Determine the equation of the tangent line to the function (² = square): f(x) = x + cos²x at x = pie/4
f'(x) = 1+ 2cosx(-sinx)
f(π/4) = π/4 + cos^2 (π/4) = π/4 + 1/2 = (2+π)/4
f'(π/4) = 1 + 2(1/√2)(-1/√2) = 1 - 2/2 = 0
so the line is horizontal and its equation is simply
y = (2+π)/4
check my arithmetic, (did not write it down first)
Thanks Reiny, I have checked it over and it seems about right to me. Thank you.
To find the equation of the tangent line to the function f(x) = x + cos²x at x = π/4, we can follow these steps:
Step 1: Find the derivative of the function f(x) with respect to x.
Step 2: Evaluate the derivative at x = π/4 to find the slope of the tangent line.
Step 3: Use the point-slope form of a linear equation to find the equation of the tangent line.
Let's go through each step in detail:
Step 1: Find the derivative of the function f(x) = x + cos²x with respect to x.
To find the derivative, we differentiate each term separately.
The derivative of x with respect to x is 1.
To find the derivative of cos²x, we can use the chain rule. First, let's rewrite cos²x as (cosx)²:
d/dx ((cosx)²) = 2(cosx) * (-sinx), using the chain rule.
= -2cosxsinx.
Therefore, the derivative of f(x) = x + cos²x with respect to x is:
f'(x) = 1 - 2cosxsinx.
Step 2: Evaluate the derivative at x = π/4 to find the slope of the tangent line.
We need to plug in x = π/4 into f'(x) to find the slope of the tangent line.
f'(π/4) = 1 - 2cos(π/4)sin(π/4)
= 1 - 2(√2/2)(√2/2) (cos(π/4) = sin(π/4) = √2/2)
= 1 - 2(1/2)(1/2)
= 1 - 1/2
= 1/2.
So, the slope of the tangent line at x = π/4 is 1/2.
Step 3: Use the point-slope form of a linear equation to find the equation of the tangent line.
Now that we have the slope of the tangent line, we can use the point-slope form:
y - y1 = m(x - x1),
where (x1, y1) is a point on the line and m is the slope.
Since we are looking for the tangent line at x = π/4, we have x1 = π/4.
To find y1, we need to plug x = π/4 into the original function f(x):
f(π/4) = π/4 + cos²(π/4)
= π/4 + (cos(π/4))²
= π/4 + (√2/2)² (cos(π/4) = √2/2)
= π/4 + 1/2²
= π/4 + 1/4
= (π + 1)/4.
So, y1 = (π + 1)/4.
Now we can write the equation of the tangent line using the point-slope form:
y - y1 = m(x - x1)
y - (π + 1)/4 = (1/2)(x - π/4)
To simplify the equation, we can multiply both sides by 4 to eliminate the fraction:
4y - (π + 1) = 2(x - π/4)
or,
4y - π - 1 = 2x - π/2
Finally, rearranging the equation:
2x - 4y = -π/2 + π + 1.
So, the equation of the tangent line to the function f(x) = x + cos²x at x = π/4 is 2x - 4y = π/2 - 1.