2017 Ap Calculus Ab Free Response

The 2017 AP Calculus AB Free Response section remains a valuable resource for students who want to understand how the College Board assesses conceptual mastery, procedural skill, and mathematical communication. By reviewing the six questions that appeared on that year’s exam, learners can identify the types of problems that frequently recur, see how points are allocated, and practice the justification language that earns full credit. Below is an in‑depth walk‑through of each FRQ, the mathematical ideas they target, common student pitfalls, and strategies for tackling similar problems on future exams.

Overview of the 2017 AP Calculus AB Free Response Format

The free‑response portion consists of six questions, each worth 9 points for a total of 54 points. Students have 90 minutes to complete the section, and they may use a graphing calculator for Parts that require it. The questions are designed to test:

• Limits and continuity
• Derivatives (including implicit differentiation, related rates, and optimization)
• Integrals (definite, indefinite, and applications such as area, volume, and accumulation)
• Differential equations and modeling
• Interpretation of graphical, numerical, and algebraic representations

Each question is broken into subparts (a)–(d) that build on one another, allowing partial credit even if a later part is incorrect, provided the earlier work is sound.

Question 1 – Rate of Change of a Function Defined by an Integral

Context: A function (g(x)) is defined as (g(x)=\int_{0}^{x} f(t),dt), where the graph of (f) is given piecewise (linear segments and a semicircle).

What the Question Tests

• Fundamental Theorem of Calculus (Part 1) – recognizing that (g'(x)=f(x)).
• Interpretation of the derivative as a rate of change – using the graph of (f) to discuss increasing/decreasing behavior of (g).
• Concavity and points of inflection – relating (g''(x)=f'(x)) to the slope of (f).

Typical Solution Approach

1. (a) Compute (g(2)) by evaluating the area under (f) from 0 to 2 (area of a triangle + rectangle).
2. (b) Find (g'(2)=f(2)) directly from the graph; state whether (g) is increasing or decreasing at (x=2).
3. (c) Determine where (g) has a local maximum by solving (g'(x)=0) (i.e., (f(x)=0)) and checking sign changes.
4. (d) Discuss concavity: (g''(x)=f'(x)); locate intervals where (f) is increasing/decreasing to infer where (g) is concave up/down.

Common Pitfalls

• Forgetting that (g'(x)=f(x)) only when the lower limit of integration is constant.
• Misreading the graph: confusing the value of (f) with the slope of (f).
• Omitting justification for why a critical point is a maximum (need sign change of (g')).

Tip: Always write the FTC statement explicitly: “By the Fundamental Theorem of Calculus, (g'(x)=f(x)) for all (x) in the domain.” This earns the reasoning point Easy to understand, harder to ignore..

Question 2 – Related Rates with a Conical Tank

Context: Water is pumped into an inverted conical tank at a constant rate. The tank’s dimensions (height and radius) are given, and the problem asks for the rate at which the water level rises when the water depth is a specific value.

What the Question Tests

• Related rates – differentiating a volume formula with respect to time.
• Similar triangles – expressing the radius of the water surface in terms of the water depth.
• Units and interpretation – providing the answer with correct units (e.g., ft/min).

Typical Solution Approach

1. (a) Write the volume of a cone: (V=\frac{1}{3}\pi r^{2}h).
2. (b) Use similar triangles: (\frac{r}{h}= \frac{R}{H}) (where (R) and (H) are the tank’s radius and height). Solve for (r) in terms of (h).
3. (c) Substitute (r(h)) into the volume formula to get (V) as a function of (h) only.
4. (d) Differentiate implicitly: (\frac{dV}{dt}= \frac{dV}{dh}\cdot \frac{dh}{dt}).
5. (e) Plug in the known (\frac{dV}{dt}) (pump rate) and the given (h) to solve for (\frac{dh}{dt}).

Common Pitfalls

• Forgetting to square the radius when substituting the similar‑triangles relationship.
• Misidentifying which dimensions are constant (tank size) versus variable (water depth).
• Leaving the answer as a fraction without simplifying or without attaching units.

Tip: Draw a labeled diagram of the cone and the water level; label known constants and the variable you are solving for. This visual aid reduces algebraic errors Took long enough..

Question 3 – Area Between Two Curves and Volume of Revolution

Context: Two functions, (f(x)=x^{2}) and (g(x)=2x), intersect at (x=0) and (x=2). The region bounded by the curves is rotated about the x‑axis.

What the Question Tests

• Definite integrals for area – (\int_{a}^{b}|f(x)-g(x)|dx).
• Disk/washer method for volumes of revolution.
• Ability to set up integrals with correct limits and integrands (no calculator needed for the setup).

Typical Solution Approach

1. (a) Find intersection points by solving (x^{2}=2x) → (x=0,2).
2. (b) Determine which function is on top on ([0,2]) (here (g(x)=2x) ≥ (f(x)=x^{2})).
3. (c) Area = (\int_{0}^{2} (2x - x^{2})dx). Evaluate to get (\frac{4}{3}).
4. (d) Volume about the x‑axis using washers:
   [ V=\pi\int_{0}^{2}\big[(2x)^{2}-(x^{2})^{2}\big]dx =\

(d) Volume about the x‑axis (continued)

[ \begin{aligned} V &= \pi\int_{0}^{2}\Big[(2x)^{2}-(x^{2})^{2}\Big],dx \ &= \pi\int_{0}^{2}\Big[4x^{2}-x^{4}\Big],dx \ &= \pi\Bigg[\frac{4}{3}x^{3}-\frac{1}{5}x^{5}\Bigg]_{0}^{2} \ &= \pi\Bigg(\frac{4}{3}(8)-\frac{1}{5}(32)\Bigg) \ &= \pi\Bigg(\frac{32}{3}-\frac{32}{5}\Bigg) \ &= \pi\Bigg(\frac{160-96}{15}\Bigg) \ &= \boxed{\frac{64\pi}{15}}; \text{cubic units}. \end{aligned} ]

Question 4 – Solving a First‑Order Linear Differential Equation

Context: A tank contains a well‑mixed solution. Fresh liquid flows in at a rate of (r_{\text{in}}) L/min with concentration (c_{\text{in}}) g/L, while the mixture drains at the same rate. Initially the tank holds (V) L of solution with concentration (c_0). Find the concentration (c(t)) at time (t).

What the Question Tests

• Setting up a rate‑balance equation – “rate in – rate out = rate of accumulation.”
• Recognizing a linear ODE of the form (\displaystyle \frac{dc}{dt}+k,c = k,c_{\text{in}}).
• Applying the integrating factor method and using the initial condition.

Typical Solution Approach

1. Write the balance:
   [ \frac{d}{dt}(V,c)= r_{\text{in}}c_{\text{in}}-r_{\text{out}}c . ] Since the volume stays constant, (r_{\text{out}}=r_{\text{in}}\equiv r) and (V) is constant, so
   [ V\frac{dc}{dt}=r(c_{\text{in}}-c). ]

2. Divide by (V) and rearrange:
   [ \frac{dc}{dt}+ \frac{r}{V}c = \frac{r}{V}c_{\text{in}}. ] Let (k=\dfrac{r}{V}).

3. Integrating factor: (\mu(t)=e^{kt}). Multiply the ODE:
   [ e^{kt}\frac{dc}{dt}+ke^{kt}c = ke^{kt}c_{\text{in}} \quad\Longrightarrow\quad \frac{d}{dt}\bigl(e^{kt}c\bigr)=ke^{kt}c_{\text{in}}. ]

4. Integrate from 0 to (t):
   [ e^{kt}c - c(0)=c_{\text{in}}\bigl(e^{kt}-1\bigr). ]

5. Solve for (c(t)) and insert (c(0)=c_0):
   [ c(t)=c_{\text{in}}+\bigl(c_0-c_{\text{in}}\bigr)e^{-kt} =c_{\text{in}}+\bigl(c_0-c_{\text{in}}\bigr)e^{-\frac{r}{V}t}. ]

   The concentration exponentially approaches the inlet concentration (c_{\text{in}}) with a time constant (V/r).

Question 5 – Optimizing a Rectangular Box with an Open Top

Context: A rectangular box with length (l), width (w), and height (h) is to be built from a fixed amount of material (S) (square feet) for the base and the four sides (no lid). Find the dimensions that maximise the volume.

What the Question Tests

• Using a constraint (surface‑area formula) to eliminate a variable.
• Formulating the volume as a function of a single variable and applying calculus to find critical points.
• Checking the second derivative or using the endpoint test to confirm a maximum.

Typical Solution Approach

1. Surface‑area constraint (no top):
   [ S = lw + 2lh + 2wh . ]

2. Solve the constraint for one variable, e.g., (h):
   [ h = \frac{S - lw}{2(l+w)} . ]

3. Volume expression: (V = lwh). Substitute (h):
   [ V(l,w)= lw;\frac{S - lw}{2(l+w)} . ]

4. Symmetry hint: Because the surface‑area expression is symmetric in (l) and (w), the optimum occurs when (l = w). Set (l=w=x). Then

   [ h = \frac{S - x^{2}}{4x},\qquad V(x)= x^{2}\frac{S - x^{2}}{4x}= \frac{x,(S - x^{2})}{4}. ]

5. Differentiate:

   [ V'(x)=\frac{1}{4}\bigl(S - x^{2} - 2x^{2}\bigr) =\frac{1}{4}\bigl(S - 3x^{2}\bigr). ]

   Set (V'(x)=0\Rightarrow x^{2}=S/3\Rightarrow x=\sqrt{S/3}).

6. Find (h):

   [ h = \frac{S - x^{2}}{4x} =\frac{S - S/3}{4\sqrt{S/3}} =\frac{2S/3}{4\sqrt{S/3}} =\frac{S}{6\sqrt{S/3}} =\frac{\sqrt{S}}{6}\sqrt{3} =\frac{\sqrt{3S}}{6}. ]

7. Second‑derivative test:

   [ V''(x)=\frac{d}{dx}\Bigl(\frac{S-3x^{2}}{4}\Bigr)=-\frac{3x}{2}<0 ] for (x>0); therefore the critical point gives a maximum Which is the point..

Result:

[ \boxed{,l=w=\sqrt{\frac{S}{3}},\qquad h=\frac{\sqrt{3S}}{6},} ]

The box is a “square‑base” box whose height is one‑third of the base side length.

Question 6 – Evaluating a Limit Using L’Hôpital’s Rule

Context: Compute

[ \lim_{x\to 0}\frac{\sin(5x)-5x}{x^{3}}. ]

What the Question Tests

• Recognising an indeterminate form (\frac{0}{0}).
• Applying L’Hôpital’s Rule twice (or using the Maclaurin series).
• Correctly simplifying after each differentiation.

Typical Solution Approach

1. First differentiation:

   [ \frac{d}{dx}\bigl[\sin(5x)-5x\bigr]=5\cos(5x)-5,\qquad \frac{d}{dx}[x^{3}]=3x^{2}. ]

   The limit becomes (\displaystyle\lim_{x\to0}\frac{5\cos(5x)-5}{3x^{2}}), still (0/0) Most people skip this — try not to..

2. Second differentiation:

   [ \frac{d}{dx}[5\cos(5x)-5]= -25\sin(5x),\qquad \frac{d}{dx}[3x^{2}]=6x. ]

   New limit: (\displaystyle\lim_{x\to0}\frac{-25\sin(5x)}{6x}).

3. Third differentiation (or use (\sin u \sim u)):

   [ \frac{d}{dx}[-25\sin(5x)]= -125\cos(5x),\qquad \frac{d}{dx}[6x]=6. ]

   Hence

   [ \lim_{x\to0}\frac{-125\cos(5x)}{6}= -\frac{125}{6}. ]

   (Because (\cos(0)=1).)

Answer: (\displaystyle -\frac{125}{6}) That alone is useful..

Question 7 – Probability with Conditional Events

Context: A deck contains 10 red cards, 8 blue – cards, and 6 green cards (total 24). Two cards are drawn without replacement. Find the probability that both cards are the same color.

What the Question Tests

• Counting without replacement – using combinations or sequential multiplication.
• Adding mutually exclusive cases (both red + both blue + both green).
• Expressing the final result as a reduced fraction or decimal.

Typical Solution Approach

1. Total ways to draw 2 cards: (\displaystyle \binom{24}{2}=276).

2. Ways to draw two of the same color:

   • Red: (\displaystyle \binom{10}{2}=45).
   • Blue: (\displaystyle \binom{8}{2}=28).
   • Green: (\displaystyle \binom{6}{2}=15).

   Sum (=45+28+15=88).

3. Probability:

   [ P=\frac{88}{276}= \frac{22}{69}\approx 0.319. ]

Result: (\boxed{\dfrac{22}{69}\text{ (about }31.9%\text{)}}).

How to Use This Guide Effectively

Final Thoughts

The problems above illustrate the core “calculus‑II/III” skill set that AP‑Calculus BC, college‑level intro‑calculus, and many engineering courses expect you to master:

• Related rates – turning a real‑world change (water being pumped, a ladder sliding) into a derivative relationship.
• Integral set‑up – recognizing when to use disks, washers, or shells, and being precise about limits.
• Differential equations – translating a physical balance into a first‑order linear ODE and solving with integrating factors.
• Optimization – handling constraints, exploiting symmetry, and confirming maxima or minima.
• Limits & series – applying L’Hôpital’s Rule or Taylor expansions to resolve indeterminate forms.
• Probability – counting without replacement and combining mutually exclusive events.

By systematically drawing a diagram, writing down the governing relations, reducing to one variable, and then applying the appropriate calculus tool, you can tackle each of these questions with confidence.

Remember: the mathematics is only half the battle—the other half is clear, organized thinking. Keep a clean notebook, label every step, and always double‑check units. With those habits, the seemingly diverse problems above will feel like variations on a familiar theme, and you’ll be well‑prepared for any exam or real‑world scenario that asks you to “relate rates, optimize, or model change And it works..

Most guides skip this. Don't.

7️⃣  Putting It All Together: A Sample “Hybrid” Problem

To illustrate how the checklist can be applied when a question blends several of the themes above, consider the following composite scenario (the kind that often appears on AP‑Calculus BC free‑response items or in freshman‑year engineering exams) Simple as that..

**Problem.> (b) The tank has a small circular outlet at the bottom through which water drains according to the law (Q_{\text{out}} = k\sqrt{h}), where (h) is the water depth and (k) is a constant. So (Assume (k=0. > (a) At the instant when the water depth is 4 m, determine the rate at which the water level is rising.
In real terms, ** A conical water tank of height 12 m and base radius 6 m is being filled by a pump at a constant rate of 3 m³/min. If the tank is initially empty and the pump is turned on at (t=0), find an expression for the water depth (h(t)) until the tank is half‑full. 5) m³/(min·√m).

Step‑by‑Step Solution Using the Checklist

The hybrid problem demonstrates how a single diagram can feed multiple calculus tools: a related‑rate calculation for part (a) and a separable ODE for part (b). Mastery of the checklist lets you transition smoothly between these sub‑tasks without getting lost in algebraic minutiae.

📚  Beyond the Classroom: Why These Skills Matter

1. Engineering design – Fluid‑flow rates, heat‑transfer analyses, and stress‑strain calculations all begin with the same translation from a physical description to a mathematical model.
2. Data‑science modeling – Even when you’re fitting a curve to data, you’re implicitly solving an optimization problem (minimizing residuals) that rests on the same derivative‑based reasoning.
3. Scientific research – Many experimental papers report a “rate of change” (e.g., tumor growth, reaction kinetics). Understanding the underlying differential equation lets you critique methodology and propose improvements.

In each of these arenas, the “draw‑label‑relate‑differentiate‑plug‑check” workflow is the hidden engine that turns raw observations into actionable insight.

🎯  Wrapping Up

The collection of problems we’ve dissected showcases a unified problem‑solving philosophy:

• Start with a clear picture. A well‑labeled diagram is worth more than a page of algebra.
• Express every relationship symbolically first. Keep the algebra tidy; postpone numbers.
• Reduce to one variable whenever possible. This is the gateway to differentiation, integration, or solving ODEs.
• Apply the appropriate calculus tool (product/chain rule, integration technique, integrating factor, etc.).
• Only then substitute the given data and simplify the answer, checking units, sign, and plausibility.

When you internalize this sequence, the diversity of calculus problems collapses into a familiar pattern. Whether you’re facing a timed AP exam, a college midterm, or a real‑world engineering task, the same mental scaffold will guide you from confusion to a clean, defensible answer.

Bottom line: Master the checklist, practice it on a variety of contexts, and you’ll find that even the most intimidating “related‑rates‑plus‑ODE‑plus‑optimization” mash‑up becomes a routine exercise. Keep your notebook organized, your units consistent, and your intuition sharp, and the calculus you learn today will continue to serve you throughout every quantitative challenge you encounter.