Common Mistakes

Calculus BC · AP · Contextual Applications of Differentiation

Focus

Solving Related Rates Problems

This topic is assessed on the AP Exam.

Frequent mistakes

### 1. Chain rule omission

What it looks/sounds like: Inner derivative is dropped on composite functions.

Why it happens: Only the outer rule is applied.

How to fix: Write g′(x) explicitly when differentiating f(g(x)).

### 2. Related rates: plug too early

What it looks/sounds like: Numeric values are substituted before differentiating.

Why it happens: The equation is not differentiated with respect to time.

How to fix: Differentiate first, then substitute known rates/values.

### 3. L’Hospital without indeterminate form

What it looks/sounds like: L’Hospital is applied to a limit that is not 0/0 or ∞/∞.

Why it happens: The rule is treated as universal.

How to fix: Check indeterminate form before applying L’Hospital.

### 4. Correct work, wrong question

What it looks/sounds like: The algebra is fine but answers a different quantity than the stem asks for.

Why it happens: Attention locks onto the first operation instead of the final ask.

How to fix: Underline the final question; after solving, verify the boxed answer matches it.

### 5. Notation and justification gaps

What it looks/sounds like: Missing units, limit language, or ‘because’ statements AP graders expect.

Why it happens: Notation is treated as decoration rather than part of the score.

How to fix: Add one justification phrase per major step (definition, theorem, or rule).

How to avoid them

1. Draw and label a related-rates diagram. 2. State L’Hospital hypotheses explicitly. 3. Check linearization point and over/under estimate.

Quick self-check

• [ ] Can set up a related-rates equation
• [ ] Can state when L’Hospital applies
• [ ] Can write a linearization formula