi/6))(tan(pi/6)) into the point-slope form, we get:

y - f(pi/6) = (4√3/3 + 2/3)(x - pi/6)

Simplifying further, we have:

y - f(pi/6) = (4√3/3 + 2/3)x - (4√3/3 + 2/3)(pi/6)

Therefore, the equation of the tangent line to the curve f(x) = (secx)(tanx) at x = pi/6 is:

y = (4√3/3 + 2/3)x - (4√3/3 + 2/3)(pi/6)

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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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