   Chapter 3.5, Problem 41E

Chapter
Section
Textbook Problem

# In the theory of relativity, the mass of a particle is m = m 0 1 − v 2 / c 2 where m 0 is the rest mass of the particle, m is the mass when the particle moves with speed v relative to the observer, and c is the speed of light. Sketch the graph of m as a function of v.

To determine

To sketch:

The curve of y

Explanation

1) Concept:

i) A domain is the set of x values that satisfy the function.

ii) To find x-intercept, put y=0, and to find y-intercept, put x=0 in the given function.

iii) Symmetry: To find symmetry, replace x by –x and check the behaviour of function. Thus, if f-x=fx, then it is an even function, so it has y-axis symmetry. If f-x=-fx, then it is an odd function, so it has x-axis symmetry. And if f-x-fxfx, then it has no symmetry.

iv) An asymptote is a tangent at infinity. To find horizontal, vertical, and slant asymptote, follow the rules.

v) A function is increasing if f'x>0  and decreasing if f'x<0 in that particular interval.

vi) The number f(c) is a local maximum value of f  if fcf(x) when x is near c and is a local minimum value of f if fc f(x) when x is near c.

vii) If f''x>0, the function is concave up and if f''x<0, the function is concave down in that particular interval. And if f''x=0, give the values of inflection points

2) Given:

m=m01-v2c2

3) Calculation:

Here, first find the domain of the given function and the x & y intercepts. Next, check the symmetry, asymptotes, intervals of increase and decrease, local maximum and minimum values, concavity, and points of inflection. Using these, sketch the curve

A. Domain:

So the domain is (0, c), since v can only be positive

B. Intercepts:

For y intercept plug v=0  in the given function and solve it

m=m0

The y-intercept is 0, m0

There are no x intercepts

C. Asymptote:

a) Horizontal asymptotes:

There are no horizontal asymptotes

b) Vertical asymptotes:

limvc-m01-v2c2=

So there are Vertical asymptotes at v=c

D

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