   Chapter 2.3, Problem 97E

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# Let f ( x ) = { x 2 + 1   if  x < 1 x + 1     if  x ≥ 1 Is f differentiable at 1? Sketch the graphs of f and f ′ .

To determine

To find:

Is f differentiable at 1 and sketch the graphs of f and f'

Explanation

Concept:

The function is differential if left and right hand derivatives are equal

Formula:

Left limit:

f-'x=limh0-fx+h-f(x)h

Right limit:

f+'x=limh0+fx+h-f(x)h

Given:

fx= x2+1      if  x<1 x+1      if x1

Calculation:

Calculate left and right hand derivatives for x=1

Left hand derivative:

f-'x=limh0-fx+h-f(x)h

f-'1=limh0-f1+h-f(1)h

=limh0- 1+h2+1-1+1h

=limh0- 1+2h+h2+1-2h

=limh0- h2+2hh

=limh0- h(h+2)h

=limh0- h+2

=2

Right hand derivative:

f+'x=limh0+fx+h-f(x)

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