1 of 8 0-day streak

Test or construct a smooth piecewise join

Question 1 of 8

No calculator

Check the branch values before comparing the gradients

f(x)=\begin{cases}-x+5,&x\le1\\-x+7,&x>1\end{cases}

The domain is all real inputs. The left branch includes the join. Work at x = 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 4 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 4 and 6 differ, so continuity fails even though the branch gradients agree