Find an equation of the tangent line to the curve f(x) = (sins)^2 + 2tanx at x= pi/4. I worked through it and got y = 5x + 5/2 for my answer, but the answer key says that it is y - 5/2 = 5(x - pi/4). Why is the pi/4 in the answer? Thanks for any help!
you say y = 5 x + 2.5
then
if x = pi/4, y = 5 pi/4 + 2.5 You say
but what is f(x) at x = pi/4 ???
sin^2 = 1/2
2 tan = 2
so f(x) = 2.5
You better take 5 pi/4 off your answer
In other words you got the slope right but did not get b right in
y = m x + b
Ok thank you so much!
You are welcome.
To find the equation of the tangent line to a curve at a given point, we need to find the slope of the curve at that point and the coordinates of the point itself.
In this case, the given curve is f(x) = sin^2(x) + 2tan(x), and we want to find the equation of the tangent line at x = pi/4.
To find the slope of the curve at x = pi/4, we take the derivative of the function f(x) with respect to x. Let's call this derivative f'(x).
First, let's find the derivative of sin^2(x):
d/dx (sin^2(x)) = 2sin(x)cos(x)
Next, let's find the derivative of 2tan(x):
d/dx (2tan(x)) = 2sec^2(x)
Now, let's find the derivative of the function f(x):
f'(x) = 2sin(x)cos(x) + 2sec^2(x)
Now that we have the derivative, we can find the slope of the curve at x = pi/4 by evaluating f'(pi/4):
f'(pi/4) = 2sin(pi/4)cos(pi/4) + 2sec^2(pi/4)
Since sin(pi/4) = cos(pi/4) = sqrt(2)/2 and sec(pi/4) = sqrt(2), we can substitute these values:
f'(pi/4) = 2(sqrt(2)/2)(sqrt(2)/2) + 2(sqrt(2))^2
= 2(1/2) + 2*2
= 1 + 4
= 5
So, the slope of the curve at x = pi/4 is 5.
Now, let's find the coordinates of the point on the curve at x = pi/4. We can substitute x = pi/4 into the original function f(x):
f(pi/4) = sin^2(pi/4) + 2tan(pi/4)
Since sin(pi/4) = 1/sqrt(2) and tan(pi/4) = 1, we have:
f(pi/4) = (1/sqrt(2))^2 + 2*1
= 1/2 + 2
= 5/2
So, the coordinates of the point on the curve at x = pi/4 are (pi/4, 5/2).
Now that we have the slope (5) and the coordinates of the point (pi/4, 5/2), we can use the point-slope form of a line to find the equation of the tangent line:
y - y1 = m(x - x1)
where m is the slope and (x1, y1) are the coordinates of the point.
Substituting the values, we have:
y - 5/2 = 5(x - pi/4)
So, the answer y - 5/2 = 5(x - pi/4) is correct. The "pi/4" is included in the answer because it represents the x-coordinate of the given point on the curve. It helps specify the exact point on the tangent line.