(a) Explanation Given information: The parabola: y = x 2 + x , the point ( 2 , − 3 ) Calculation: Note that all points on the parabola y = x 2 + x are of the form ( x , x 2 + x ) Assume that the tangent at ( a , a 2 + a ) passes through ( 2 , − 3 ) We know that the slope of tangent at any point is the derivative at that point. Differentiate y = x 2 + x , to get d y d x = 2 x + 1 Therefore, the slope of tangent at ( a , a 2 + a ) is 2 a + 1 → ( 1 ) Since the tangent passes through the points ( a , a 2 + a ) and ( 2 , − 3 ) , we can write the slope of the tangent as ( a 2 + a ) − ( − 3 ) a − 2 = a 2 + a + 3 a − 2 → ( 2 ) Using (1) and (2), we can write a 2 + a + 3 a − 2 = 2 a + 1 a 2 + a + 3 = ( 2 a + 1 ) ( a − (b) To determine To show: there is no line through the point ( 2 , 7 ) that is tangent to the parabola.

15th Edition

ISBN: 9781305177291

Author: James Stewart, Troy Day

Publisher: Cengage Learning

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Question

Chapter 3.3, Problem 68E

(a)

To determine

To find: equations of both lines through the point (2,−3) that are tangent to the parabola y=x2+x .

(b)

To determine

To show: there is no line through the point (2,7) that is tangent to the parabola.

Expert Solution & Answer

Chapter 3 Solutions

Biocalculus: Calculus For Life Sciences

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equations of both lines through the point ( 2 , − 3 ) that are tangent to the parabola y = x 2 + x .

Solution Summary: The author explains that the slope of the tangent at any point is the derivative at that point.

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To find: equations of both lines through the point (2,−3) that are tangent to the parabola y=x2+x .

To show: there is no line through the point (2,7) that is tangent to the parabola.

Chapter 3 Solutions