CramX Logo
RegisterLog in
Back to FlashcardsMathematics / AP Calculus AB: 7.4.2 The Ladder Problem

AP Calculus AB: 7.4.2 The Ladder Problem

Mathematics6 CardsCreated 8 months ago

This content explores related rates problems involving implicit differentiation, such as determining how fast the top of a sliding ladder falls given the rate its base moves away from the wall. It also covers finding the rate of change of the distance between a moving particle on a curve and the origin using derivatives.

Report

The Ladder Problem

  • Related rate problems involve using a known rate of change to find an associated rate of change.

  • Use implicit differentiation when you cannot write the dependent variable in terms of the independent variable.

Rate to track your progress ✦

Tap or swipe ↕ to flip
Swipe ←→Navigate
1/6

Key Terms

Term
Definition

The Ladder Problem

  • Related rate problems involve using a known rate of change to find an associated rate of change.

  • Use implicit differentiatio...

note

  • When a ladder slides down a wall, the rate at which it falls downward is not necessarily equal to the rate at which the base of the ladder ...

Suppose a particle is moving from left to right along the graph of y = x^ 2. Find the rate of change of the distance between the particle and the origin at the instant x = 5 if the particle moves horizontally at a constant rate of 10 units / second. (In other words, dx / dt = 10)

100 units / second

A winch on a motionless truck 6 feet above the ground is dragging a heavy load (see diagram).
If the winch pulls the cable at a constant rate of 1.5 feet / second, how quickly is the load moving on the ground when it is 11 feet from the truck?

1.7 feet / second

Jim, who is 6 ft tall, is walking directly away from a 15 ft lamppost at a rate of 4 ft per sec. What is the rate of change in the length of Jim’s shadow when he is 8 ft from the base of the lamppost?

8/3 feet/second

A 10 ft ladder is being pulled away from
a wall at a rate of 3 ft/sec. What is the rate
of change in the area beneath the ladder
when the ladder is 6 ft from the wall?

dA/dt=214 ft^2/sec

Related Flashcard Decks

Mathematics

Apex 2.2.3 Quiz: Graphing Functions

This deck covers key concepts from graphing functions, focusing on understanding graphical representations and their corresponding equations.

9 cards
View Deck
Mathematics

8 Basic Trigonometric Identities

This deck covers the fundamental trigonometric identities, including reciprocal, ratio, and Pythagorean identities.

8 cards
View Deck
Mathematics

AP Calculus AB: 9.1.1 Antidifferentiation

This content provides an overview of antidifferentiation—the process of finding a function whose derivative is a given function. It introduces the concept of antiderivatives, outlines how to reverse common derivative rules, and includes simple examples to illustrate the idea.

8 cards
View Deck
Mathematics

AP Calculus AB: 9.1.2 Antiderivatives of Powers

This content explains how to find antiderivatives of functions involving powers of 𝑥, emphasizing the concept of indefinite integrals and the constant of integration 𝐶. It highlights the idea that functions can have infinitely many antiderivatives and includes example problems demonstrating how to evaluate integrals involving powers and rational expressions.

9 cards
View Deck
Mathematics

AP Calculus AB: 9.2.2 Integrating Polynomials by Substitution

This content explains how to integrate polynomial expressions using integration by substitution, especially when dealing with composite functions. It outlines the step-by-step substitution process, emphasizes correct variable handling, and highlights the importance of rewriting the final answer in terms of the original variable along with including the constant of integration.

12 cards
View Deck
Mathematics

AP Calculus AB: 9.3.4 Choosing Effective Function Decompositions

This section focuses on how to strategically choose a substitution when performing integration by substitution. It explains that the best choice for 𝑢 is typically a part of the integrand whose derivative is also present, making the integral easier to evaluate. It also covers how to simplify trigonometric expressions using identities and emphasizes that effective decomposition is key to simplifying complex integrals.

6 cards
View Deck
Mathematics

AP Calculus AB: 9.3.1 Integrating Composite Trigonometric Functions by Substitution

This content covers the technique of integration by substitution applied to composite trigonometric functions. It explains how to identify the inner function for substitution, handle constant multiples, and ensure complete substitution. Worked examples demonstrate how substitution simplifies complex integrals, with emphasis on converting back to the original variable and verifying results through differentiation.

12 cards
View Deck
Mathematics

Conversion Factors and Problem Solving

This deck covers essential concepts related to rounding rules, significant figures, metric conversions, and basic geometry calculations.

18 cards
View Deck
Mathematics

Unit 1 Progress Check: MCQ Part A

This deck covers 18 questions and answers from a unit progress check, focusing on calculus concepts such as limits, derivatives, and function behavior.

18 cards
View Deck
AP Calculus AB: 7.4.2 The Ladder Problem
TermDefinition

The Ladder Problem

  • Related rate problems involve using a known rate of change to find an associated rate of change.

  • Use implicit differentiation when you cannot write the dependent variable in terms of the independent variable.

note

  • When a ladder slides down a wall, the rate at which it falls downward is not necessarily equal to the rate at which the base of the ladder moves away from the wall.

  • Suppose you are given a 13-foot ladder and you are told that the ladder is moving 6 feet per second away from the wall when the ladder is 12 feet away from the wall. What is the rate of change in the y-direction?

  • This is another example of a related rate. The rate at which a ladder falls downward depends on the rate at which the ladder moves away from the wall.

  • You want to know the rate of change in the y-direction when the ladder is 12 feet from the wall.

  • You are told that the rate of change in the x-direction when the ladder is 12 feet from the wall is 6 feet per second.

  • Because the Pythagorean theorem relates the distance away from the wall to the distance to the top of the ladder, the rates of change can be related by taking a derivative.

  • Notice that you can express the derivative of y with respect to time in terms of the derivative of x with respect to time, the x-value, and the y-value.

  • Substitute to find the value of dy/dt.

Suppose a particle is moving from left to right along the graph of y = x^ 2. Find the rate of change of the distance between the particle and the origin at the instant x = 5 if the particle moves horizontally at a constant rate of 10 units / second. (In other words, dx / dt = 10)

100 units / second

A winch on a motionless truck 6 feet above the ground is dragging a heavy load (see diagram).
If the winch pulls the cable at a constant rate of 1.5 feet / second, how quickly is the load moving on the ground when it is 11 feet from the truck?

1.7 feet / second

Jim, who is 6 ft tall, is walking directly away from a 15 ft lamppost at a rate of 4 ft per sec. What is the rate of change in the length of Jim’s shadow when he is 8 ft from the base of the lamppost?

8/3 feet/second

A 10 ft ladder is being pulled away from
a wall at a rate of 3 ft/sec. What is the rate
of change in the area beneath the ladder
when the ladder is 6 ft from the wall?

dA/dt=214 ft^2/sec