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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

Show that the function is continuous on its domain. State the domain.

h(x) = xesin x

To determine

To state: The domain of the function.

Explanation

Given:

The function h(x)=xesinx.

Calculation:

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

The sine function is defined for every real numbers. So that, the exponential of sinx is also defined in

The polynomial function x is defined in .

Therefore, the product of polynomial and exponential function is defined in .

Thus, the domain of the function h(x) is .

To show: The function is continuous on its domain.

Theorems used:

1. 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.

2. (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.

4. 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.

Proof:

Consider the function h(x) is of the form h(x)=p(x)q(x) where p(x)=x and q(x)=esinx

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