Question

For what values of x does the graph of f have a horizontal tangent? (Use n as your integer variable. Enter your answers as a comma-separated list.)
f(x) = x − 2 sin(x)

Answer 1

"Horiz. tangent line" implies that the derivative / slope of the tangent line is 0.
Find the derivative and set it equal to 0. Solve for x.

Answer 2

Hi cam4km! To find the values of x for which the graph has a horizontal tangent, you need to calculate the derivative of the function f
\[f(x)=x-2\times \sin (x) \]
\[f \prime = (x)\prime - (2\times \sin (x)) \prime\]
\[f \prime = (x)\prime - 2 \times (\sin (x)) \prime \]
\[f \prime = 1 - 2 \times \cos(x)\]

Then, you calculate the values of x for which the derivative is 0.

\[1 - 2 \times \cos(x) = 0\]
\[2 \times \cos(x) \ = 1 \]
\[\cos(x) = \frac{ 1 }{ 2 }\]

So, for all values of x for which the cos(x) is equal to 1/2, the tangent line to the graph f will be horizontal, because its derivative is 0.

In this link you can find a graph of both the initial function f and its derivative. You can see that when the red graph (the derivative) touches the x axis, is 0 on the y axis, the blue graph, if you draw a tangent line at that point of the graph, it will be horizontal.


http://graphsketch.com/?eqn1_color=1&eqn1_eqn=x-2*sin(x)&eqn2_color=2&eqn2_eqn=1-2*cos(x)&eqn3_color=3&eqn3_eqn=&eqn4_color=4&eqn4_eqn=&eqn5_color=5&eqn5_eqn=&eqn6_color=6&eqn6_eqn=&x_min=-17&x_max=17&y_min=-10.5&y_max=10.5&x_tick=1&y_tick=1&x_label_freq=5&y_label_freq=5&do_grid=0&do_grid=1&bold_labeled_lines=0&bold_labeled_lines=1&line_width=4&image_w=850&image_h=525


In this link (below), you can see that the cycle goes on because cos(x) is cyclic, there are multiple values of x for which its derivative is 0, so there are multiple points of x for which its tangent line is horizontal.

I attach the images of the graphs found in the links (made with graphsketch.com)

Hope it helps!
Estefania.

Answer 3

This is the link to the second graph

http://graphsketch.com/?eqn1_color=1&eqn1_eqn=x-2*sin(x)&eqn2_color=2&eqn2_eqn=1-2*cos(x)&eqn3_color=3&eqn3_eqn=&eqn4_color=4&eqn4_eqn=&eqn5_color=5&eqn5_eqn=&eqn6_color=6&eqn6_eqn=&x_min=-50&x_max=50&y_min=-20&y_max=20&x_tick=1&y_tick=1&x_label_freq=5&y_label_freq=5&do_grid=0&do_grid=1&bold_labeled_lines=0&bold_labeled_lines=1&line_width=4&image_w=850&image_h=525