AP Calculus AB 2015 Free Response- Solutions and Analysis

AP Calculus AB 2015 Free Response: What You Need to Know

The 2015 AP Calculus AB Free Response section tested students on derivatives, integrals, and the relationship between them. If you're studying for the exam, analyzing past FRQs is one of the most effective ways to understand what the College Board expects.

No fluff here. Let's get into it.

Section Structure: What You're Facing

The 2015 FRQ section had 6 problems total. Questions 1-3 were shorter, worth 2-3 points each. Questions 4-6 were longer, worth 4-5 points each. You had 45 minutes for the entire section.

Problem Breakdown

Question 1: Particle Motion 🏃

This was a classic particle motion problem. You were given velocity v(t) and had to find position, distance traveled, and acceleration.

What they tested:

• Integrating velocity to get position
• Finding total distance vs. displacement
• Evaluating definite integrals
• Connecting acceleration to velocity

Common mistakes:

• Confusing distance traveled with displacement — always use absolute value when integrating speed
• Forgetting to add the initial position when finding position from velocity
• Not checking where the particle changes direction

The key insight: if v(t) changes sign, split your integral at those points. That's where most students lost points.

Question 2: Graph Analysis 📈

You had to work with a graph of f(x) and answer questions about f, f', and definite integrals.

What they tested:

• Reading information from derivative graphs
• Connecting graph features to function behavior
• Setting up integrals based on area
• Understanding the First Derivative Test

The trap: Students often tried to reconstruct the original function from the derivative graph. You don't need to do that. Use the graph as given.

If the problem asks for f(c) and you only have f'(x) graphed, look for a point where you can set up a definite integral. The relationship is:

f(b) - f(a) = ∫f'(x)dx from a to b

Use it.

Question 3: Rate Problems

This was a related rates question disguised as a word problem. Water draining from a tank, or a ladder sliding down a wall — something like that.

What they tested:

• Setting up rates of change relationships
• Implicit differentiation
• Evaluating derivatives at specific points

The approach that works:

1. Write an equation relating the variables
2. Take d/dt of both sides
3. Plug in known values
4. Solve for the requested rate

The mistake most people make is skipping step 1 and trying to set up the derivative directly. Write the geometric relationship first.

Question 4: Differential Equations 📊

This question gave you a differential equation and asked you to solve it or interpret its meaning.

What they tested:

• Slope fields
• Separation of variables
• Particular solutions using initial conditions
• Euler's Method (possibly)

When you see dy/dx = something, check if variables separate. If yes, integrate both sides. If no, the question is probably about slope fields or interpretation.

For slope fields:

At point (x, y), the slope is whatever the differential equation gives you. Plot enough points to see the pattern. Don't connect dots — these are short line segments showing direction.

Question 5: Integration Applications

Area under a curve, average value, or accumulated change. These problems are formula-driven once you set them up correctly.

Average value formula:

f(c) = (1/(b-a)) × ∫f(x)dx from a to b

Students forgot to divide by the interval width constantly. The integral gives you accumulated change. Divide to get the average rate or value.

For area problems:

If the region crosses the x-axis, split the integral. Positive area above the axis, negative below. If they ask for total area, take absolute values.

Question 6: Mean Value Theorem 🔑

This was the hardest problem on the 2015 exam. It tested your understanding of the Mean Value Theorem for Derivatives.

The theorem states: if f 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)

What the question asked:

Find a value c satisfying the theorem, or prove that one exists. You had to calculate the average rate of change over the interval, then find where the derivative equals that value.

How to approach it:

1. Verify the conditions (continuity, differentiability)
2. Calculate (f(b) - f(a)) / (b - a)
3. Set f'(x) equal to that number
4. Solve for x

If the derivative equation is too hard to solve algebraically, you might need to use the Intermediate Value Theorem on f'(x) or identify the value by inspection.

Scoring Distribution: What Scores Looked Like

To get a 5, you needed around 70-75% of total points. Free response typically counts for 50% of your score.

Getting Started: How to Use Past FRQs

Don't just read solutions. That's passive and doesn't build skills.

The method that works:

1. Time yourself — 45 minutes for all 6 questions
2. Work through each problem completely
3. Grade yourself using the official rubrics
4. Identify exactly which parts you lost points on
5. Rework those specific parts until they're automatic

Focus your review on the question types you find hardest. If related rates trip you up, do 10 related rates problems. If integration applications are weak, drill those.

Common Patterns Across AP Calculus AB Free Response

The College Board recycles concepts. Watch for these:

• Particle motion — appears nearly every year
• Area/volume — solid of revolution or cross-sections
• Rate problems — related rates or accumulation
• Differential equations — separation of variables or slope fields
• Mean Value Theorem — appears every 2-3 years

If you can handle these topics cold, you're in good shape.

Final Thoughts

The 2015 free response wasn't trickier than other years. The questions tested standard concepts: derivatives, integrals, and the connections between them.

What trips students up isn't the calculus — it's the algebra. Simplifying fractions, solving for variables, and handling negative signs are where points disappear.

Before your exam, practice the algebra as much as the calculus.