1 of 8 0-day streak

Test or construct a smooth piecewise join

Question 1 of 8

No calculator

Use exact branch rules to decide whether the join is smooth

f(t)=\begin{cases}-t+2,&t\le-1\\-t+2,&t>-1\end{cases}

The domain is all real inputs. The left branch includes the join. Work at t = -1

(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

\(L(-1)\) \(R(-1)\) \(L'(-1)\) \(R'(-1)\)

(b)1 mark

Choose the complete justification

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

Differentiable: the branch values both equal 3 and the branch gradients both equal -1

Continuous but not differentiable: the branch values agree, but the branch gradients -1 and -1 differ

Not differentiable: the branch values 3 and 3 differ, so continuity fails even though the branch gradients agree