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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.

Theorem 4

Chapter 2.5, Problem 28E, Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its , example  1

Chapter 2.5, Problem 28E, Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its , example  2

Theorem 5

Chapter 2.5, Problem 28E, Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its , example  3

Theorem 7

Chapter 2.5, Problem 28E, Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its , example  4

Theorem 9

Chapter 2.5, Problem 28E, Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its , example  5

Chapter 2.5, Problem 28E, Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its , example  6

R ( t ) = e sin t 2 + cos π t

To determine

To state: The domain and explain the function is continuous at every number in its domain.

Explanation

Given:

The function R(t)=esint2+cosπt.

Theorems used:

4. If f and g are continuous at a and c is a constant, then the following functions are also continuous at a:

(i) f+g (ii) fg (iii) cf (iv) fg (v) fg if g(a)0.

5. (a) Any polynomial is continuous everywhere; that is, it is continuous on =(,).

(b) Any rational function is continuous wherever it is defined; that is, it is continuous on its domain.

7. The functions such as “Polynomials, rational functions, root functions, trigonometric functions, inverse trigonometric functions, exponential functions and logarithmic functions” are continuous at every number in their domains.

9. If g is continuous at a and f is continuous at g(a), then the composite function fg given by (fg)(x)=f(g(x)) is continuous at a.

Calculation:

The domain is the set of all input values of the function for which the function is real and defined.

Consider the denominator of the function R(t) and equate to zero to obtain the undefined points.

Since the denominator of R(t) is 2+cosπt, the undefined points are obtained as shown below:

2+cosπt=0cosπt=2

Since cosπt cannot be smaller than 1 for real solutions because the range of cost lies between minus one and plus one. So there is no undefined points in .

Therefore, the domain of the function is the set of all real numbers or {t|t}.

And, the interval notation of the domain of R(t)  is (,).

Consider the function R(t) of the form R(t)=p(t)q(t) where p(t)=esint and q(t)=2+cosπt

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