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(t)=\begin{cases}-2t,&t\le0\\-3t,&t>0\end{cases}

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

Not differentiable: the branch values 0 and 0 differ, so continuity fails even though the branch gradients agree

Continuous but not differentiable: the branch values agree, but the branch gradients -2 and -3 differ

Differentiable: the branch values both equal 0 and the branch gradients both equal -2

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