Find the equation of the tangent line to
f(x)=sec x - 2 cos x at (pi/3, 1).
f' = sec x tan x + 2 sin x
at x = pi/3 which is 60 deg
f' = m = (2 * sqrt 3) + 2 (sqrt 3/2)
so m = 3 sqrt 3
so y = 3 sqrt 3 * x + b
at (pi/3 , 1)
1 = 3 (pi/3) sqrt 3 + b
b = 1 - pi sqrt 3
so
y = (3 sqrt 3) x + (1-pi sqrt 3)
or approx
y = 5.20 x - 4.44
To find the equation of the tangent line to the function f(x) = sec(x) - 2cos(x) at the point (π/3, 1), we will use the concept of the derivative.
The derivative of a function represents the rate of change of the function at any given point. By finding the derivative of f(x), we can determine the slope of the tangent line at that point.
Step 1: Find the derivative of f(x)
To find the derivative of f(x) = sec(x) - 2cos(x), we will differentiate each term separately using the rules of differentiation.
The derivative of sec(x) is given by:
d/dx [sec(x)] = sec(x)tan(x)
The derivative of cos(x) is given by:
d/dx [cos(x)] = -sin(x)
Therefore, the derivative of f(x) = sec(x) - 2cos(x) is:
f'(x) = sec(x)tan(x) - 2sin(x)
Step 2: Calculate the value of the derivative at x = π/3
To find the slope of the tangent line at the point (π/3, 1), we need to evaluate the derivative f'(x) at x = π/3.
substituting x = π/3 into the derivative:
f'(π/3) = sec(π/3)tan(π/3) - 2sin(π/3)
Using trigonometric identities:
1/tan(π/3) = sqrt(3) and sin(π/3) = sqrt(3)/2
f'(π/3) = 2 * sqrt(3)/2 - 2 * sqrt(3)/2 = 0
So, the slope of the tangent line at x = π/3 is 0.
Step 3: Write the equation of the tangent line
The equation of a line can be written in the form y = mx + b, where m represents the slope of the line and b represents the y-intercept.
Since the slope of the tangent line at (π/3, 1) is 0, the equation of the tangent line will be in the form y = b, where b is the y-coordinate of the point (π/3, 1).
Therefore, the equation of the tangent line to f(x) = sec(x) - 2cos(x) at (π/3, 1) is y = 1.