AP Calculus AB Unit 5 Test- Review and Preparation
What Unit 5 Actually Covers
Unit 5 is called Analytical Applications of Derivatives. It's where calculus stops being abstract and starts being useful. You learn to extract real information about functions just by looking at their derivatives.
The test will hit you with:
- Finding relative and absolute extrema
- Using the Mean Value Theorem to prove things
- Solving optimization problems
- Related rates questions
- L'Hospital's Rule for limits
- Motion along a line (position, velocity, acceleration)
If your derivative skills from Unit 3 are weak, Unit 5 will expose that immediately. Fix that first.
Theorems You Must Know Cold
These show up every year. Memorize the conditions. Know when you can apply them.
Extreme Value Theorem
If a function is continuous on a closed interval [a,b], then it attains both an absolute maximum and absolute minimum on that interval. That's it. No exceptions to the conditions.
Mean Value Theorem (MVT)
If a function is continuous on [a,b] and differentiable on (a,b), then there exists at least one c in (a,b) where:
f'(c) = [f(b) - f(a)] / (b - a)
This is the big one. It connects average rate of change to instantaneous rate of change. You'll use it to justify answers or prove that a function has a certain property.
Rolle's Theorem
This is just MVT when f(a) = f(b). The conclusion becomes f'(c) = 0. Useful for proving existence of critical points.
Fermat's Theorem
If f has a local extremum at c and f' exists, then f'(c) = 0. Don't confuse this with the converse—that's false. Having f'(c) = 0 doesn't guarantee a local extremum.
First and Second Derivative Tests
These are your tools for classifying critical points.
First Derivative Test
Sign of f' changes from positive to negative at c → local maximum. Negative to positive → local minimum. No change → not an extremum.
Second Derivative Test
If f''(c) > 0 → local minimum. If f''(c) < 0 → local maximum. If f''(c) = 0 or doesn't exist → test is inconclusive, go back to the first derivative test.
The second derivative test is faster when it works. The first derivative test always works. Know both.
Optimization: The Money Maker
Optimization problems are guaranteed to be on your test. Here's the process:
- Identify what needs to be maximized or minimized
- Write a function for the quantity in terms of one variable
- Use the constraint to eliminate a variable
- Find the derivative, set it equal to zero, solve
- Verify it's actually a max or min (first or second derivative test)
- Check endpoints if the domain is closed and bounded
Common traps:
- Forgetting to check endpoints
- Not using the constraint correctly
- Solving for the wrong variable first
Related Rates: The Setup Is Everything
Related rates questions have a specific structure. You need to:
- Identify all given rates and what you're solving for
- Write an equation that relates the variables
- Implicitly differentiate with respect to time
- Plug in known values and solve
These problems are 90% setup, 10% algebra. If your equation is wrong, nothing else matters. Practice setting them up until it's automatic.
Common related rates patterns:
- Pythagorean theorem (ladder, shadow, distance problems)
- Similar triangles
- Volume formulas (spheres, cones, cylinders)
- Area formulas (expanding circles, rectangles)
L'Hospital's Rule
This handles limits that give you 0/0 or ∞/∞. The rule states:
lim f(x)/g(x) = lim f'(x)/g'(x)
You can apply it repeatedly until you get a determinate form. Common mistakes:
- Using it when the limit isn't indeterminate
- Confusing the quotient rule with L'Hospital's Rule
- Forgetting to check if the limit exists after applying
It only works for 0/0 and ∞/∞. Other indeterminate forms need algebra first.
Motion Along a Line
Position s(t), velocity v(t) = s'(t), acceleration a(t) = v'(t) = s''(t). This connects everything.
Key relationships:
- Object moving right: v(t) > 0. Moving left: v(t) < 0
- Speed = |v(t)|
- Object speeding up when v and a have the same sign
- Object slowing down when v and a have opposite signs
- Total distance = integral of |v(t)|
- Displacement = integral of v(t)
Quick Reference: Key Theorems Comparison
| Theorem | Conditions | Conclusion |
|---|---|---|
| Extreme Value | Continuous on [a,b] | Absolute max and min exist |
| Mean Value | Continuous [a,b], differentiable (a,b) | f'(c) = [f(b)-f(a)]/(b-a) |
| Rolle's | Continuous [a,b], differentiable (a,b), f(a)=f(b) | f'(c) = 0 for some c |
| Fermat's | f has local extremum at c, f'(c) exists | f'(c) = 0 |
How to Actually Prepare
Don't just read notes. Here's what works:
- Drill derivative rules until you can take derivatives without thinking. Unit 5 assumes this is automatic.
- Practice optimization setups separately from solving. Write the equation first, check your work, then differentiate.
- Time yourself on past FRQs. You have roughly 15 minutes per question. If you're spending 25 minutes, you need to speed up.
- Know your calculator. For Unit 5, you'll use it for graphing, finding zeros, calculating derivatives at points, and numerical integration.
Most common mistakes on test day:
- Forgetting to state the conditions for theorems
- Misidentifying what the question is asking for
- Not answering in the units requested
- Rushing the setup on related rates
What to Focus On Tonight
If you're cramming, prioritize:
- Finding and classifying critical points (FDT and SDT)
- Setting up optimization problems
- Mean Value Theorem interpretations
- Motion along a line (speeding up/slowing down)
These four topics cover roughly 70% of the free response questions. Get those down first.
Unit 5 rewards people who understand the concepts, not just the procedures. Know why the theorems work. Know what the derivative actually tells you about a function. That's the difference between a 3 and a 5.