   Chapter 11.6, Problem 34E Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Solutions

Chapter
Section Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

In Exercises 31–42, use implicit differentiation to find (a) the slope of the tangent line and (b) the equation of the tangent line at the indicated point on the graph. (Round answers to four decimal places as needed.) If only the x-coordinate is given, you must also find the y-coordinate. [HINT: See Example 2 and 3.] 2 x 2 + x y = 3 y 2 , ( − 1 , − 1 )

(a)

To determine

To calculate: The slope of the tangent line to the graph of the equation 2x2+xy=3y2 at the point (1,1) using the implicit differentiation.

Explanation

Given Information:

The equation is 2x2+xy=3y2 and the point is (1,1).

Formula used:

Slope of tangent to the graph of f(x) at point (x1,y1) is given by dydx|(x1,y1).

The derivative of function f(x)=un using the chain rule is f(x)=ddx(un)=nun1dudx, where u is the function of x.

Constant multiple rule of derivative of function f(x) is f'(cx)=cf'(x) where c is constant.

Product rule of derivative of differentiable functions, f(x) and g(x) is

ddx[f(x)g(x)]=f(x)g(x)+f(x)g(x).

Calculation:

Consider the equation, 2x2+xy=3y2

Find the slope of tangent line to the equation 2x2+xy=3y2 by determining the derivative.

Then, take ddx of both sides of the equation,

ddx(2x2+xy)=ddx(3y2)ddx(2x2)+ddx(xy)=ddx(3y2)

Apply constant multiple rule and the product rule of derivative,

2ddx(x2)+(dxdxy+xdydx)=3ddx(y2)2ddx(x2)+(1y+xdydx)=3ddx(y2)2ddx(x2)+y+xdyd

(b)

To determine

To calculate: The equation of the tangent line to the graph of the equation 2x2+xy=3y2 at the point (1,1).

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