For how many values of x will the tangent lines to y=4sinx and y=x^2/2 be parallel?
A. 0
B. 1
C. 3
D. 4
E. Infinite
To determine the number of values of x for which the tangent lines to the curves y=4sinx and y=x^2/2 are parallel, we need to find the condition under which the slopes of the two tangent lines are equal.
First, let's find the slopes of the tangent lines to the curve y=4sinx. The derivative of y=4sinx with respect to x is dy/dx = 4cosx. Therefore, the slope of the tangent line to y=4sinx at any point is 4cosx.
Next, let's find the slope of the tangent line to the curve y=x^2/2. The derivative of y=x^2/2 with respect to x is dy/dx = x. Therefore, the slope of the tangent line to y=x^2/2 at any point is x.
To find the values of x for which the two slopes are equal, we need to equate 4cosx and x and solve for x.
4cosx = x
To solve this equation, we need to use numerical methods or graphing technology. Let's use a graphing calculator or software to find the solution.
Plot the two curves y=4sinx and y=x^2/2 on the same graph. Then, find the intersection point(s) of the two graphs.
Once you find the intersection point(s), that would be the value(s) of x for which the tangent lines to the two curves are parallel.
Count the number of intersection points you find. This will give you the number of values of x for which the tangent lines are parallel.
Now that we have walked through the steps to find the solution, you can proceed to use a graphing calculator or software to determine the number of intersection points and select the corresponding answer choice from the options provided.