The College Board’s Advanced Placement (AP) Calculus BC exam is a rigorous assessment of students’ understanding of advanced calculus concepts. The 2019 AP Calculus BC free-response questions (FRQs) were particularly challenging, testing students’ problem-solving skills and conceptual knowledge in depth.
FRQ 1: Area Between Two Curves
Problem: Find the area of the region bounded by the curves $y = x^3 – 2x^2 + 1$ and $y = x^2 – x + 1$.
Solution: This problem requires students to use integration to find the net area between the two curves.
- Sketch the region: Plot both curves to determine the bounds of the region.
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Set up the integral: Subtract the area below the lower curve from the area above the upper curve:
$$A = \int_a^b (y_{\text{upper}} – y_{\text{lower}}) dx$$
- Evaluate the integral: Find the indefinite integrals of both functions and subtract them to obtain the net area.
FRQ 2: Volume of a Solid of Revolution
Problem: Set up an integral to find the volume of the solid generated by rotating the region bounded by the curves $y = x$ and $y = x^2$ about the $x$-axis.
Solution: This problem requires students to use the disk method to calculate the volume of the solid.
- Sketch the region: Determine the bounds of the region and visualize the solid that will be generated.
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Integral formula: The volume using the disk method is given by:
$$V = \pi \int_a^b [f(x)]^2 dx$$
- Evaluate the integral: Find the intersection points of the two curves and substitute the appropriate function into the integral to calculate the volume.
FRQ 3: Partial Derivatives and Optimization
Problem: Consider the function $f(x, y) = x^2 + y^2 – 4x – 6y + 1$.
- Find the partial derivatives of $f$.
- Find the critical points of $f$.
- Determine if each critical point is a local maximum, local minimum, or saddle point.
Solution: This problem tests students’ understanding of partial derivatives and their applications in optimization.
- Partial derivatives: Compute the partial derivatives of $f$ with respect to $x$ and $y$.
- Critical points: Find the points where both partial derivatives are zero or undefined.
- Second partial derivatives: Compute the second partial derivatives of $f$.
- Classification: Determine the nature of each critical point using the discriminant of the second partial derivatives.
FRQ 4: Integration by Parts
Problem: Evaluate the integral $\int x e^{2x} dx$.
Solution: This problem requires students to use integration by parts, a technique used to integrate products of functions.
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Identify the parts: Select appropriate functions for $u$ and $dv$.
$$u = x, dv = e^{2x}$$
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Integrate by parts: Use the integration by parts formula to transform the integral:
$$uv – \int v du$$
- Simplify and continue: Evaluate the remaining integral and combine it with the original integral.
FRQ 5: Related Rates and Chain Rule
Problem: A particle moves along the curve $y = x^3 + 2x$. At a certain moment, $x = 2$ and $\frac{dy}{dt} = 6$. Find $\frac{dx}{dt}$.
Solution: This problem requires students to apply the chain rule and related rates to determine the rate of change of $x$ with respect to $t$.
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Differentiate implicitly: Differentiate both sides of the equation $y = f(x)$ with respect to $t$.
$$\frac{dy}{dt} = \frac{d}{dt}(x^3 + 2x)$$
- Substitute given values: Substitute the given values of $x$ and $\frac{dy}{dt}$ into the differential equation.
- Solve for $\frac{dx}{dt}$: Rearrange the equation to solve for $\frac{dx}{dt}$.
Benefits of Studying FRQs
Thoroughly preparing for the AP Calculus BC FRQs offers numerous benefits to students, including:
- Deepens conceptual understanding: FRQs require students to analyze and apply calculus concepts in complex situations, fostering a deeper understanding of the material.
- Develops problem-solving skills: FRQs challenge students to think critically and solve multi-step problems, enhancing their problem-solving abilities.
- Prepares for college and beyond: The analytical and problem-solving skills acquired from FRQs are essential for success in higher-level mathematics, physics, and engineering courses.
- Improves test-taking strategies: Practicing FRQs familiarizes students with the types of questions they will encounter on the AP exam, allowing them to develop effective test-taking strategies.
Effective Strategies for Success
To excel in the AP Calculus BC FRQs, students should adopt the following effective strategies:
- Start early: Begin preparing well before the exam date, allowing ample time for practice and review.
- Review course material: Ensure thorough understanding of all calculus concepts covered in class.
- Practice regularly: Solve as many practice FRQs as possible to develop proficiency.
- Analyze solutions: Study the solutions to FRQs, identifying the steps and techniques used to solve them.
- Time yourself: Practice solving FRQs under timed conditions to improve speed and accuracy.
- Seek help when needed: Don’t hesitate to ask your teacher or classmates for assistance with challenging problems.
- Stay positive: Believe in your abilities and approach the exam with confidence.
The 2019 AP Calculus BC FRQs were designed to test students’ mastery of advanced calculus topics. By understanding the problem-solving techniques and conceptual frameworks required to solve these problems, students can enhance their preparation for the exam and develop valuable analytical and problem-solving skills that will benefit them in higher-level mathematics and STEM fields.