History
The man credited with developing the concept of related rates.
Who is Rev. William Ritchie?
The reason related rates can be useful.
What is: simplifying complex calculus topics and making them easier to understand?
Derive the Pythagorean Theorem as if you were using it to solve a related rate.
What is:
2x*dx/dt + 2y*dy/dy = 0
or
x*dx/dt + y*dy/dy = 0
(Multiple choice)
The radius of a circle is decreasing at a constant rate of of 0.5 cm/sec. In terms of the circumference C, what is the rate of change of the area of the circle, in square centimeters per second? (no calculators)
a) -(0.1)(pi)C
b) -0.5C
c) 0.5(pi)C
d) (0.1)C/2(pi)
e) (0.5)^2*(pi)C
What is (B)?
The reason Rev. William Ritchie decided to create related rates problems.
What is: making complex calculus problems easier to understand?
An issue with the application of related rates.
What are actual real world scenarios that students can use them in?
Derive the area of a circle as if you were using it to solve a related rate.
What is:
dA/dt = 2(pi)r(dr/dt)
A person whose height is 6 feet is walking away from the base of a streetlight along a straight path at a rate of 8 ft/sec. If the height of the streetlight is 18 feet, what is the rate at which the person's shadow is lengthening? (no calculators)
What is 4 ft/sec?
The man who made related rates popular.
Who is Elias Loomis?
The basic four-step method of solving most related rates problems.
What is identifying what you know and don't know, deriving various formulas, plugging in constants and known variables, and using this to determine the desired rate of change?
Derive the volume of a cone as if you were using to solve a related rate.
Formula: (⅓ 𝜋*r^2*h)
What is:
dV/dh = ((pi)r^2)/3
A 17 foot ladder is sliding down a building at a constant rate of 4 ft/min. How fast is the base of the ladder moving away from the building when the base of the ladder is 12 feet away from the building? (calculators required)
What is 4.0138 ft/min?
The first appearance of related rates in a math textbook was in this book.
What is "Principles of the Differential and Integral Calculus?"
Two most common types of related rates problems.
What are shadow problems and ladder problems?
Derive the area of a triangle as if you were using it to solve a related rate.
Formula: (1/2)*bh
What is:
dA/dt = 1/2(dB/dt*h+dH/dt*b)
The radius of a circle is increasing at a constant rate of 0.2 meters per second. What is the rate of increase in the area of the circle at the instant when the circumference is 60(pi)? (no calculators)
What is 16(pi) m/sec^2?
The reason related rates problems did not appear in math textbooks until the middle of the 19th century.
What is: calculus was still widely debated.
The reason so many students struggle with related rates.
What is their level of difficulty and teachers struggling to teach them?
Derive the surface area of a cylinder as if you were using it in a related rate problem.
Formula: A=2(pi)rh+2(pi)r^2
What is:
dA/dt = 2pi((dr/dt*h)+(dH/dt*r)) +4(pi)r*dr/dt
The height of a tree, in meters, can be modeled by the function G, given by G(x) = 200x/1+x, where x is the diameter of the base of the tree, in meters. When the tree is 60 meters tall, the diameter of the base of the tree is increasing at a rate of 0.07 meters m/yr. According to this model, what is the rate of change of the height of the tree with respect to time, in meters per year, at the the time when the tree is 60 meters tall? (calculators required)
What is 6.86 m/yr?