Question

How do you find the tangent line approximation to `f(x)=cos(x)` at `x=pi/4` ?

Answer 1

In this problem we need to follow 5 steps.

1) Find the value of y given the x-value of `pi/4`. This gives you the point of tangency.

2) Find the derivative of `f(x)`

3) Substitute in the x-value of `pi/4` to find the numeric slope.

4) Substitute the slope and the `x` and `y` values to find the y-intercept.

5) Use the y-intercept and slope to find the equation of the of the tangent line.

`f(x)=cos(x)`

`f(pi/4)=cos(pi/4)=sqrt(2)/2`

Point of tangency: `(pi/4,sqrt(2)/2)`

`f'(x)=-sin(x)`

`f'(pi/4)=-sin(pi/4)=-sqrt(2)/2 => slope or m`

`y=mx+b`

`sqrt(2)/2=(-sqrt(2)/2)(pi/4)+b`

`sqrt(2)/2=((-sqrt(2)pi)/8)+b`

`((sqrt(2)pi)/8)+sqrt(2)/2=b`

`1.2625=b`

`y=mx+b`

`y=(-sqrt(2))/2x+1.2625 =>`Equation of the tangent line