6.2 Related Rates Suppose we have two variables and (in most problems the letters will be different, but for now let's use and) which are both changing with time. A 'related rates' problem is a problem in which we know one of the rates of change at a given instant—say, —and we want to find the other rate at that instant. AP Calculus AB – “Related Rates” Solving Related Rates Problems 1. Name the variables and constants. Determine what you are asked to find. Write an equation(s) that relates the variables. Implicitly differentiate with respect to time. Evaluate and solve for the unknown.
This video lesson explores the concept of Related Rates, which is the study of what is happening over time.
To solve problems with Related Rates, we will need to know how to differentiate implicitly, as most problems will be formulas of one or more variables.
But this time we are going to take the derivative with respect to time, t, so this means we will multiply by a differential for the derivative of every variable!
But don’t let this confused you, as there are only four basic steps when problem-solving with Related Rates:
- Write down every given information or quantity, even what you’re looking for, and make a sketch.
- Determine the equation needed to solve for your unknown quantity.
- Take the derivative implicitly, not forgetting to multiply by your differential for every single variable!
- Plug in all of your given information or quantities and solve.
And here’s a big hint…
…most of the time all we have to do is find an equation that relates the rate we’re looking for to a rate that we already know, as Paul’s Online Notes so nicely states.
This means we need to look for geometric shapes, known formulas, and ratios.
Ladder Sliding Down a Wall
Together we will solve these 8 classic questions:
- Pebble Dropped In Pond
- Air Being Pumped Into A Balloon
- Edges Of Cube Expanding
- Radar Tracking Station & Airplane Flight Path
- Sand Falling onto a Conical Pile
- Water Pouring into a Conical Tank
- Boat Pulled into a Dock
- Ladder Sliding Down a Wall
How to Solve Related Rates – Video
1 hr 35 min
- Overview of Related Rates + Tips to Solve Them
- 00:02:58 – Increasing Area of a Circle
- 00:12:30 – Expanding Volume of a Sphere
- 00:21:15 – Expanding Volume of a Cube
- 00:26:32 – Calculate the Speed of an Airplane
- 00:39:13 – Conical Sand Pile
- 00:51:19 – Conical Water Tank
- 00:59:59 – Boat & Winch
- 01:09:13 – Ladder Sliding Down A Wall
Related Rates
Related Rates – 2 Examples
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6.2 Related Ratesap Calculus Worksheet
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