For x≠0, the slope of the tangent to y=xcosx equals zero whenever:
(a) tanx=-x
(b) tanx=1/x
(c) sinx=x
(d) cosx=x
Please help. I have a final tomorrow and I am working diligently to understand every type of problem that may show up on my test. Thank you very much.
To determine when the slope of the tangent to the curve y = xcos(x) is equal to zero, we need to find the values of x where the derivative of the function is equal to zero.
The first step is to find the derivative of the function y = xcos(x). We can do this using the product rule, which states that the derivative of the product of two functions u(x) and v(x) is given by:
d(uv)/dx = u'v + uv'
In this case, u(x) = x and v(x) = cos(x), so the derivative of u(x) with respect to x is 1 and the derivative of v(x) with respect to x is -sin(x).
Therefore, the derivative of y = xcos(x) is:
dy/dx = (1)(cos(x)) + (x)(-sin(x))
= cos(x) - xsin(x)
To find the values of x where the derivative is equal to zero, we set dy/dx = 0:
cos(x) - xsin(x) = 0
To solve this equation, we can't apply direct algebraic methods. Instead, we can utilize numerical or graphical methods to find approximate solutions.
One way to find approximate solutions is to plot the curves of y = xcos(x) and y = 0 on a graphing tool or software. The points where the two curves intersect will be the approximate solutions to the equation cos(x) - xsin(x) = 0.
Another method is to use a numerical solver, such as Newton's method or the bisection method, which will iteratively find an approximation to the roots of the equation.
Now, let's analyze the given answer choices:
(a) tan(x) = -x: This equation is not directly related to the derivative of y = xcos(x), as it involves a different trigonometric function.
(b) tan(x) = 1/x: Similarly, this equation is not directly related to the derivative of y = xcos(x), as it also involves a different trigonometric function.
(c) sin(x) = x: This equation is not directly related to the derivative of y = xcos(x), as it is a different equation altogether.
(d) cos(x) = x: This equation resembles the equation we obtained for the derivative. If we set cos(x) - xsin(x) = 0, we can see that this answer choice is a possible candidate.
Therefore, the correct answer is (d) cos(x) = x.
However, please note that it is important to verify this solution by actually finding the values of x where the derivative is zero using numerical or graphical methods, as explained earlier, to obtain precise solutions.