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Essential Calculus: Early Transcendentals
Found in: Page 632
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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Short Answer

Explain why each function is continuous or discontinuous.

  1. The outdoor temperature as a function of longitude, latitude and time
  2. Elevation (height above sea level) as a function of longitude, latitude and time.
  3. The cost of a taxi ride as a function of distance traveled and time.

According to the definition of continuous function, the outdoor temperature as a function of longitude, latitude and time is continuous.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Definition of is continuous or discontinuous function

Continuous function:-

Continuous function is a function such that a continuous variation (that is a change without a jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value.

Discontinuous function:-

Discontinuous function is a function such that a continuous variation (that is a change with a jump) of the argument induces a discontinuous variation of the value of the function. This means that there are abrupt changes in value.

Step 2: Check whether the function is continuous or discontinuous

The outdoor temperature as a function of longitude, latitude and time is continuous.

Small changes in longitude, latitude or time can produce only small changes in temperature, as the temperature does not jump abruptly from one value to another.

Therefore, the outdoor temperature as a function of longitude, latitude and time is continuous.

Answer (b):

According to the definition of discontinuous function, Elevation (height above sea level) as a function of longitude, latitude and time is discontinuous.

Step 1: Definition of is continuous or discontinuous function

Continuous function:-

Continuous function is a function such that a continuous variation (that is a change without a jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value.

Discontinuous function:-

Discontinuous function is a function such that a continuous variation (that is a change with a jump) of the argument induces a discontinuous variation of the value of the function. This means that there are abrupt changes in value.

Step 2: Check whether the function is continuous or discontinuous

Elevation is not necessarily continuous. If we think of a cliff with a sudden drop off, a very small change in longitude or latitude can produce a comparatively large change in elevation, without all the intermediate values being attained. Elevation can jump from one value to another.

Therefore, Elevation (height above sea level) as a function of longitude, latitude and time is discontinuous.

Answer (c):

According to the definition of discontinuous function, the cost of a taxi ride as a function of distance traveled and time is discontinuous.

Step 1: Definition of is continuous or discontinuous function

Continuous function:-

Continuous function is a function such that a continuous variation (that is a change without a jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value.

Discontinuous function:-

Discontinuous function is a function such that a continuous variation (that is a change with a jump) of the argument induces a discontinuous variation of the value of the function. This means that there are abrupt changes in value.

Step 2: Check whether the function is continuous or discontinuous

The cost of a taxi ride is usually discontinuous. The cost normally increases in jumps, so small changes in distance traveled or time can produce a jump in cost. A graph of the function would show breaks in the surface.

Therefore, the cost of a taxi ride as a function of distance traveled and time is discontinuous.

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Equation 6 is a formula for the derivative \(\frac{{dy}}{{dx}}\) of a function defined implicitly by an equation \(F(x,y) = 0\), provided that \(F\) is differentiable and \({F_y} \ne 0\). Prove that if \(F\) has continuous second derivatives, then a formula for the second derivative of \(y\) is

\(\frac{{{d^2}y}}{{d{x^2}}} = - \frac{{{F_{xx}}F_y^2 - 2{F_{xy}}{F_s}{F_y} + {F_{yy}}F_x^2}}{{F_y^3}}\)

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Let \(g(x,y) = \cos (x + 2y)\)

  1. Evaluate \(g(2, - 1)\)
  2. Find the domain of g
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(a) Determine the rate of change of the volume of the box whose length \(l\) increase from \(1\;{\rm{m}}/{\rm{s}}\) to \(2\;{\rm{m}}/{\rm{s}}\), width \(w\) increase from \(2\;{\rm{m}}/{\rm{s}}\) to \(2\;{\rm{m}}/{\rm{s}}\) and height \(h\) decrease from \(2\;{\rm{m}}/{\rm{s}}\) to \(3\;{\rm{m}}/{\rm{s}}\).

(b) Determine the rate of change of the surface of the box whose length \(l\) increase from \(1\;{\rm{m}}/{\rm{s}}\) to \(2\;{\rm{m}}/{\rm{s}}\), width \(w\) increase from \(2\;{\rm{m}}/{\rm{s}}\) to \(2\;{\rm{m}}/{\rm{s}}\) and height \(h\) decrease from \(2\;{\rm{m}}/{\rm{s}}\) to \(3\;{\rm{m}}/{\rm{s}}\).

(c) Determine the rate of change of the length of a diagonal of the box whose length \(l\) increase from \(1\;{\rm{m}}/{\rm{s}}\) to \(2\;{\rm{m}}/{\rm{s}}\), width \(w\) increase from \(2\;{\rm{m}}/{\rm{s}}\) to \(2\;{\rm{m}}/{\rm{s}}\) and height \(h\) decrease from \(2\;{\rm{m}}/{\rm{s}}\) to \(3\;{\rm{m}}/{\rm{s}}\).

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