Notes

Hello! I am Octavio Hernandez and this is my video for related rates problems

I will begin with introducing some background information about related rates problems.

We should first ask what are related rates? To start off, we must recall that derivatives represent rates of change. In related rates problems, students are given one or more rates and are asked to find the values of one or more rates, often at specific points. Students may be given the values of functions at specific points too to provide information needed to solve a problem. Lastly, students may have to use their own background knowledge to solve a particular related rates problem, often students will have to use the pythagorean theorem or various equations for area or volume. Related rates problems will require you to notice the relationships between different rates, hence the name related rates.

Here is a list of the equations, formulas, and concepts that are critical in being able to solve related rates problems. A lot of the time, related rates problems will involve a triangle, which means one must know the pythagorean theorem and the formula for the area of a triangle. Sometimes, you will be asked about angles of triangles, so be sure to know how sine, cosine, and tangent are defined. You may asked to use the area of a circle or volume of a sphere. Additionally, some problems will involve different objects moving towards or away from one another, hence the need to remember the relationship between position, velocity, and acceleration as derivatives of one another. Lastly, the chain rule will be important in inputting rates given by the problem to the formulas and equations shown above, since you may often have to apply the chain rule to find the derivative.

Now that you have some preliminary or background information about related rates, we can move onto some example prompts.

The first prompt is from the AP Calculus textbook titled Calculus: Single and Multivariable 6th Edition. This particular prompt can be found on page 237.

11. The average cost per item, C, in dollars, of manufacturing a quantity q of cell phones is given by
C=(a/q)+b where a, b are positive constants
a) Find the rate of change of C as q increases. What are its units?
b) If production increases at a rate of 100 cell phones per week, how fast is the average cost changing? Is the average cost increasing or decreasing?

(after finishing problem) -- Since we are never given a definite value for q, we cannot go any further. This problem tries to get at the basics of related rates, it shows you how you will be inputting values and rates into formulas to acquire an answer.

The next prompt is from Khan Academy. You can access similar prompts through Khan Academy as well as more instructional and educational videos about related rates narrated by Sal Khan.

Two cars are driving away from an intersection in perpendicular directions.
The first car's velocity is 5 meters per second and the second car's velocity is 8 meters per second.
At a certain instant, the first car is 15 meters from the intersection and the second car is 20 meters from the intersection.
What is the rate of change of the distance between the cars at that instant (in meters per second)?


The last prompt is again from the 6th edition of Calculus: Single and Multivariable. This prompt can be found on page 239.
32. The metal frame of a rectangular box has a square base. The horizontal rods in the base are made out of one metal and the vertical rods out of a different metal. If the horizontal rods expand at a rate of 0.001 cm/hr and the vertical rods expand at a rate of 0.002 cm/hr, at what rate is the volume of the box expanding when the vase has an area of 9cm2 and the volume is 180cm3?