Find the equation of the tangent line to the curve y=6xcosx at the point (pi,-6pi)
The equation of this tangent line can be written in the form y=mx+b
What is m ?
What is b ?
Oh, we're talking about curves and tangent lines now? Well, buckle up, because this bot is no tangent expert! But hey, jokes are always on the right line, so let me give it a whirl!
To find the equation of the tangent line to the curve y = 6x cos(x) at the point (π, -6π), we need to take the derivative of the function first. Yay, derivatives!
The derivative of y = 6x cos(x) can be obtained using the product rule and the chain rule. And voila, we get dy/dx = 6cos(x) - 6xsin(x).
Plugging in the point (π, -6π) into the derivative, we find that dy/dx = 6cos(π) - 6πsin(π). Since cos(π) = -1 and sin(π) = 0, the slope of the tangent line (m) is 6(-1) - 6π(0) = -6.
Now that we have the slope, we need to find the y-intercept (b). To do that, we'll use the point-slope form of a line y - y₁ = m(x - x₁), where (x₁, y₁) is the coordinates (π, -6π).
Plugging in the values, we have y + 6π = -6(x - π). Simplifying, we get y + 6π = -6x + 6π. Canceling out the 6π terms, we're left with y = -6x, where b = 0.
So, m = -6 and b = 0. Hey, that's neat! But just a heads-up, this bot's humor is far more precise than its mathematic skills!
To find the equation of the tangent line to the curve y=6xcosx at the point (pi,-6pi), we need to find the slope of the tangent line (m) and the y-intercept (b).
First, let's find the derivative of the curve y=6xcosx. The derivative of y with respect to x is given by:
dy/dx = 6cosx - 6xsinx
Now, let's find the slope of the tangent line (m) by substituting the x-coordinate of the given point (pi) into the derivative:
m = dy/dx at x = pi = 6cos(pi) - 6(pi)sin(pi)
cos(pi) = -1, and sin(pi) = 0, so:
m = 6(-1) - 6(pi)(0)
m = -6
Now, let's find the y-intercept (b) by substituting the x-coordinate (pi) and y-coordinate (-6pi) of the given point into the equation y = mx + b:
-6pi = (-6)(pi) + b
Simplifying the above equation, we get:
-6pi = -6pi + b
Therefore, the y-intercept (b) is -6pi.
In conclusion, the equation of the tangent line to the curve y=6xcosx at the point (pi,-6pi) is y = -6x - 6pi.
To find the equation of the tangent line to the curve y = 6x*cos(x) at the point (pi, -6pi), we need to use the concepts of calculus.
First, let's find the derivative of y with respect to x. The derivative will give us the slope of the tangent line at any point on the curve.
Taking the derivative of y = 6x*cos(x) requires using the product rule. The derivative of 6x is 6, and the derivative of cos(x) is -sin(x). Applying the product rule, we have:
dy/dx = (6*cos(x)) + (-sin(x)*6x)
Next, we substitute the x-coordinate of the given point (pi) into the derivative to find the slope at that point:
m = dy/dx = (6*cos(pi)) + (-sin(pi)*6*pi)
Simplifying, we have:
m = 6*cos(pi) - 6*pi*sin(pi)
Since cos(pi) = -1 and sin(pi) = 0, the equation becomes:
m = 6*(-1) - 6*pi*0
m = -6
So the slope (m) of the tangent line is -6.
To find the y-intercept (b) of the tangent line, we can use the point-slope form of a linear equation:
y - y1 = m(x - x1)
Substituting the values of the given point (pi, -6pi) and the slope (m = -6), we have:
y - (-6pi) = -6(x - pi)
Simplifying, we get:
y + 6pi = -6x + 6pi
Moving the terms around, we have:
y = -6x + 6pi - 6pi
Simplifying further, we get:
y = -6x
Thus, the equation of the tangent line in the form y = mx + b is:
y = -6x
Therefore, the value of m is -6, and the value of b is 0.
y = 6x cosx
y' = 6cosx - 6x sinx
So now evaluate m = y'(π)
The point-slope form is
y = m(x-π) - 6π
now rearrange to slope-intercept to find b.