What is the equation of the line tangent to # f(x)=secx - 2cosx # at # x=pi/3#?

Answer 1

Calculate the derivative (slope of the line), and find #f(pi/3)# so you have a point for point slope form to find the line equation.

#y = (3sqrt3)x + 1 - (sqrt3)pi#

To find the equation of the line in question, we need both a point, and a slope. The point will be found by simply finding #f(pi/3)#
#f(pi/3) = sec (pi/3) - 2 cos(pi/3) = 1/cos(pi/3)-2cos(pi/3)#
We know, either from use of a calculator or from learning the values of the trig functions for angles such as #npi, npi/2, npi/3, npi/4, npi/6#, that #cos(pi/3) = 1/2#. Thus...
#f(pi/3) = 1/(1/2) - 2(1/2) = 2-1 = 1#
Thus the point #(pi/3, 1)# is where our tangent line touches.
For the slope, we must calculate the derivative of the function, and then find the exact derivative at #pi/3#
#(df)/dx = secxtanx + 2sinx#
If #x=pi/3...#
#(df)/dx (x=pi/3) = sec (pi/3) tan (pi/3) + 2 sin (pi/3) = 1/(1/2) * ((sqrt3)/2)/(1/2) + 2(sqrt3)/2 = 2*sqrt3 + sqrt3 = 3sqrt3 #
Thus, our tangent line is of the form #y=(3sqrt3) x + c_1#
Since #y(pi/3)=1...#, use point slope form...
#y-1 = 3sqrt3 (x-pi/3) = 3sqrt3x - pisqrt3 -> y = 3sqrt3x + 1 - sqrt3pi#

The tangent line equation is therefore:

#y = (3sqrt3)x + 1 - (sqrt3)pi#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

The equation of the line tangent to f(x) = sec(x) - 2cos(x) at x = pi/3 is y = sqrt(3) - 2.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7