AP Calculus AB Practice Questions | Answers + Explanation

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Planning to take the AP Calculus AB exam soon? This article is packed with everything you need to ace the exam. 

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AP Calculus AB is known to be one of the more difficult AP courses, with a reputation for its challenging curriculum and historically lower pass rates. It requires students to understand calculus concepts deeply and have rigorous problem-solving skills. 

This difficulty is why students must use AP Calculus AB practice questions to help them navigate their exam preparation.

In this article, we’ll break down what you can expect in the AP Calculus AB exam, provide a few practice questions you can use to prepare, and offer key tips to help streamline your study sessions. 

AP Calculus AB Exam Overview

AP Calculus AB is designed to provide high school students with a thorough understanding of fundamental calculus principles to prepare them for college-level calculus. 

The exam has two main sections: Multiple Choice (MC) and Free Response (FRQ). The MC section has 45 questions total, with 30 to be completed in 60 minutes without a calculator and 15 more in 45 minutes with a graphing calculator allowed. The FRQ section has 6 multi-part questions to be completed in a total of 105 minutes, with 75 minutes for Part A and 30 minutes for Part B.

AP Calculus AB Practice Questions and Answers

The best way to prepare for this exam is to complete as many practice questions as possible. Here are a few AP Calculus AB Practice questions and answers you can use to help you prepare for the exam.

Question 1:

=

A) e

B) 1

C) e^h

D) e⁴

E) 4^e

Explanation

Let f(x) = e^x. The given limit may be written as follows: as x ----> h

Lim (e⁴ e^h - e⁴) / h = lim (e^{4 + h} - e⁴) / h = limit [ f(4+h) – f (4) ] / h

which is the definition of the first derivative of f(x) = e^x at x = 4. Hence as x ---> h

lim (e⁴ e^h - e⁴) / h = e ⁴

The answer is D.

Question 2

Curve C is described by the equation 0.25x² + y² = 9. Determine the y coordinates of the points on curve C whose tangent lines have a slope equal to 1.

A) -3 sqrt (5) / 5, 3 sqrt (5) / 5

B) - sqrt(35) / 2 , sqrt(35) / 2

C) -3, 3

D) - sqrt(2) / 2 , sqrt(2) / 2

E) -3 sqrt(2) , 3 sqrt(2)

Explanation

Let us calculate the first derivative. Differentiate both sides of the given equation

0.25 (2x) + 2 y y ' = 0

y ' = - 0.5 x / (2 y)

We now solve the given equation 0.25 x² + y² = 9 for x

x = + or - sqrt [ (9 - y²) / 0.25 ]

Substitute x in y ' = - 0.5 x / (2 y) by + or - sqrt [ (9 - y²) / 0.25 ]

y ' = - 0.5 (+ or - sqrt [ (9 - y²) / 0.25 ] / (2 y)

The slope of the tangent is equal to 1. Hence

- 0.5 (+ or - sqrt [ (9 - y²) / 0.25 ] )/ (2 y) = 1

Solve the above for y. Two solutions

y = 3 sqrt(5) / 5 , y = - 3 sqrt(5) / 5

The answer is A.

Question 3

Find the solution to the differential equation dy/dx = cos(x) / y², where y(π/2) = 0.

A) y = (3 sin(x) - 3)

B) y = sin(x) - 1

C) y = (3 sin(x) - 3)^1/3

D) y = (3 sin(x) - 3)³

E) y = (3 sin(x) - 3)^-1/3

Explanation

The variable in the given differential equation may be separated as follows

y² dy = cos(x) dx

Integrate both sides

∫ y² dy = ∫ cos(x) dx

(1/3) y³ = sin(x) + C , constant of integration

We now use the condition y(π/2) = 0 to find the constant C

(1/3) y³(π/2) = sin(π/2) + C

0 = 1 + C

C = - 1

Substitute C by -1 in (1/3) y³ = sin(x) + C and solve for y

(1/3) y³ = sin(x) - 1

y³ = 3(sin(x) - 1)

y = (3 sin(x) - 3)^1/3

The answer is C.

Question 4

 A) cos⁵(x) + C

B) -(1/5)sin⁵(x) + C

C) sin⁵(x) + C

D) -(1/5)cos⁵(x) + C

E) -5cos⁵(x) + C

Explanation

Let u = cos x and therefore du/dx = - sin x. 

We now substitute cos x by u and sin x by -du/dx in the given integral. Hence

∫cos⁴(x)sin(x) dx = ∫ u⁴ (-du/dx) dx

= - ∫ u⁴du

= (-1/5)u⁵ + C , C constant of integration

= (-1/5)cos⁵(x) + C

The answer is D.

Question 5

A) 2sin(4x² + 1)

B) 2sin(x² + 1)

C) sin(x² + 1)

D) 2 sin(4x² + 1) - 2 sin(3² + 1)

E) 2 sin(4x²)

Explanation

Let u = 2x and therefore du/dx = 2 or dx = du / 2. Hence the given integral becomes

d/dx ∫ ₃^2xsin (t² + 1) dt = 2 d/du ∫ ₃^usin (t² + 1) dt

using the fundamental theorem of calculus, we obtain

= 2 sin (u² + 1)

Substitute u by 2x

= 2 sin (4x² + 1)

The answer is A.

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Question 6

A) 100

B) 108

C) 110

D) 112

E) 114

Explanation

We first analyze the signs of the expressions 4 - x and 2 - 2x between the limits of integration 0 and 10. 4 - x changes sign at x = 4 and 2 - 2x changes sign at x = 1.

for x between 0 and 4: 4 - x is positive and hence |4 - x| = 4 - x

for x between 4 and 10: 4 - x is negative and hence |4 - x| = -(4 - x)

for x between 0 and 1: 2 - 2x is positive and hence |2 - 2x| = 2 - 2x

for x between 1 and 10: 2 - 2x is negative and hence |2 - 2x| = -(2 - 2x)

We now rewrite the given integral as a sum of two integrals as follows.

∫ ₀¹⁰ (|4 - x|+|2 - 2x|) dx =

∫ ₀¹⁰ (|4 - x|) dx + ∫ ₀¹⁰ (|2 - 2x|) dx

We now calculate each of the individual integrals above as follows.

∫ ₀¹⁰ (|4 - x|) dx = ∫ ₀⁴ (4 - x) dx + ∫ ₄¹⁰ -(4 - x) dx = 8 + 18 = 26

and

∫ ₀¹⁰ (|2 - 2x|) dx = ∫ ₀¹ (2 - 2x) dx + ∫ ₁¹⁰ -(2 - 2x) dx = 1 + 81 = 82

We now have

∫ ₀¹⁰ (|4 - x|) dx + ∫ ₀¹⁰ (|2 - 2x|) dx = 26 + 82 = 108

The answer is B.

Question 7

Evaluate the integral:

A) (5 + x^3/4)¹⁰

B) (x^3/4)¹⁰

C) (1/10) (5 + x^3/4)¹⁰

D) (1/10) (5 + x^3/4)¹⁰ / x^1/4

E) (2/15) (5 + x^3/4)¹⁰

Explanation

Let u = 5 + x^3/4 and therefore du/dx = (3/4) 1/x^1/4 and substitute in the given integral

∫ (5 + x^3/4)⁹ / (x^1/4) dx = ∫ [ ( u⁹ ) / (x^1/4) ] (4/3) x^1/4 du

= (4/3) ∫ u⁹ du

= (4/3) (1/10) u¹⁰

= (2/15) (5 + x^3/4)¹⁰

The answer is E.

Question 8

Given that function h is defined by:

find h'(x).

A) (3x² / (x⁶ + 2x³ + 2) + 2)

B) 4 (arctan (x³ + 1) + 2x)³ (3x² / (x⁶ + 2x³ + 2) )

C) 4 (arctan(x³ + 1) + 2x)³

D) 4 (3x² / (x⁶ + 2x³ + 2) + 2)

E) (1/4)(arctan(x³ + 1) + 2x)³

Explanation

Let u = arctan(x³ + 1) + 2x. Hence function h can be written as

h(x) = u⁴ h '(x) = 4 u³ u'

We now let v = arctan(x³ + 1) and calculate u '

u ' = (v ')( 1 / (1 + v²) )

= (3x²) / (1 + (x³ + 1)²)

= (3x²) / (x⁶ + 2x³ + 2)

Hence

h '(x) = 4 (arctan(x³ + 1) + 2x)³ (3x²) / (x⁶ + 2x³ + 2)

The answer is B.

Question 9 

The set of all points (ln(t - 2), 3t), where t is a real number greater than 2, is the graph of

A) y = ln(x/3 - 2)

B) y = 3x

C) x = ln(y - 2)

D) y = 3(e^x + 2)

E) y = ln(x)

Explanation

The given parametric equations may be written as

x(t) = ln (t - 2) and y(t) = 3t

Solve y(t) = 3t for t

t = y / 3

Substitute t by y / 3 in x(t) = ln (t - 2)

x = ln(y / 3 - 2)

Solve for y

y/3 - 2 = e^x y = 3 ( e^x + 2 )

The answer is D.

Question 10 

Let P(x) = 2 x³ + K x + 1. Find K if the remainder of the division of P(x) by x - 2 is equal to 10.

A) -7/2

B) 2/7

C) 7/2

D) -2/7

E) K cannot be determined

Explanation

The Remainder theorem states that the division of P(x) by x - 2 is equal to P(2). Hence

P(2) = 2 (2)³ + K (2) + 1 = 10

Solve for K

K = - 7/2

The answer is A.

Tips to Prepare for The AP Calculus AB Exam

AP courses are rigorous and challenging. After taking AP courses for a whole academic year, the next thing to face is the exams. It is normal for students to worry about passing their AP exams, especially AP Calculus AB. However, individual students can boost their scores when they self-study for the exam.

Below are some tips to prepare for the AP Calculus AB Exam:

1. Master the core concepts like limits, derivatives, and integrals by doing a lot of practice questions and problems. Review the must-know formulas and theorems.

2. Use online AP Calculus resources for content review, AP calculus AB practice questions with detailed explanations, and full-length practice tests.

3. Take official practice exams under timed, test-like conditions to get comfortable with the format and pacing. Review mistakes to identify and improve weak areas.

4. Know how to use your graphing calculator efficiently for the parts of the exam where it is allowed. Be adept at key functions like graphing functions, finding zeros, and computing derivatives/integrals numerically.

5. Start studying several months before the exam and create a weekly study plan. Consistent and active preparation over an extended period is key rather than cramming.

6. As exam day nears, focus on solving many FRQs from previous exams to gain familiarity with the different question types and skills tested.

7. Go into the exam confident in your preparation. Think positively and treat it like any other test you have successfully taken.

8. Memorize important formulas that will be on the test. The best-case scenario is practicing for all questions before the exam so you’re not surprised.

Remember that success in this exam is a result of the combination of these tips. 

FAQs: AP Calculus AB Questions

AP Calculus AB exams come with a lot of questions from the students. Here are some frequently asked questions about this AP course:

1. Is AP Calc AB Really Hard?

AP Calculus AB is considered one of the most difficult AP classes, with a difficulty rating of 5.5 out of 10 from alumnae. As the 12th hardest AP course, it covers advanced mathematical concepts including limits, derivatives, and integrals.

2. Is It Hard to Get a 5 on AP Calculus AB?

Earning a top score of 5 on the demanding AP Calculus AB exam requires strong conceptual knowledge, problem-solving skills, and test prep strategies. In 2023, 22% of exam takers who took AP Calculus AB got a 5.

3. How Long is the AP Calculus AB Exam?

The AP Calculus AB exam is 3 hours and 15 minutes long. Proper time management across the different sections is key to scoring well.

4. How Long Should I Study for the AP Calculus AB Exam?

Ideal study plans dedicate 2-4 hours per day over a 2-3 week period leading up to the test date. The exact number of study hours needed will vary based on current comprehension of course material and concepts.

5. Is AP Calculus AB Worth It?

Taking AP Calculus in high school can seem daunting, but the academic and financial benefits make it ultimately worth the time and effort for motivated students aiming for careers in STEM fields.

Key advantages of AP Calculus include:

• Earn college math credits: Score high enough on the AP exam to gain 3-6 credits at most colleges, saving on tuition.
• Stand out for college admissions: Rigorous calculus coursework demonstrates academic ambition that colleges favorably view.
• Build math proficiency: Develop conceptual knowledge and problem-solving skills that prepare for college-level calculus and science courses.
• Explore STEM interests: Get early exposure to advanced math topics applicable across science and engineering domains.

6. What Happens If You Fail AP Calculus AB?

For students who fail the AP Calculus AB exam by scoring a 1 or 2, the options are to either retake the test or move on without college credit. For non-STEM majors, moving on may be advisable instead of repeating a full-year AP course.

Final Thoughts

While AP Calculus AB has a reputation for being a demanding course, with diligent studying it is very much conquerable for motivated students. By dedicating time to actively work through AP Calculus AB practice questions from official study guides, students can master the problem-solving abilities and conceptual knowledge required for success.

By internalizing must-know content and testing themselves under exam-like conditions, students will be equipped to earn a passing score come test day. With a commitment to consistent preparation, a strong AP Calculus AB exam performance is within reach.