Explanation Given: The equation y 4 + x y = x 3 − x + 2 The point ( 1 , 1 ) The graph: Formula used: Equation of tangent line is given as: y − y 1 = m ( x − x 1 ) Key Concepts Used: Derivative of a function y = f ( x ) gives the rate of change of y with respect to x . Thus, the derivative of a function at a point is equivalent to slope of the tangent line at that point. Using that, equation of tangent line is given by y − y 1 = m ( x − x 1 ) Calculations: We are given with the equation y 4 + x y = x 3 − x + 2 First of all, we find the values of y when x = 0 ⇒ y 4 + ( 0 ) y = 0 3 − 0 + 2 ⇒ y 4 + 0 = 2 ⇒ y 4 = 2 ⇒ y 4 = 2 ⇒ y = ± 2 4 Next, we find d y d x for the equation y 4 + x y = x 3 − x + 2 . On taking derivative of this equation on both the sides, we get: ⇒ d d x ( y 4 + x y ) = d d x ( x 3 − x + 2 ) ⇒ 4 y 3 d y d x + x d y d x + ( 1 ) y = 3 x 2 − 1 + 0 ⇒ 4 y 3 d y d x + x d y d x + y = 3 x 2 − 1 ⇒ ( 4 y 3 + x ) d y d x = 3 x 2 − y − 1 ⇒ d y d x = 3 x 2 − y − 1 4 y 3 + x Now we have to find the value of this derivative at the points ( 0 , 2 4 ) and ( 0 , − 2 4 ) . ⇒ d y d x | ( 0 , 2 4 ) = 3 ( 0 ) 2 − 2 4 − 1 4 ( 2 4 ) 3 + 0 = − 2 4 − 1 4 ( 2 4 ) 3 ⇒ d y d x | ( 0 , 2 4 ) = ( − 2 1 4 − 1 ) 2 1 4 4 ( 2 ) = ( − 2 1 4 ⋅ 2 1 4 − 2 1 4 ) 8 ⇒ d y d x | ( 0 ,

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Question

Chapter 3.8, Problem 67E

To determine

The value of $\frac{d y}{d x}$ at two points on the graph of the given function where $x = 0$ .

Also, the equation of tangent line to the curve at the given point $(1, 1)$ .

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Chapter 3.8, Problem 66E

Chapter 3.8, Problem 68E

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Chapter 3 Solutions

Calculus: Early Transcendentals

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The value of d y d x at two points on the graph of the given function where x = 0 . Also, the equation of tangent line to the curve at the given point ( 1 , 1 ) .

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The value of dydx at two points on the graph of the given function where x=0. Also, the equation of tangent line to the curve at the given point (1,1).

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The value of $\frac{d y}{d x}$ at two points on the graph of the given function where $x = 0$ .

Also, the equation of tangent line to the curve at the given point $(1, 1)$ .