How to find all points on a curve where the tangent line is parallel

Question

I was given this homework problem, but I'm not even sure what it's asking me to do. 

 

Find all points on the curve y=8tanx, [-π/2,π/2], where the tangent line is parallel to y=16x. Sketch the curve and tangents together.

 

Thank you for your help! 

Answer 1

You have a curve and a line, and want to find out at which points the curve has the same slope at the line. The word tangent has two meanings, and uses both different meanings in this problem, which may be what is confusing.

 

tangent(1)- one of the six trigonometric functions, the ratio of sine/cosine

tangent(2)- a line that touches a curve at a single point, and has the same slope as the curve at that point.

 

This problem asks to find the tangent(2) line to the tangent(1) curve that is parallel to the line y=16x. Two lines that are parallel have the same slope, so you're looking for the tangent(2) that has a slope of 16 (y=mx+b; m=16 here.)

 

To find the slope of the tangent(1) curve, you take the derivative of the function.

y = 8 tan(x)

y' = 8 sec²(x)

 

You're looking for the values of x where y' = 16.

16 = 8 sec²(x)

 

Remember that sec(x) is the reciprocal of cos(x), which is an even function cos(x) = cos(-x), so you'll find two angles (one +, one -) which satisfy this equation.

 

Hope this helps, and please comment if you have further questions.

Answer 2

The slope of the line is 16.  So you want the derivative of 8 tan(x) to be the same magnitude.

 

y' =8 sec²x

 

Set y' = 16 and then solve for x.   Good luck.