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Chapter 3: Problem 33

a. Find an equation of the line tangent to the given curve at a. b. Use a graphing utility to graph the curve and the tangent line on the same set of axes. $$y=\frac{x+5}{x-1} ; \quad a=3$$

Short Answer

Expert verified
Answer: The equation of the tangent line to the given curve at the point where x = 3 is \(y = -\frac{3}{2}x + \frac{17}{2}\).

Step by step solution

01

Find the derivative of the curve with respect to x

We have to differentiate the given function, y = (x+5)/(x-1). For that, we will use the quotient rule: \((\frac{u}{v})' = \frac{uv' - vu'}{v^2}\), where u = x + 5 and v = x - 1. So, first differentiate both u and v w.r.t x: $$u' = \frac{d}{dx}(x + 5) = 1$$ $$v' = \frac{d}{dx}(x - 1) = 1$$ Now, apply the quotient rule for y: $$y' = \frac{(x - 1)(1) - (x + 5)(1)}{(x - 1)^2}$$
02

Evaluate y'(a) to find the slope of tangent line

Evaluate the derivative of y with respect to x at the point a = 3: $$y'(3) = \frac{(3 - 1)(1) - (3 + 5)(1)}{(3 - 1)^2}$$ $$ = \frac{2 - 8}{2^2}$$ $$ = \frac{-6}{4}$$ $$ = -\frac{3}{2}$$ The slope of the tangent line is -3/2.
03

Find the y-coordinate of the curve at a = 3

Evaluate the given equation at x = 3 to find the value of y-coordinate: $$y(3) = \frac{3+5}{3-1}$$ $$ = \frac{8}{2}$$ $$ = 4$$ So, the point on the curve at a = 3 is (3, 4).
04

Use slope-point form to find the equation of the tangent line

Use the slope-point form of a line, which is given by: \(y - y_1 = m(x - x_1)\). Here, \((x_1, y_1)\) = (3, 4) and slope, m = -3/2. Plug in the values into the slope-point form equation: $$y - 4 = -\frac{3}{2}(x - 3)$$
05

Simplify the equation of the tangent line

Simplify the equation from the previous step: $$y - 4 = -\frac{3}{2}x + \frac{9}{2}$$ $$y = -\frac{3}{2}x + \frac{17}{2}$$ The equation of the tangent line is: \(y = -\frac{3}{2}x + \frac{17}{2}\).
06

Graph the curve and tangent line using a graphing utility

Now, using a graphing utility, graph the given curve \(y = \frac{x+5}{x-1}\) and the tangent line \(y = -\frac{3}{2}x+\frac{17}{2}\) on the same set of axes. Observe that the tangent line only touches the curve at the point a = 3 (3, 4) as expected.

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