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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

 (a) Use the identity for tan ( x y ) (see Equation 14b in Appendix D) to show that if two lines L 1 and L 2 intersect at an angle α , then

tan α = m 2 m 1 1 + m 1 m 2

Where m 1 and m 2 are the slopes of L 1 and L 2 , respectively.

(b) The angle between the curves C 1 and C 2 at a point of intersection P is defined to be the angle between the tangent lines to C 1 and C 2 at P (if these tangent lines exist). Use part (a) to find, correct to the nearest degree, the angle between each pair of curves at each point of intersection.

(i) y = x 2 and y = ( x 2 ) 2

(ii) x 2 y 2 = 3 and x 2 4 x + y 2 + 3 = 0

Part (a):

To determine

To show: If two lines L1 and L2 intersect at an angle Α, then

tanΑ = m2 - m11 + m1m2 where m1 and m2 are the slopes of L1 and L2 respectively, by using identity for  tan(x-y)

Explanation

1) Concept:

tan(x-y) = tanx - tany1 +tanxtany

If two lines L1 and L2 have slopes m1 and m2 and angles of inclination θ1 and θ2 shown in the following figure.

Here,  slope of line L1  is m1 = tanθ1 and slope of line L2  is m2 = tanθ2

2) Given:

Two lines L1 and L2 intersect at an angle Α, then we need to show that

tanΑ = m2 - m11 + m1m2

where m1 and m2 are the slopes of L1 and L2, respectively

3) Calculation:

In triangle ABC,  A+ B+ C  =1800

 Α+ θ1+  1800- θ2  =1800

Subtract  1800 from both sides,

Α+ θ1 - θ2  = 0

By adding θ2 on both sides,

Α+ θ1  = θ2

Then, subtract θ1 f

To determine

Part (b):

To find: The angle between each pair of curves at each point of intersection correct to the nearest degree using part (a).

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