- Topic
- Related rates
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Related rates practice question 1
Two quantities and both change with time and are related by . Which one of the following can be worked out from the information given?
Differentiating in time links the two rates. Test each choice: does it give every quantity that link needs, all at the same instant?
Differentiate the relation: . Count what appears on the right-hand side.
Theory UBC Math 100 §5CLP-1 §3.2Key results
Worked solution
Strategy
Differentiating the relation shows exactly which pieces of information are needed: the two rates are linked through the current value of Q, so one rate alone is never enough.
Working
Chain rule, because Q depends on t.
To find dP/dt at t = 0 you need both dQ/dt and Q at t = 0. Two rates with no value, a value from another instant, or values with no rate each leave an unknown in the equation.
Answer
The answer is dP/dt, given dQ/dt and the value of Q: dP/dt needs both.
Check
The same dQ/dt = 1 gives dP/dt = 3 when Q = 1 and 12 when Q = 2, so the value of Q at that instant is needed along with its rate.
Where this goes wrong
Expecting one rate to determine the other. The relation between the rates involves the current value of Q, which is why every related-rates problem tells you the values at the instant in question.
Leans on
The chain rule, from The chain rule and logarithms
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