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Chapter 2: Q. 47 (page 222)

Describe the derivatives of each of the piecewise-defined functions in Exercises 45–48.

f(x)={x2,x<1;1+lnx,x≥1}

Short Answer

Expert verified

The derivative of function is


f'(x)=2x,x<1DNE,x=11x,x>1

Step by step solution

01

Step 1. Given Information 

The given function is f(x)={x2,x<1;1+lnx,x≥1}

02

Step 2. Calculation 

From the differentiation rules,

ddx(x2)=2xddx(1+lnx)=1xLetg(x)=x2,h(x)=1+lnxAndg'(x)=2x,h'(x)=1xCheckg'(1)=h'(1)g'(1)=2(1)=2h'(1)=11=1

It follows that the function is not continuous at x=1.

Thus, the derivative of function is,

f'(x)=2x,x<1DNE,x=11x,x>1

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Most popular questions from this chapter

Differentiate f(x)=3x+x2in three ways. When you have completed all three parts, show that your three answers are the same:

(a) with the chain rule

(b) with the product rule but not the chain rule

(c) without the chain or product rules.

For each function f(x)and interval a,bin Exercises 81-86, use the Intermediate Value Theorem to argue that the function must have at least one real root on a,b. Then apply Newton’s method to approximate that root.

f(x)=x3+1,a,b=-2,1

Use the definition of the derivative to find f'for each function fin Exercises 39-54

role="math" localid="1648290170541" f(x)=x-1x+3

In Exercises 69–80, determine whether or not f is continuous and/or differentiable at the given value of x. If not, determine any left or right continuity or differentiability. For the last four functions, use graphs instead of the definition of the derivative.

fx=x2-3,ifx<3x+2,ifx≥3,x=3

Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=x-12(x2-1)3

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