2004 Ap Calc Ab Frq Form B Solutions

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AP Calculus AB FRQ Form B Solutions: A Comprehensive Guide

The AP Calculus AB Free Response Questions (FRQ) Form B is a challenging exam that tests students' knowledge and understanding of calculus concepts. In this article, we will provide a comprehensive guide to solving the 2004 AP Calculus AB FRQ Form B, including step-by-step solutions and explanations.

Understanding the Exam Format

The AP Calculus AB FRQ Form B exam consists of six questions, each designed to test different aspects of calculus. The exam is divided into two sections: Section I, which contains four questions, and Section II, which contains two questions. Students have 90 minutes to complete the exam.

Section I: Multiple-Choice Questions

Section I consists of four multiple-choice questions, each worth 3 points. The questions cover a range of topics, including limits, derivatives, and integrals.

Solution to Question 1

Question 1: If f(x) = x^2 - 4x + 3, find f'(2).

Step-by-step solution:

1. Use the power rule to find the derivative of f(x). f'(x) = d/dx (x^2 - 4x + 3) = 2x - 4
2. Evaluate f'(2) by substituting x = 2 into the derivative. f'(2) = 2(2) - 4 = 0

Solution to Question 2

Question 2: Find the area under the curve y = x^2 from x = 0 to x = 4.

Step-by-step solution:

1. Use the definite integral to find the area under the curve. ∫[0,4] x^2 dx = (1/3)x^3 | [0,4] = (1/3)(4^3) - (1/3)(0^3) = 64/3
2. Evaluate the definite integral to find the final answer. The area under the curve is 64/3.

Solution to Question 3

Question 3: Find the equation of the tangent line to the curve y = x^3 at the point (2, 8).

Step-by-step solution:

1. Use the point-slope form to find the equation of the tangent line. y - 8 = m(x - 2)
2. Find the slope of the tangent line by finding the derivative of y = x^3. dy/dx = 3x^2
3. Evaluate the derivative at x = 2 to find the slope. m = 3(2)^2 = 12
4. Substitute the slope into the point-slope form to find the equation of the tangent line. y - 8 = 12(x - 2)

Solution to Question 4

Question 4: Find the limit as x approaches infinity of (2x^2 + 3x) / (x^2 + 1).

Step-by-step solution:

1. Use the limit laws to simplify the expression. lim (x→∞) (2x^2 + 3x) / (x^2 + 1) = lim (x→∞) (2 + 3/x) / (1 + 1/x^2)
2. Evaluate the limit by substituting x = ∞ into the simplified expression. lim (x→∞) (2 + 3/x) / (1 + 1/x^2) = 2 / 1 = 2

Section II: Free-Response Questions

Section II consists of two free-response questions, each worth 9 points. The questions cover a range of topics, including optimization problems and parametric equations.

Question 5

Question 5: A cylindrical can is to be designed to hold 1000 cubic centimeters of volume. Find the dimensions of the can that will minimize the surface area.

Step-by-step solution:

1. Use the volume formula to express the height of the can in terms of the radius. V = πr^2h 1000 = πr^2h h = 1000 / (πr^2)
2. Use the surface area formula to express the surface area in terms of the radius. A = 2πrh + 2πr^2 A = 2πr(1000 / (πr^2)) + 2πr^2 A = 2000/r + 2πr^2
3. Find the critical points of the surface area function by taking the derivative and setting it equal to zero. dA/dr = -2000/r^2 + 4πr 0 = -2000/r^2 + 4πr 2000/r^2 = 4πr r^3 = 500 / π r = (500 / π)^(1/3)
4. Evaluate the second derivative to determine the nature of the critical point. d^2A/dr^2 = 4000/r^3 + 4π

0 for r = (500 / π)^(1/3) The critical point corresponds to a minimum.

Solution to Question 6

Question 6: A particle moves along the x-axis according to the equation x(t) = t^3 - 2t^2 + t + 1, where x is the position of the particle at time t. Find the velocity and acceleration of the particle at time t = 2.

Step-by-step solution:

1. Use the definition of velocity to find the velocity function. v(t) = dx/dt = d/dt (t^3 - 2t^2 + t + 1) = 3t^2 - 4t + 1
2. Evaluate the velocity function at t = 2 to find the velocity. v(2) = 3(2)^2 - 4(2) + 1 = 12 - 8 + 1 = 5
3. Use the definition of acceleration to find the acceleration function. a(t) = dv/dt = d/dt (3t^2 - 4t + 1) = 6t - 4
4. Evaluate the acceleration function at t = 2 to find the acceleration. a(2) = 6(2) - 4 = 12 - 4 = 8

Conclusion

In conclusion, the 2004 AP Calculus AB FRQ Form B exam requires students to apply their knowledge of calculus concepts to solve a variety of problems. By following the step-by-step solutions provided in this guide, students can improve their understanding of the material and develop the skills needed to succeed on the exam.