Q.Total Revenue is given by $TR=100Q-2Q^{2}$.

Apply the power rule to differentiate TR with respect to Q to get MR, evaluate at Q=10, then cross-check using TR values at Q=9 and Q=11.

(a) Applying the power rule term by term to $TR=100Q-2Q^{2}$:

• Derivative of $100Q$ (i.e. $100Q^{1}$): $100\times1\times Q^{0}=100$.
• Derivative of $-2Q^{2}$: $-2\times2\times Q^{1}=-4Q$.

(b) At $Q=10$:

Verification by the central-difference method: compute TR at $Q=9$ and $Q=11$ and take the average change:

$\dfrac{TR(11)-TR(9)}{2}=\dfrac{858-738}{2}=\dfrac{120}{2}=60$ …

As an additional cross-check, a ONE-SIDED difference, $TR(11)-TR(10)$, gives $858-800=58$ — close to but not exactly 60, because a one-sided difference only approximates the derivative rather than matching it exactly; this confirms the CENTRED difference (usin …

A common error is forgetting the power rule's exponent-reduction step and writing the derivative of $-2Q^2$ as $-2Q$ (dropping the multiplication by the original exponent 2) instead of the correct $-4Q$ …