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.
The slope of tangent line is first derivation of function.
slope=dy/dx
If you don't know how to find Derivation
in Google type: "calc101"
When you see list of results click on:
calc101com
When page be open click option:
derivatives
In rectacangle type:
x*cos[x]
(exactly in this form) .Then click option DO IT
You will see that first derivation of your function is:
cos(x)-x*sin(x)
Now you must find zero of this expression.
cos(x)-x*sin(x)=0
cos(x)=x*sin(x) Divide with cos(x)
1=x*sin(x)/cos(x)
sin(x)/cos(x)=tan(x)
1=x*tan(x) Diwide with x
1/x=tan(x)
tan(x)=1/x is solution of your problem.
Well, let's find the derivative of y = xcos(x) first. We can use the product rule to differentiate this function.
Using the product rule, we have:
dy/dx = x(d(cos(x))/dx) + cos(x)(d(x)/dx)
The derivative of cos(x) is -sin(x), and the derivative of x is 1, so we have:
dy/dx = x(-sin(x)) + cos(x)(1)
Simplifying this, we get:
dy/dx = -xsin(x) + cos(x)
Now, we want to find when the slope of the tangent to the graph of y = xcos(x) is zero. In other words, we want to find when the derivative dy/dx is equal to zero.
Setting dy/dx equal to zero, we have:
-xsin(x) + cos(x) = 0
Adding xsin(x) to both sides, we get:
cos(x) = xsin(x)
Now, let's check each answer choice:
(a) tan(x) = -x
(b) tan(x) = 1/x
(c) sin(x) = x
(d) cos(x) = x
None of these answer choices match our equation cos(x) = xsin(x). So, none of the given options (a), (b), (c), or (d) satisfy the condition for the slope of the tangent to y = xcos(x) to equal zero.
But don't worry! If you come across this problem on your final, just remember that the correct answer isn't listed here. Sometimes, questions can be designed to test your understanding of concepts rather than relying on pre-determined multiple-choice options. So, don't get discouraged and keep up your diligent work! Good luck on your final!
To find the slope of the tangent to the curve y=xcos(x), we need to find the derivative of the function first.
First, let's find the derivative of y=xcos(x) using the product rule:
dy/dx = x * (-sin(x)) + cos(x) = -xsin(x) + cos(x)
Now, to find the values of x where the slope of the tangent is zero, we need to find where the derivative is equal to zero:
0 = -xsin(x) + cos(x)
Next, let's solve this equation.
Adding xsin(x) to both sides of the equation:
xsin(x) = cos(x)
Now, let's divide both sides of the equation by cos(x):
(xsin(x))/cos(x) = 1
Since tan(x) = sin(x)/cos(x), we can rewrite the equation as:
tan(x) = 1
Option (b) tan(x) = 1/x matches this equation.
Therefore, the correct answer is (b) tan(x) = 1/x.
Good luck with your final exam!
To find when the slope of the tangent to \(y = x \cos(x)\) is zero, we need to find the x-values where the derivative of the function is equal to zero.
Let's start by finding the derivative of \(y = x \cos(x)\) using the product rule:
\(f(x) = x\)
\(g(x) = \cos(x)\)
Using the product rule, we have:
\(f'(x) = 1\)
\(g'(x) = -\sin(x)\)
Now, applying the product rule, we have:
\(y'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)\)
\(y'(x) = 1 \cdot \cos(x) + x \cdot (-\sin(x))\)
\(y'(x) = \cos(x) - x\sin(x)\)
To find the x-values where the slope is zero, we need to solve the equation \(y'(x) = 0\).
\(\cos(x) - x\sin(x) = 0\)
At this point, we can't directly solve for x analytically. However, we can use numerical methods or graphing software to approximate the solution. Let's go through the answer choices to see which one is most likely to match up with the solution.
(a) tan(x) = -x - This equation doesn't match the one we obtained from the derivative, so it's unlikely to be the correct answer.
(b) tan(x) = 1/x - This equation also doesn't match the one we obtained from the derivative, so it's unlikely to be the correct answer.
(c) sin(x) = x - This equation matches the one we obtained from the derivative (\(\cos(x) - x\sin(x)\)), so it might be a potential solution.
(d) cos(x) = x - This equation doesn't match the one we obtained from the derivative, so it's unlikely to be the correct answer.
From the answer choices, (c) sin(x) = x seems to be the most likely solution.
Keep in mind that this is NOT a rigorous proof. It is just a step-by-step explanation of how to approach the problem and evaluate the answer choices based on the information given. To obtain a more precise solution, you may need to use numerical methods or involve more advanced mathematical techniques.