Determine an equation of the line tangent to the curve [tex]y=\sqrt{x-8}[/tex] at the point with x-coordinate 9.
A. [tex]-x+2 y+7=0[/tex]
B. [tex]x-2 y+8=0[/tex]
C. [tex]x-2 y+7=0[/tex]
D. [tex]-x+2 y+8=0[/tex]
Answer (2)
Find the y-coordinate at x = 9 : y = 9 − 8 = 1 .
Calculate the derivative: d x d y = 2 x − 8 1 .
Evaluate the derivative at x = 9 to find the slope: m = 2 1 .
Use the point-slope form to find the tangent line equation: x − 2 y + 7 = 0 , so the answer is x − 2 y + 7 = 0 .
Explanation
Problem Analysis We are given the curve y = x − 8 and we want to find the equation of the tangent line at the point where x = 9 .
Finding the Point First, we need to find the y -coordinate of the point on the curve where x = 9 . Substituting x = 9 into the equation, we get
y = 9 − 8 = 1 = 1 .
So the point is ( 9 , 1 ) .
Finding the Derivative Next, we need to find the derivative of the function y = x − 8 with respect to x . We can rewrite the function as y = ( x − 8 ) 2 1 . Using the power rule, we have
d x d y = 2 1 ( x − 8 ) − 2 1 = 2 x − 8 1 .
Finding the Slope Now, we need to evaluate the derivative at x = 9 to find the slope of the tangent line at that point.
m = d x d y ∣ x = 9 = 2 9 − 8 1 = 2 1 1 = 2 1 .
So the slope of the tangent line at the point ( 9 , 1 ) is 2 1 .
Finding the Equation of the Tangent Line Now we use the point-slope form of a line, y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) = ( 9 , 1 ) and m = 2 1 .
y − 1 = 2 1 ( x − 9 ) .
Multiplying both sides by 2, we get
2 ( y − 1 ) = x − 9
2 y − 2 = x − 9
x − 2 y − 9 + 2 = 0
x − 2 y − 7 = 0
x − 2 y = 7
x − 2 y + 7 = 0
Final Answer Comparing this equation to the given options, we see that option c) x − 2 y + 7 = 0 is the correct answer.
Examples
Understanding tangent lines is crucial in fields like physics and engineering. For example, when analyzing the motion of a projectile, the tangent line to the trajectory at any given point indicates the instantaneous velocity vector. Similarly, in electrical engineering, the tangent line to a voltage or current curve can help determine the rate of change of these quantities, which is essential for circuit analysis and design. By finding the equation of a tangent line, we can make accurate approximations and predictions about the behavior of complex systems.
The tangent line to the curve y = x − 8 at the point where x = 9 has the equation x − 2 y + 7 = 0 . This results from finding the point on the curve, calculating the derivative for the slope, and applying the point-slope form of a linear equation. Therefore, the answer is option C.
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