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Q. 2

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A function that is its own fourth derivative.

(b) A function whose domain is larger than the domain of its derivative.

(c) Three non-logarithmic functions that are transcendental, but whose derivatives are algebraic.

Short Answer

Expert verified

(a) the function f(x)=sinxis its own fourth derivative.

(b) the domain of the function f(x)=sin-1xis larger than the domain of its derivative

(c) transcendental functions whose derivatives are algebraic are following.

f(x)=sin-1xf(x)=tan-1xf(x)=sec-1x

Step by step solution

01

Step 1. Given information.

(a) A function is its own fourth derivative.

(b) Domain of a function is larger than the domain of its derivative.

(c) derivatives of transcendental non-logarithmic functions are algebraic.

02

Step 2. Part (a)

Consider a function f(x)=sinx.

Find the fourth derivative.

role="math" localid="1648764216410" f'(x)=ddxsinxf'(x)=cosxf''(x)=ddxcosxf''(x)=-sinxf'''(x)=ddx-sinxf'''(x)=-cosxf''''(x)=ddx-cosxf''''(x)=--sinx=sinx

So fourth derivative of the functionf(x)=sinxissinx.

03

Step 3. Part (b)

Consider a function f(x)=sin-1x.

The domain of the function f(x)=sin-1xis -1,1.

Derivative of the function.

f'(x)=ddxsin-1xf'(x)=11-x2

Domain of role="math" localid="1648764574229" f'(x)=11-x2is -1,1.

-1,1>-1,1

So the domain of the function role="math" localid="1648764542847" f(x)=sin-1xis larger than the domain of its derivative f'(x)=11-x2.

04

Step 4. Part (c)

Consider a transcendental function f(x)=sin-1x.

Derivative of function.

f'(x)=ddxsin-1xf'(x)=11-x2

Consider a transcendental functionf(x)=tan-1x.

Derivative of function.

f'(x)=ddxtan-1xf'(x)=11+x2

Consider a transcendental function f(x)=sec-1x.

Derivative of function.

f'(x)=ddxsec-1xf'(x)=1xx2-1

Derivatives of all functions are algebraic.

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Use (a) the hโ†’0definition of the derivative and then

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