Find the equation of the tangent line to the curve y=5xcosx at the point (pi,–5pi).
The equation of this tangent line can be written in the form y=mx+b where
m=
and b=
y=5x cos x
y'=slope m= 5cosx-5xsinx at x=pi
y'=m=-5
line equation: y= -5x+b
now to find b, you know the point(PI,-5PI) is on it, so
-5pi--5PI+b so b=0
To find the equation of the tangent line, we need to find the slope (m) of the tangent line and the y-intercept (b).
To find the slope (m), we can use the derivative of the function y = 5x*cos(x). The derivative of this function can be found using the product rule.
First, let's find the derivative of cos(x) which is -sin(x).
Next, let's find the derivative of 5x which is 5.
Applying the product rule, we have:
dy/dx = 5*cos(x) - 5x*sin(x)
Now, let's substitute x = pi into the equation:
dy/dx = 5*cos(pi) - 5*pi*sin(pi)
Since cos(pi) = -1 and sin(pi) = 0, the equation becomes:
dy/dx = 5*(-1) - 5*pi*0
dy/dx = -5
Therefore, the slope (m) of the tangent line is -5.
Now, let's find the y-intercept (b) by substituting the point (pi, -5pi) into the equation y = mx + b.
Using the point (pi, -5pi), we have:
-5pi = -5*pi + b
Simplifying the equation, we have:
-5pi = -5pi + b
Therefore, the y-intercept (b) is -5pi.
So, the equation of the tangent line is:
y = -5x - 5pi
To find the equation of the tangent line to the curve y = 5x*cos(x) at the point (pi, -5pi), we can follow these steps:
Step 1: Find the derivative of the function y = 5x*cos(x) using the product rule or chain rule for differentiation. Let's call this derivative dy/dx.
Step 2: Substitute the x-coordinate and evaluate dy/dx at x = pi.
Step 3: Use the point-slope form of the equation of a line, y - y₁ = m(x - x₁), where (x₁, y₁) is the given point.
Step 4: Substitute the slope m found in Step 2, and the given point (pi, -5pi), into the point-slope form to find the equation of the tangent line.
Now let's go through these steps in detail:
Step 1: Differentiate y = 5x*cos(x) with respect to x.
The derivative of y with respect to x (dy/dx) can be found using the product rule:
dy/dx = 5*cos(x) - 5x*sin(x).
Step 2: Evaluate dy/dx at x = pi.
Substitute x = pi into the derivative expression found in Step 1:
dy/dx = 5*cos(pi) - 5*pi*sin(pi).
cos(pi) = -1 and sin(pi) = 0.
dy/dx = 5*(-1) - 5*pi*0
dy/dx = -5
So, the slope of the tangent line (m) is -5.
Step 3: Use the point-slope form and substitute the given values.
The point-slope form equation is y - y₁ = m(x - x₁), where (x₁, y₁) is the point (pi, -5pi).
Substitute x₁ = pi, y₁ = -5pi, and m = -5 into the equation:
y - (-5pi) = -5(x - pi).
Simplify:
y + 5pi = -5x + 5pi.
Step 4: Simplify the equation.
The constant 5pi on both sides cancels out:
y = -5x + 5pi - 5pi.
Simplify further:
y = -5x.
Therefore, the equation of the tangent line to the curve y = 5x*cos(x) at the point (pi, -5pi) is y = -5x.
In summary, the equation of the tangent line is y = -5x, where m = -5 and b = 0.