Find an equation of the tangent line to the curve [tex]$y=18 \arccos \left(\frac{x}{2}\right)$[/tex] at the point ([tex]$1,6 \pi$[/tex]).
Answer (1)
Find the derivative of the function $y=18
arccos(\frac{x}{2}) : \frac{dy}{dx} = -\frac{18}{\sqrt{4 - x^2}}$.
Evaluate the derivative at x = 1 to find the slope: m = − 6 3 .
Use the point-slope form of the line with the point ( 1 , 6 π ) : y − 6 π = − 6 3 ( x − 1 ) .
Rewrite the equation in slope-intercept form: y = − 6 3 x + 6 π + 6 3 .
y = − 6 3 x + 6 π + 6 3
Explanation
Problem Analysis We are given the curve y = 18 arccos ( 2 x ) and asked to find the equation of the tangent line at the point ( 1 , 6 π ) . To do this, we need to find the derivative of the function, evaluate it at x = 1 to find the slope of the tangent line, and then use the point-slope form of a line to find the equation of the tangent line.
Finding the Derivative First, we find the derivative of y = 18 arccos ( 2 x ) with respect to x . Recall that the derivative of arccos ( u ) is − 1 − u 2 1 d x d u . In our case, u = 2 x , so d x d u = 2 1 . Therefore, the derivative of y with respect to x is: d x d y = 18 ⋅ ( − 1 − ( 2 x ) 2 1 ⋅ 2 1 ) = − 1 − 4 x 2 9 = − 4 4 − x 2 9 = − 2 4 − x 2 9 = − 4 − x 2 18
Evaluating the Derivative Next, we evaluate the derivative at x = 1 to find the slope of the tangent line at the point ( 1 , 6 π ) .
d x d y x = 1 = − 4 − 1 2 18 = − 3 18 = − 3 18 3 = − 6 3 So, the slope of the tangent line at the point ( 1 , 6 π ) is − 6 3 .
Finding the Tangent Line Equation Now, we use the point-slope form of a line, y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is the point ( 1 , 6 π ) . Plugging in the values, we get: y − 6 π = − 6 3 ( x − 1 ) y = − 6 3 x + 6 3 + 6 π Thus, the equation of the tangent line is y = − 6 3 x + 6 π + 6 3 .
Final Answer The equation of the tangent line to the curve y = 18 arccos ( 2 x ) at the point ( 1 , 6 π ) is:
y = − 6 3 x + 6 π + 6 3
Examples
Understanding tangent lines is crucial in physics, especially when analyzing motion. For example, if you're tracking the trajectory of a ball thrown in the air, the tangent line at any point on its path indicates the instantaneous velocity of the ball at that moment. The slope of this line gives you the rate of change of the ball's position, helping you predict its future location or understand its speed at a specific point.
