2. a) Determine the coordinates of all point(s) on the graph of y = 2x ln x where the tangent line isperpendicular to the line x+4y-7=0.b) Find the equation of the tangent line to the graph of f(x) = xe* + e* + 1 at the point where x =1.

Answered: 2. a) Determine the coordinates of all…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

2. a) Determine the coordinates of all point(s) on the graph of y = 2x ln x where the tangent line is
perpendicular to the line x+4y-7=0.
b) Find the equation of the tangent line to the graph of f(x) = xe* + e* + 1 at the point where x =
1.

Expert Solution

Step 1

Part_a_ Determine the coordinates of all point(s) on the graph of $y = 2 x \ln x$ where the tangent line is perpendicular to the line $x + 4 y - 7 = 0$ .

Solution:-

1. Slope of line $x + 4 y - 7 = 0$

Slope of line perpendicular to the line $a x + b y + c = 0$ is given as $m = \frac{b}{a}$

Substitute a=1 and b=4 in $m = \frac{b}{a}$

$m = \frac{4}{1}$

$m = 4$ Equation 1

2. Slope of the tangent line of $y = 2 x \ln x$ is given as $\frac{d y}{d x}$

$\frac{d y}{d x} = \frac{d}{d x} (2 x \ln x)$

$\frac{d y}{d x} = 2 \ln x \frac{d}{d x} (x) + 2 x \frac{d}{d x} (\ln x)$

$\frac{d y}{d x} = 2 \ln x + 2 x \times \frac{1}{x}$

$\frac{d y}{d x} = 2 \ln x + 2$

Since, the tangent line is perpendicular to the line $x + 4 y - 7 = 0$

Therefore, $\frac{d y}{d x} = 4$

$4 = 2 \ln x + 2$

$2 \ln x = 4 - 2$

$2 \ln x = 2$

$\ln x = \frac{2}{2}$

$\ln x = 1$

Take antilog in both sides

$x = e$ Equation 2

Substitute $x = e$ in $y = 2 x \ln x$

$y = 2 \times e \times \ln e$

$y = 2 e \times 1$

$y = 2 e$ Equation 2

Coordinates of the point $(e, 2 e)$

Hence, the coordinates of all point on the graph of $y = 2 x \ln x$ where the tangent line is perpendicular to the line $x + 4 y - 7 = 0$ is $(e, 2 e)$