2002 Ap Calc Ab Frq Form B Solutions

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The Advanced Placement (AP) Calculus AB exam is a challenging test that assesses a student's understanding of calculus concepts, including limits, derivatives, and integrals. The Free-Response Questions (FRQs) section of the exam requires students to provide detailed solutions to a series of problems. In this article, we will go through the solutions to the 2002 AP Calc AB FRQ Form B.

Understanding the Format of the AP Calc AB FRQ

The AP Calc AB FRQ consists of six questions, each with multiple parts. Students are required to provide detailed solutions to each part, showing all their work and explaining their reasoning. The questions are designed to test a student's understanding of calculus concepts, as well as their ability to apply those concepts to solve problems.

Section 1: Limits and Continuity

The first section of the FRQ deals with limits and continuity. These are fundamental concepts in calculus, and students must have a strong understanding of them in order to succeed in the course.

Question 1

This question requires students to find the limit of a function as x approaches a certain value. The function is given as:

f(x) = (x^2 - 4) / (x - 2)

Students must use their knowledge of limits to simplify the function and find the limit as x approaches 2.

Solution:

Using the factoring method, we can rewrite the function as:

f(x) = (x + 2)(x - 2) / (x - 2)

Canceling out the common factor (x - 2), we get:

f(x) = x + 2

Now, we can find the limit as x approaches 2:

lim (x→2) f(x) = lim (x→2) (x + 2) = 2 + 2 = 4

Section 2: Differentiation

The second section of the FRQ deals with differentiation. This is a critical concept in calculus, and students must be able to apply it to a variety of problems.

Question 2

This question requires students to find the derivative of a function using the power rule and the sum rule.

f(x) = 3x^2 + 2x - 5

Students must use their knowledge of differentiation to find the derivative of the function.

Solution:

Using the power rule, we can find the derivative of the first term:

d(3x^2)/dx = 6x

Using the sum rule, we can find the derivative of the second term:

d(2x)/dx = 2

The derivative of the constant term (-5) is 0.

Combining the results, we get:

f'(x) = 6x + 2

Section 3: Applications of Derivatives

The third section of the FRQ deals with applications of derivatives. This includes finding the maximum and minimum values of a function, as well as determining the rate at which a quantity is changing.

Question 3

This question requires students to find the maximum value of a function using the first derivative test.

f(x) = x^3 - 6x^2 + 9x + 2

Students must use their knowledge of derivatives to find the critical points of the function and determine which one corresponds to the maximum value.

Solution:

Using the first derivative test, we can find the critical points of the function:

f'(x) = 3x^2 - 12x + 9

Setting the derivative equal to 0, we get:

3x^2 - 12x + 9 = 0

Solving for x, we get:

x = 1 or x = 3

Evaluating the function at these points, we get:

f(1) = 4 f(3) = -4

Since f(1) is greater than f(3), the maximum value of the function occurs at x = 1.

Section 4: Integrals

The fourth section of the FRQ deals with integrals. This includes finding the definite integral of a function, as well as applying the fundamental theorem of calculus.

Question 4

This question requires students to find the definite integral of a function using the substitution method.

∫(2x + 1)^5 dx

Students must use their knowledge of integrals to find the antiderivative of the function.

Solution:

Using the substitution method, we can let u = 2x + 1. Then, du/dx = 2.

du = 2dx

Substituting this into the integral, we get:

∫u^5 du

Evaluating the integral, we get:

∫u^5 du = (1/6)u^6 + C

Substituting back in for u, we get:

∫(2x + 1)^5 dx = (1/6)(2x + 1)^6 + C

Conclusion

The 2002 AP Calc AB FRQ Form B is a challenging test that requires students to demonstrate their understanding of calculus concepts. By providing detailed solutions to each question, students can show their knowledge and application of limits, derivatives, and integrals. We hope that this article has been helpful in providing solutions to the 2002 AP Calc AB FRQ Form B.