1 of 8 0-day streak

Test or construct a smooth piecewise join

Question 1 of 8

No calculator

Separate continuity from equality of the two branch gradients

f(x)=\begin{cases}2x-2,&x<0\\2x-2,&x\ge0\end{cases}

The domain is all real inputs. The right branch includes the join. Work at x = 0

(a)4 marks

Let L and R denote the left and right branch polynomials, extended to the join for this calculation. Enter their values and derivatives there. For a jump, these branch derivatives are not the one-sided derivatives of the piecewise function itself

\(L(0)\) \(R(0)\) \(L'(0)\) \(R'(0)\)

(b)1 mark

Choose the complete justification

Continuous but not differentiable: the branch values agree, but the branch gradients 2 and 2 differ

Differentiable: the branch values both equal -2 and the branch gradients both equal 2

Differentiable whenever each branch can be differentiated, regardless of the join

Not differentiable: the branch values -2 and -2 differ, so continuity fails even though the branch gradients agree