Answered: Find an equation of the line tangent to…

Math

Calculus

Find an equation of the line tangent to each of the following curves at the given point. 4x 11, 12 y = x? + 3

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Topics In Analytic Geometry

Section6.2: Introduction To Conics: parabolas

Problem 4ECP: Find an equation of the tangent line to the parabola y=3x2 at the point 1,3.

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Question

Find an equation of the line tangent to each of
the following curves at the given point.
4x
11, 12
y =
x? + 3

Expert Solution

Step 1

The given equation of curve is $y = \frac{4 x}{x^{2} + 3} \dots \dots (1)$

Now differentiate the equation (1) w.r.t. $' x'$ by using quotient rule.

$\begin{array}{rcl} \frac{d y}{d x} & = & \frac{d}{d x} (\frac{4 x}{x^{2} + 3}) \\ = & \frac{(x^{2} + 3) 4 - (4 x) (2 x)}{{(x^{2} + 3)}^{2}} \\ = & \frac{12 - 4 x^{2}}{{(x^{2} + 3)}^{2}} \end{array}$

Step 2

Find the slope of tangents at point $(11, 12)$ ,

$\begin{array}{rcl} {\frac{d y}{d x}}_{(11, 12)} & = & \frac{12 - 4 {(11)}^{2}}{{({(11)}^{2} + 3)}^{2}} \\ = & - \frac{472}{15376} \\ = & - \frac{59}{1922} \end{array}$

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