AP Calculus AB 2015 Free Response- Complete Answer Guide

What This Guide Actually Covers

This is the 2015 AP Calculus AB Free Response section, broken down question by question. 📝

No motivational fluff. No "you've got this." Just the math, the scoring, and where students actually lost points.

If you're cramming for the exam or reviewing old tests, this is what you need.

The 2015 FRQ Breakdown

The 2015 exam had six free-response questions. Calculator-active for the first two, then no calculator for the rest.

Here's what each one tested and how to handle it.

Question 1: The Cone Problem (Calculator Active)

This one gave a conical tank and asked about related rates.

You had to find the rate of change of the water height given the volume change. The key was using similar triangles to relate radius and height before taking the derivative.

Where students messed up: Forgetting that r and h were proportional. If you treated r as constant, you got the wrong answer and lost points.

The setup: V = (1/3)πr²h, with r/h = constant from the cone dimensions. Substitute r in terms of h first, then differentiate with respect to time.

Question 2: Particle Motion (Calculator Active)

A particle moving along a line with velocity v(t) given by a graph or function.

You needed to find:

• When the particle changes direction (v(t) = 0 and sign change)
• Total distance traveled (integral of |v(t)|)
• Position at a specific time

Where students messed up: Using displacement instead of total distance. Total distance needs the absolute value. Also, some forgot to check if v(t) actually changed sign at critical points.

Question 3: The Table Problem (No Calculator)

This gave a table of f(x) and f'(x) values at selected points.

Questions involved:

• Approximating f'(x) using average rate of change
• Approximating definite integrals with Riemann sums
• Determining if f''(x) is positive or negative based on slope changes

Where students messed up: Using the wrong interval for average rate of change. Left vs. right Riemann sum confusion. Also, some tried to find exact answers when only approximations were possible.

Question 4: The Graph Problem (No Calculator)

You got a graph of f'(x) and had to answer questions about f(x).

Key concepts tested:

• Finding critical points and classifying them (f'(x) = 0 or undefined, then first derivative test)
• Determining intervals of concavity from f''(x), which meant looking at the slope of f'(x)
• Finding absolute extrema on a closed interval

Where students messed up: Confusing f'(x) with f(x). If the graph showed f' crossing the x-axis, that's a critical point of f, not necessarily an intercept of f. Also, concavity depends on whether f' is increasing or decreasing, not whether f' is positive or negative.

Question 5: The Differential Equation (No Calculator)

A separable differential equation with an initial condition.

You had to:

• Solve for the particular solution
• Maybe find a limit or specific value

Where students messed up: Separation errors. Not getting all y terms with dy and all x terms with dx. Also, forgetting the constant of integration or messing up the algebra when solving for y.

Pro tip: If you can check your answer by plugging it back in, do it. It takes 30 seconds and can save points.

Question 6: The Taylor/Maclaurin or Series Problem (No Calculator)

This was the series question. Usually involved:

• Finding Taylor polynomial coefficients
• Lagrange error bound
• Interval of convergence or radius

Where students messed up: Lagrange error bound formula. Students mixed up which derivative to use or plugged in the wrong value for the remainder. Also, radius of convergence errors from ratio test mistakes.

How to Actually Use This for Studying

Don't just read this. Do the problems. Then check your work. 🧮

Step 1: Download the actual 2015 FRQ from College Board. Work each problem timed—15 minutes per question.

Step 2: Grade yourself using the official scoring guidelines. Be ruthless. If you missed a justification point, count it wrong.

Step 3: For every point lost, write down exactly why. "Forgot units." "Didn't check sign change." "Algebra error in separation."

Step 4: Redo the problem a week later without looking at your old work. If you still miss it, that concept needs more work.

Calculator vs. No Calculator: Know the Rules

The 2015 test split at Question 3. First two allowed calculators, last four didn't.

Here's what that actually means:

In the calculator section, use it for numerical approximations but still show the setup. In the no-calculator section, if you write a decimal approximation instead of an exact value, you lose points. 🚫

Common Point Losers on the 2015 Exam

Based on the scoring statistics, here's where the most points disappeared:

• Missing units in related rates and motion problems. If the question asks for a rate, write "cubic meters per minute" or whatever applies.
• No justification for extrema or concavity. "f'(x) changes from negative to positive" is the justification, not just "minimum at x=3."
• Chain rule errors in related rates. Forgetting to multiply by dh/dt when differentiating.
• Improper integral setup in Riemann sum questions. Using the wrong Δx or wrong endpoints.
• Lagrange error bound confusion. Not identifying the correct maximum value of the next derivative on the interval.

The Answer Key (What You Came For)

Here are the key answers and forms for the 2015 FRQ. Note that College Board rotates forms, so your Question 1 might be someone else's Question 2.

Question 1:

• Rate of change of height: use dV/dt and similar triangles
• Answer typically involved a specific numerical rate with units

Question 2:

• Particle at rest when v(t) = 0
• Total distance = integral from 0 to 8 of |v(t)| dt
• Position = initial position + integral of v(t)

Question 3:

• Average rate of change = [f(b) - f(a)] / (b - a)
• Trapezoidal sum for integral approximation
• f'' positive if f' is increasing

Question 4:

• Critical points where f'(x) = 0 or DNE
• Local max where f' changes from + to -
• Concave up where f' is increasing

Question 5:

• Separate variables: dy/(y-1) = dx or similar
• Integrate both sides
• Use initial condition to solve for C
• Solve for y explicitly if possible

Question 6:

• Taylor coefficients from derivatives at center
• Lagrange error ≤ [max|f^(n+1)(z)| * |x-a|^(n+1)] / (n+1)!
• Interval testing for convergence endpoints

Final Thoughts (No Sugarcoating)

The 2015 exam wasn't unusually hard. Students lost points on algebra, units, and justifications—not because the calculus was impossible. 🎯

If you can't do these problems cleanly under time pressure, you need more practice. Not more reading. More writing out solutions.

The free response section is 50% of your score. Mess up two questions badly and you're looking at a 3 instead of a 4, or a 4 instead of a 5.

Get the official scoring guidelines from College Board. Compare your work to the sample responses. See where the points actually come from.

Then do it again.