   Chapter 2.6, Problem 46E

Chapter
Section
Textbook Problem

# Show that the sum of the x- and y-intercepts of any tangent line to the curve x + y = c is equal to c.

To determine

To show:

The sum of x and y intersect of any tangent to the given curve is equal to c

Explanation

1) Concept:

Slope of the tangent line is a derivate of the curve at that point.

2) Formula:

ddxxn=nxn-1 Intercept form of equation of a line if x-intercept of a line is a and y-intercept is b then equation of the line is xa+yb=1

3) Given:

x+y=c

4) Calculations:

Differentiate x+y=c with respect to x to get

ddxx+y=c

ddxx+ddxy=ddxc

12x+12ydydx=0

12ydydx=-12x

1ydydx=-1x

dydx=-yx

Let a,b be a point on the curve then

Slop of tangent at

a,b=dydxa,b=-ba

By using Slope point form the equation of the tangent line at (a,b) is

y-b=-ba (x-a)

Multiplying both sides by a

ya-ba=-x

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