If the equation of the tangent line to the curve y=9cosx
at the point on the curve with x-coordinate 3pi/4 is written in the form y=mx+b then m=? and b=?
m is the slope of the tangent,
m= d/dx (9cosx)=-9sinx
you are given x, so you can compute m.
then, knowing x,y, put that into y=mx+b to solve for b.
x would be 3pi/4 correct?
What would 3pi/4 equal approximately? I tried to do -9 * sin * 3pi/4 but I'm not getting the correct answer.
3pi / 4 is a memory value (use the 45 - 45 - 90 right triangle). If you're not getting the correct answer inputting it into a calculator, you may be in degree mode.
To find the equation of the tangent line to the curve y = 9cos(x) at the point on the curve with x-coordinate 3π/4, we first need to find the derivative of the function.
The derivative dy/dx represents the slope of the tangent line at any point on the curve.
Now let's find the derivative of y = 9cos(x):
dy/dx = d(9cos(x))/dx
To differentiate cos(x), we use the chain rule:
d(cos(x))/dx = -sin(x)
Now we multiply by the constant factor of 9:
dy/dx = 9(-sin(x))
The slope of the tangent line at any point on the curve is given by dy/dx. To find the slope at x = 3π/4, we substitute this value into the derivative:
dy/dx = 9(-sin(3π/4))
To simplify, we know that sin(3π/4) = √2/2:
dy/dx = 9(-√2/2)
Now we have the slope, which is equal to m, in the equation y = mx + b.
So, m = -9√2/2.
To find the value of b, we need the y-coordinate of the point on the curve with x = 3π/4. We can plug this value into the original function y = 9cos(x):
y = 9cos(3π/4)
Since cos(3π/4) = -1/√2, we have:
y = 9(-1/√2) = -9/√2
Now we have both the slope (m) and the y-coordinate (y) at the given x-coordinate.
To find b, we can rearrange the equation y = mx+b and solve for b:
b = y - mx = (-9/√2) - (-9√2/2)
To simplify:
b = (-9/√2) + (9√2/2) = -9/√2 + 9√2/2
To get the final answer, rationalize the denominator:
b = -9/√2 + 9√2/2 * √2/√2 = (-9√2 + 9√4) / 2√2
b = (9√4 - 9√2) / (2√2) = (9*2 - 9√2) / (2√2) = (18 - 9√2) / (2√2)
Therefore, m = -9√2/2 and b = (18 - 9√2) / (2√2).